User:Chjoaygame/sandbox/archive 2

Mario Bunge

"the ontologist is concerned with all of the factual domains: he is a generalist not a fragmentarian." (Vol. 3, p. 1.)

"We shall agree to call an ordered triple C = (S, P, D) an ontological framework iff S is a set of statements in which occur only the predicate constants in the predicate family P, which includes a nonempty set 0 of basic ontological concepts (i.e. categories), and the reference class of every P in P is included in the universe or domain D of hypothesized entities, or objects assumed to exist (in our case concrete objects or things). Though no substitute for an ontological theory, an ontological framework serves as a matrix for any number of ontological systems or theories, and has thus a guiding or heuristic power." (Vol. 3, p.12.)

Philosopher Mario Bunge has been cited above as an ontologist. The question has been raised above as to whether ontology is about things, and about some world or universe or domain of discourse. Here is a quote from Bunge.

"The following list of ontological principles occurring in scientific research (Bunge, 1974d) must suffice here:

Ml There is a world external to the cognitive subject. If there were no such world it would not be subject to scientific inquiry. Rather we would resort to introspection or to pure mathematics instead of attempting to discover the unknown beyond the self.

M2 The world is composed of things. Consequently the sciences of reality (natural or social) study things, their properties and changes. If there were real objects other than things it would be impossible to act upon them with the help of other things.

M3 Forms are properties of things. There are no Platonic Forms in themselves flying above concrete things. This is why (a) we study and modify properties by examining things and forcing them to change, and (b) properties are represented by predicates (e.g. functions) defined on domains that are, at least in part, sets of concrete objects. (Think of fertility, defined on the set of organisms.)

M4 Things are grouped into systems or aggregates of interacting components. There is no thing that fails to be a part of at least one system. There are no independent things: the borders we trace between entities are often imaginary. What there really is, are systems — physical, chemical, living, or social.

M5 Every system, except the universe, interacts with other systems in certain respects and is isolated from other systems in other respects. Totally isolated things would be unknowable. And if there were no relative isolation we would be forced to know the whole before knowing any of its parts." (Mario Bunge, TREATISE ON BASIC PHILOSOPHY, Volume 3, ONTOLOGY I: THE FURNITURE OF THE WORLD, pp. 16–17; D. REIDEL PUBLISHING COMPANY DORDRECHT — HOLLAND/BOSTON — U.S.A., ISBN — 13: 978 90 277 0785 7, 1977. Further principles continue this list but are perhaps not necessary to clarify the present questions.)

Perhaps some comment may help?

References

• Dirac, P. A. M. (1925). "The fundamental equations of quantum mechanics". Proceedings of the Royal Society A. 109 (752): 642–653. doi:10.1098/rspa.1925.0150.

Bra–ket notation

Books that do not use the bra-ket notation for Hilbert spaces, but are largely about such spaces.

• Dieudonné, Jean (1960), Foundations of Modern Analysis, Academic Press.
• Reed, Michael; Simon, Barry (1980), Functional Analysis, Methods of Modern Mathematical Physics, Academic Press, ISBN 0-12-585050-6.
• Rudin, Walter (1987), Real and Complex Analysis (3rd ed.), McGraw-Hill, ISBN 0-07-100276-6.

Books that do use the bra-ket notation for Hilbert spaces, and are largely about such spaces.

• Hewitt, Edwin; Stromberg, Karl (1965), Real and Abstract Analysis, New York: Springer-Verlag.

Feynman III, 8—8:

For consistency we will always use the ket, writing |ψ⟩, to identify a state. (It is, of course an arbitrary choice; we could equally well have chosen to use the bra, ⟨ψ|.)

8—14:

This means the same as we meant by (8.25), namely, that the amplitude to find χ at the time t + ∆t, is

$\langle \chi |\psi (t+\Delta t)\rangle =\langle \chi |U(t+\Delta t,t)|\psi (t)\rangle .\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(8.32)$

8—19:

At some later time, there is some chance that it will be found in state |2⟩.

9—4:

Well, this is just the amplitude to find the state |Φ⟩ in a new state |II⟩ in which the amplitudes of the original base states are equal. That is, writing C_II = ⟨II|Φ⟩, we can abstract the |Φ⟩ away from Eq. (9.4)—because it is true for any Φ—and get

⟨II| = ⟨1| + ⟨2|,

which means the same as

|II⟩ = |1⟩ + |2⟩.             (9.5)

The amplitude for the state |II⟩ to be in the state |1⟩ is ...

von Neumann's projection postulate

von Neumann (1932), pp. 200–201:

( P.) The probability that in the state φ the quantities with the operators R₁, ... ,R_l take on values from the respective intervals I₁ ... I_l is

|| E₁(I₁) ... E_l(I_l) φ ||²

where E₁(λ), ..., E_l(λ) are the resolutions of the identity belonging to R₁, ... ,R_l respectively.

...we postulate P. for all commuting R₁, ... ,R_l . Then the E₁(I₁), ..., E_l(I_l) commute, and therefore E₁(I₁) ... E_l(I_l) is a projection (THEOREM 14. in II. 4.), and the probability in question becomes

P  =  || E₁(I₁) ... E_l(I_l) φ ||²  =  (E₁(I₁) ... E_l(I_l) φ , φ)

Chjoaygame (talk) 04:21, 21 March 2016 (UTC)

wave function

$\underbrace {|\Psi \rangle } _{\text{state ket}}=\underbrace {\overbrace {\sum _{s_{z\,1},\ldots ,s_{z\,N}}} ^{{\text{discrete}} \atop {\text{labels}}}\overbrace {\int \limits \limits _{R_{N}}d^{3}\mathbf {r} _{N}\cdots \int \limits \limits _{R_{1}}d^{3}\mathbf {r} _{1}} ^{\text{continuous labels}}} _{\text{superposing weighted basis kets}}\,\underbrace {\overbrace {\Psi } ^{{\text{state}} \atop {\text{label}}}(\overbrace {\mathbf {r} _{1},\ldots ,\mathbf {r} _{N},s_{z\,1},\ldots ,s_{z\,N}} ^{\text{eigenvalues of basis observables}})} _{{\text{component of state ket = complex number}} \atop {\text{= value of scalar product }}\langle {\text{basis bra }}|\,\Psi \rangle \,\,\,}\underbrace {|\overbrace {\mathbf {r} _{1},\ldots ,\mathbf {r} _{N},s_{z\,1},\ldots ,s_{z\,N}} ^{\text{eigenvalues of basis observables}}\rangle } _{\text{basis ket}}$

$\underbrace {|\Psi \rangle } _{\text{state ket}}=\underbrace {\overbrace {\sum _{s_{z\,1},\ldots ,s_{z\,N}}} ^{{\text{discrete}} \atop {\text{labels}}}\overbrace {\int \limits \limits _{R_{N}}d^{3}\mathbf {r} _{N}\cdots \int \limits \limits _{R_{1}}d^{3}\mathbf {r} _{1}} ^{\text{continuous labels}}} _{\text{superposing weighted basis kets}}\,\underbrace {\overbrace {\Psi } ^{{\text{state}} \atop {\text{label}}}(\overbrace {\mathbf {r} _{1},\ldots ,\mathbf {r} _{N},s_{z\,1},\ldots ,s_{z\,N}} ^{{\text{eigenvalues of basis observables}} \atop {\mathord {\sim }}{\text{ argument of wave function}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,})} _{{{{\text{component of state ket}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \atop {\text{= value of scalar product }}\langle \,{\text{basis bra }}|\,\Psi \,\rangle \,\,} \atop {\text{= value of wave function}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \atop {\text{= weight of basis ket}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\underbrace {|\overbrace {\mathbf {r} _{1},\ldots ,\mathbf {r} _{N},s_{z\,1},\ldots ,s_{z\,N}} ^{{\text{eigenvalues of basis observables}} \atop {\mathord {\sim }}{\text{ label of basis ket}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\rangle } _{\text{basis ket}}$

$\underbrace {|\Psi \rangle } _{\text{state ket}}=\underbrace {\overbrace {\sum _{s_{z\,1},\ldots ,s_{z\,N}}} ^{{\text{discrete}} \atop {\text{basis labels}}}} _{{{\text{superposing}} \atop {\text{weighted}}} \atop {\text{basis kets}}}\,\underbrace {\overbrace {\Psi } ^{{\text{state}} \atop {\text{label}}}(\overbrace {s_{z\,1},\ldots ,s_{z\,N}} ^{{{{\text{eigenvalues of}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \atop \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{basis observables}}} \atop {\mathord {\sim }}{\text{ wave-function argument}}})} _{{{{{{\text{weight of basis ket}}\,\,\,\,\,} \atop {\text{= component of state ket}}} \atop {\text{= value of scalar product}}} \atop \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{i.e.}}\langle \,{\text{basis bra}}|\,\Psi \,\rangle } \atop {\text{= value of wave function}}\,}\underbrace {|\overbrace {s_{z\,1},\ldots ,s_{z\,N}} ^{{{\,\,\,\,\,{\text{eigenvalues of}}}\,\,\,\,\,\,\,\,\,\,\,\, \atop \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{basis observables}}} \atop {\mathord {\sim }}{\text{ label of basis ket}}\,\,\,\,}\rangle } _{\text{basis ket}}$

species

London & Bauer (p. 257 of 1983):

"Quantum mechanics, truly a "theory of species," is perfectly adapted to this experimental task."

A wave function in quantum mechanics is a mathematical object that represents a particular pure quantum state of a specific isolated system of one or more particles. It is a central entity in quantum mechanics. The most common symbols for a wave function are the Greek letters ψ or Ψ (lower-case and capital psi).

A single wave function describes the entire system, covering at once all the particles in it. For the one state, however, there are many different wave functions, each giving its respective version, or representative description, in a chosen coordinate system. Each of the different representatives encodes all observable items of information about the state, for example the average momentum of a particle, the different versions being mutually interconvertible by one-to-one mathematical transformations. In general, a physical observation of a quantum system in a particular state gives a partly unpredictable result. The wave function, however, encodes probabilities of the possible results, through mathematical operators called observables. Each of the many possible modes of physical observation is encoded by its own mathematical observable. Each such observable generates a representation that encompasses all the possible particular states of the specific system. For a particular state and coordinate system, for each representation there is just one representative wave function.

In non-relativistic quantum mechanics, disregarding spin, wave functions are solutions of the Schrödinger equation for the specific system, in a chosen representation, and coordinate system. This equation governs their dynamical behavior over time. Because the Schrödinger equation is mathematically a wave equation, a wave function behaves in some respects like other waves, such as water waves or waves on a string. This explains the name "wave function". The wave of the wave function, however, is not a wave in ordinary physical space; it is a wave in an abstract mathematical "space", known as "configuration space", and in this respect it differs fundamentally from water waves or waves on a string.

index or coordinate

Von Neumann p. 27: "This analogy may seem entirely formal, but in reality this is not so. The indices ν and ν′ can also be regarded as coordinates in a state space, that is, if we interpret them as quantum numbers."

quantal

Forbidden or denied by you know who but used by

• Fowler & Guggenheim (1956).

Wave function

• Corresponding to the two examples in the first item, to a particular state there correspond two wave functions, Ψ(x, S_z) and Φ(p, S_y), both describing the same state. For each choice of maximal commuting sets of observables for the abstract state space, there is a corresponding representation with its function space of wave functions.
• Each choice of representation should be thought of as specifying a unique function space composed of its corresponding wave functions. The several spaces are isomorphic. They are, however, best thought of as distinct. They may, for example, be regarded as distinct copies of a common underlying set of square integrable functions.

Physical meaning

There are two physically distinct kinds of quantum state. They are the pure states and the mixed states.

Pure states

The present sub-section is focused on the pure states.

Dirac

A pure state is a state that is respectively pure with respect to every member of some maximal set of compatible observables. That means some particular maximal set, at least; perhaps more than one maximal set. It is not pure with respect to each member of every maximal set of compatible observables. There are maximal sets with respect to some member of which it is not pure. If it is pure with respect to some member A of some maximal set of compatible observables, then it must be impure with respect to the observable that is conjugate to A, namely the observable B such that AB − BA = iħ. The emergence of the beam from a B analyzer shows how it is in a superposition of B states. If however a detector is put in some output channel of the B analyzer, then that channel contains particles pure with respect to observable B, not pure with respect to an A analyzer, because that sub-beam is not formed by re-assembly of the whole original beam that was pure with respect to an A analyzer.

It is, therefore, the beam that is pure or impure. A single particle can be considered to be in a pure state only by virtue of its membership of a pure beam.

To be verified as pure, a particle must pass through two copies of a definite type of analyzer and come through each by way of the respective copy-same channels. The first analyzer purifies the beam and thereby creates and certifies the pure state, and the second verifies the certified purity. The only way the particles can all come out of one and the same channel of the verifying copy of the analyzer is for their source to have been the purifying first copy of the analyzer. If an incompatible analyzer is put between the purifying and verifying analyzers, then, in order to have a pure state reach the verifying analyzer, the sub-beams from the incompatible analyzer must be re-assembled intact before they reach the verifying analyzer.

Lamb

"I mentioned that systems can exist in states. However, in general, this means that some process has prepared the system to be in that state. It is very unlikely that a given complicated system will have a definite wave function. The best we can do is try to describe a system by a statistical distribution or ensemble of states. The density matrix of Landau, von Neumann, and Dirac is ideally suited for describing mixtures. If a system has a wave function, one speaks of a "pure case". Otherwise one has a "statistical mixture". A pure case is all too easily converted into a mixture by any small erratic disturbance, while a mixture can never be put back into a pure case except by some process of selection of a member of the ensemble which happens to be in a desired state. Such a selection process involves external interactions with the system that completely wipe out the memory of the past.

...

"The cat–box system is not really an isolated system, because there is an observer who will open the box. Even if it were an isolated system, it could not be assigned a wave function, but only a mixture of wave functions."

Surely Lamb is wrong in claiming that all memory is lost, if the word memory has any semblance of its ordinary meaning. The Feynman re-assembly proves this, if it is valid. It is not the process of selection that wipes out all memory: it is detection, an irreversible process.

Pauli

Pauli (1958/1980), p. 68:

"Wave equations of this type form the basis for all experiments, which deal with the deflections of molecular beams in external force fields."

Dual states

There are two physically distinct kinds of quantum pure state. They are (a) states as prepared without regard to how they will be observed; and (b) states as observed without regard to how they were prepared. Mathematically, they are signified as (x) state vectors and (y) their duals. The quantum theory is symmetrical between (x) and (y), and so it is a matter of convenience whether (x) is chosen for (a) or for (b), and consequently (y) for (b) or for (a). In Dirac's bra–ket notation, it may be convenient to choose to denote (a), states as prepared, by kets, and then (b), states as observed, by bras. (It would be valid, but not customary, to make the alternative choice, with kets as output states, mutatis mutandis.)

In this area of physics, it is customary to use a special terminology, as follows. In this custom, the term 'inner product' is not widely used; the scalar product is widely used instead.

A scalar product is between a vector and a dual vector, for example between a bra and a ket.

Orthogonality is defined between bras, or between kets. In this custom, for example, $|A\rangle$ and $|B\rangle$ are orthogonal just when $\langle A|B\rangle =0$. This is in some ways physically convenient. It is fundamental to quantum mechanical observation that the output channels of a quantum analyzer are all mutually orthogonal. To verify orthogonality between two output channels of a given analyzer, an experiment needs two copies of it, the first to prepare one state as coming from one channel, say $|B\rangle$, and the second to detect the other state as coming from the other channel, say $\langle A|$. If the system always fails to pass through both relevant analyzer channels, not only are the vectors said to be orthogonal, but so also are the states.

Dirac

"Then the number φ corresponding to any  $|A\rangle$ may be looked upon as the scalar product of that  $|A\rangle$ with some new vector, there being one of these new vectors for each linear function of the ket vectors  $|A\rangle$."

"A bra vector is considered to be completely defined when its scalar product with every ket vector is given."

"In ordinary space, from any two vectors one can construct a number—their scalar product—which is a real number and is symmetrical between them. In the space of bra vectors or the space of ket vectors, from any two vectors one can again construct a number—the scalar product of one with the conjugate imaginary of the other—but this number is complex and goes over into the conjugate complex number when the two vectors are interchanged. There is thus a kind of perpendicularity in these spaces, which is a generalization of the perpendicularity in ordinary space. We shall call a bra and a ket vector orthogonal if their scalar product is zero, and two bras or two kets will be called orthogonal if the scalar product of one with the conjugate imaginary of the other is zero. Further me shall say that two states of our dynamical system are orthogonal if the vectors corresponding to these states are orthogonal."

Gottfried

"It is necessary, first, to associate a dual vector to every ket in a one-to-one manner, called a bra, denoted by the symbol  $\langle \,|$ ; and second, to define the scalar products as being between bras and kets."

"Although we shall frequently say that  $|\alpha \rangle$ and  $|\beta \rangle$ are orthogonal if  $\langle \beta |\alpha \rangle \,=\,0$ , it should be remembered that the scalar product is only defined between a vector and a dual vector."

Again, Gottfried does not bother with the inner product as distinct from the scalar product.

Bibliography for cited texts

• Auletta, G., Fortunato, M., Parisi, G. (2009). Quantum Mechanics, Cambridge University Press, Cambridge UK, ISBN 978-0-521-86963-8.
• Dirac, P.A.M. (1930). The Principles of Quantum Mechanics, 1st edition, Oxford University Press, Oxford UK.
• Dirac, P.A.M. (1935). The Principles of Quantum Mechanics, 2nd edition, Oxford University Press, Oxford UK.
• Dirac, P.A.M. (1947). The Principles of Quantum Mechanics, 3rd edition, Oxford University Press, Oxford UK.
• Dirac, P.A.M. (1958). The Principles of Quantum Mechanics, 4th edition, Oxford University Press, Oxford UK.
• Feynman, R.P., Leighton, R.B., Sands, M. (1963). The Feynman Lectures on Physics, Volume 3, Addison-Wesley, Reading MA, available here.
• Gottfried, K. (1966). Quantum Mechanics, volume 1, Fundamentals, W.A. Benjamin, New York.
• Gottfried, K., Yan, T.-M. (2003). Quantum Mechanics: Fundamentals, 2nd edition, Springer, New York, ISBN 978-0-387-22023-9.
• Landau, L.D.; Lifshitz, E. M. (1977). Quantum Mechanics: Non-Relativistic Theory. Vol. Vol. 3 (3rd ed.). Pergamon Press. ISBN 978-0-08-020940-1. {{cite book}}: |volume= has extra text (help) Online copy
• Schwartz, M.D. (2014), Quantum Field Theory and the Standard Model, Cambridge University Press, Cambridge UK, ISBN 978-1-107-03473-0.

section

I am responding to the above comment

The meaning of that is perfectly clear: in QM, states where the photon is in different places can be superposed, much like states with different polarization can be superposed in classical optics. A little later he points out that the fact that states where the photon is in different places can be superposed is a specific example of a general principle, namely that any two (or more) states can be superposed. Contrary to what you assert above, there is no restriction on that whatsoever, nor is there any such thing as "incompatible wavefunctions", nor is there any problem superposing a state specified in terms of position with one specified in terms of momentum.

I have no worry about superposition of states where the photon is in different places, in agreement, I think, with you and Dirac; the photon originated in one beam, prepared so as to define a space of position-specified states. Different position states are compatible. What worries me is the idea, for an electron, of superposing a position-specified state with a momentum-specified state. I think those two states, in general, come from differently prepared beams, and cannot superpose. You say I am mistaken about that, as I read you.

Perhaps you will very kindly enlighten me. How does one superpose a position-specified and a momentum-specified state? I am asking what is the physical experimental procedure? Dirac says on page 12 (see below) that a mathematical procedure is always possible, but the use of it depends on the special physical conditions under consideration. What does he mean by that?

Dirac 1st edition

You wrote that I didn't provide enough context to show what Dirac was talking about. Here is some context. I don't know how much is allowed by copyright. I don't know if this context will help.

P. 8, the context:

When a state is formed by the superposition of two other states, it will have properties that are in a certain way intermediate between those of the two original states and that approach more or less closely to those of either of them according to the greater or less 'weight' attached to this state in the superposition process. The new state is completely defined by the two original states when their relative weights in the superposition process are known, together with a certain phase difference, the exact meaning of weights and phases being provided in the general case by the mathematical theory of the next chapter. In the case of polarization of the photon their meaning is that provided by classical optics, e.g. when two perpendicularly plane polarized states are superposed with equal weights, the new state may be circularly polarized in either direction or linearly polarized at an angle π/4, or else elliptically polarized, according to the phase difference. This, of course, is true only provided the two states that are superposed refer to the same beam of light, i.e. all that is known about the position and momentum of a photon in either of these states must be the same for each.

Dirac 4th edition

P. 8, near the bottom, and page 9:

So long as the photon is partly in one beam and partly in the other, interference can occur when the two beams are superposed, but this possibility disappears when the photon is forced entirely into one of the beams by an observation. The other beam then no longer enters into the description of the photon, so that it counts as being entirely in the one beam in the ordinary way for any experiment that may be subsequently performed on it.

To be partly in one beam and partly in the other, I read Dirac as saying the photon must have come from one original beam that was split into two by a beam-splitter or whatever.

P. 12:

Any state may be considered as the result of a superposition of two or more other states, and indeed in an infinite number of ways. Conversely any two or more states may be superposed to give a new state. The procedure of expressing a state as the result of superposition of a number of other states is a mathematical procedure that is always permissible, independent of any reference to physical conditions, like the procedure of resolving a wave into Fourier components. Whether it is useful in any particular case, though, depends upon the special physical conditions under consideration.

Bransden & Joachain

"... the superposition principle, according to which a linear superposition of possible wave functions is also a possible wave function."

• Bransden, B.H., Joachain, C.J. (1983). Physics of Atoms and Molecules, Wiley, New York, ISBN 0-582-44401-2.

A wave function has a specified generalized quantum configuration space. It is related to an abstract state vector, but is not the same thing. A state vector is understood as being representable in any one of many generalized quantum configuration spaces (with associated basis of states), but only one at a time, that is to say only one in a particular beam of pure states. I think wave functions from different bases cannot be superposed. It seems I am being told I am mistaken in that? B & J speak of "possible wave functions". I think they mean 'possible in the chosen basis'.

superposition a relation

"The general principle of superposition of quantum mechanics ... requires us to assume that between these states there exist peculiar relationships such that whenever the system is definitely in one state we can consider it as being partly in each of two or more other states."

"The nature of the relationships which the superposition principle requires to exist between the states of any system is of a kind that cannot be explained in terms of familiar physical concepts."

"The assumption of superposition relationships between the states leads to a mathematical theory in which the equations that define a state are linear in the unknowns."

• Dirac, P.A.M. (1958). The Principles of Quantum Mechanics, 4th edition, Oxford University Press, Oxford UK.

method of exposition

There are two ways of presenting the basic formalism of quantum mechanics, which are here regarded as the concrete and the abstract ways. The present article above uses the abstract, or symbolic, way, in terms of Dirac's bras and kets, or state vectors. The alternative is the concrete, or explicitly presented, way, in terms of wave functions. It may be useful here to give some information about both ways, and about how the two are related.

Dirac

"With regard to the mathematical form in which the theory can be presented, an author must decide at the outset between two methods. There is the symbolic method, which deals directly in an abstract way with the quantities of fundamental importance (the invariants, etc., of the transformations) and there is the method of coordinates or representations, which deals with sets of numbers which correspond with such quantities. The second of these has usually been used for the presentation of quantum mechanics (in fact it has been used practically exclusively with the exception of Weyl's book Gruppentheorie und Quantenmechanik.) It is known under one or the other of the two names 'Wave Mechanics' and 'Matrix Mechanics' according to which physical things receive emphasis in the treatment, the states of a system or its dynamical variables."

• Dirac, P.A.M. (1930). The Principles of Quantum Mechanics, 1st edition, Oxford University Press, Oxford UK.

Messiah

"Of the various ways of introducing the Quantum Theory, the one which uses the general formalism is undoubtedly the most elegant and the most satisfactory. However, it requires the handling of a mathematical formalism whose abstract character runs the risk of obscuring the underlying physical reality. Wave mechanics, which utilizes the more familiar language of waves and partial differential equations, lends itself better to a first encounter. Furthermore, it is in that form that the Quantum Theory is most frequently used in elementary applications. That is why we shall begin with a general outline of wave mechanics."

"One is thus led to build up the entire Quantum Theory by starting directly from the vector concept without reference to the particular representation which can be made thereof."

• Messiah, A. (1961). Quantum Mechanics, volume 1, translated by G.M. Temmer from the French Mécanique Quantique, North-Holland, Amsterdam.

Cohen-Tannoudji

"Similarly. we have chosen to use state spaces and Dirac notation from the very beginning. This avoids the useless repetition which results from presenting the more general bra ket formalism only after having developed wave mechanics uniquely in terms of wave functions. In addition, a belated change in the notation runs the risk of confusing the student, and casting doubts on concepts which he has only just acquired and not yet completely assimilated."

• Cohen-Tannoudji, C., Diu, B., Laloë, F. (1973/1977). Quantum Mechanics, translated from the French by S.R. Hemley, N. Ostrowsky, D. Ostrowsky, second edition, volume 1, Wiley, New York, ISBN 0-471-16432-1.
• Cohen-Tannoudji, C.; Diu, B.; Laloë, F. (1977) [1973]. Quantum Mechanics. Vol. 1 (2nd ed.). New York: Wiley. ISBN 0-471-16432-1.. Translated from the French by S.R. Hemley, N. Ostrowsky, D. Ostrowsky.

Weinberg

"The viewpoint of this book is that physical states are represented by vectors in Hilbert space, with the wave functions of Schrödinger just the scalar products of these states with basis states of definite position. This is essentially the approach of Dirac’s “transformation theory.” I do not use Dirac’s bra-ket notation, because for some purposes it is awkward, but in Section 3.1 I explain how it is related to the notation used in this book. In any notation, the Hilbert space approach may seem to the beginner to be rather abstract, so to give the reader a greater sense of the physical significance of this formalism I go back to its historic roots."

• Weinberg, S. (2013). Lectures on Quantum Mechanics, Cambridge University Press, Cambridge UK, ISBN 978-1-107-02872-2.

Terminology

The two modes of presentation lead to a muddle of terminology. For example, Gottfried writes:

"A variety of names are associated with the vectors we have just defined, and we shall use them all rather indiscriminately. The objects  |a′〉, etc., are frequently referred to as state vectors, or simply vectors, and occasionally they are even called states. We shall also use the Dirac terminology:  |a′〉 is called a ket, and  〈a′ |  a bra."

Some Wikipedia articles use the term 'quantum state' to refer to the mathematical abstractions that may also be called bras and kets, but, from the present viewpoint, that do not actually specify particular definite physical states. In this sense, the term 'state vector' might be regarded as misnamed; the term 'state generating vector' might be clearer, but is not customary. These mathematical abstractions are devices for mathematical manipulations. They are abstractions in the sense that they are derived from physical pure states in two logical moves: the first is from a physical pure state to a wave function; the next, second, logical move is from the wave function to a Hilbert space vector that has forgotten which generalized quantum configuration space the wave function refers to. The second move is possible because the various generalized quantum configuration spaces of interest can be derived from one another by standard linear transformations, and in that sense are equivalent to one another, so that for the relevant mathematical manipulations, it doesn't matter which is the concrete basis. The state vector (bra or ket) does not specify a particular physical pure state. It determines only the convenient mathematical object that can be manipulated as if it referred to a particular physical pure state; one might say that it refers to a species, regardless of its particular definite physical state, rather than to a particular definite individual physical state of that species, and that, for the relevant mathematical purposes, the species is an adequate reference. Thus a wave function is a different mathematical object from a state vector, but a state vector can for mathematical purposes be regarded as representing a wave function. To recover a wave function from a state vector, one must supply a particular generalized quantum configuration state, and the wave functions that form the corresponding vector space basis.

Thus one might say that a state vector (bra or ket) is a mathematical object that can be used, subject to the supply of further data, to generate a wave function that refers to a particular definite physical pure state, or that can be used to generate a presentation of a particular definite physical pure state, but does not by itself specify a physical state. Nevertheless, for many purposes, it is customary, as indicated by Gottfried, and reasonable because of the transformation property, to speak of, and think of, and manipulate mathematically, a state vector as if it did refer to a particular definite physical pure state.

This indefiniteness of the abstraction is also described as due to the state vector's referring to the system in isolation, that is to say "unobserved". One may add 'and unprepared'. Its "state" is not actually specified in physical terms, but is nevertheless symbolically formulated.

For example, Newton writes "A vector in Hilbert space therefore describes the state of a system not only as perfectly specified in physical isolation but also its readiness for any potential correlation with other systems and states."

And: "It follows from the second postulate that a measurement is, in effect, identical to the preparation of a state: by measuring the variable  A and finding the value  A, we have prepared the system in an eigenstate of the operator  A with eigenvalue  A."

• Gottfried, K. (1966). Quantum Mechanics, volume I, Fundamentals, W.A Benjamin, New York.
• Newton, R.G. (2002). Quantum Physics: a Text for Graduate students, Springer, New York, ISBN 0-387-95473-2.

quantum state

concrete and abstract quantum states

There are two distinct meanings to the term quantum state. Here it is convenient to use two distinct respective verbal forms for them. A concrete state is a physical object that is physically preparable and detectable, and is uniquely represented by a wave function. An abstract state, also called a point in abstract state space, is a mathematical object that encodes all generic information about the possible concrete states, or their respective wave functions, of a species of quantal entity or system. A concrete state can in itself be pure or mixed, but is not open to be acted on by a density operator, while an abstract state in itself is not open to such a distinction, but can be rendered pure or mixed under the action of a density operator. A pure concrete state corresponds one-to-one with a wave function. A mixed concrete state corresponds with the effect of a density operator on a point in abstract state space.

The abstract state leaves undefined the chosen experimental arrangement or configuration space. Thus it contains less information than does a concrete state specified with a chosen configuration space. Indeed, the abstractive process that generates the abstract state space is just the forgetting of the specification of the experimental arrangement. The utility of the abstraction is that for mathematical calculations, it supports one and the same method, no matter which configuration space is specifying the concrete state. Thus, some mathematically inclined writers feel comfortable in saying that the point in abstract state space "gives the state". Of course it doesn't give the exact physical state, but it is near enough for mathematical purposes.

McIntyre

"Because the quantum state vectors are abstract, it is hard to say much about what they are, other than how they behave mathematically and how they lead to physical predictions. ... You may have seen $\psi (x)$ used before as a quantum mechanical wave function. However, the state vector or ket $|\psi \rangle$ is not a wave function. Kets do not have any spatial dependence as wave functions do. "

• McIntyre, D.H. (2012). Quantum Mechanics: a Paradigms Approach, Pearson Addison–Wesley, Boston, ISBN 978-0-321-76579-6.

Kets are not functions, and have as domains neither positional nor momental generalized quantum configuration spaces. Wave functions do have such domains. The stripping off of a domain is a main reason why quantum state vectors are recognized as abstract. They lack a degree of detail that is supplied in a wave function. The absence of detail is part of their advantage for mathematical manipulation.

Moreover, though they have morphisms with wave functions, the "representatives" of Dirac are arrays of numbers, not functions with domains and ranges in the sense that is usual for wave functions in their native form. By some writers, the representatives are described as "scalar products" and at the same time as "wave functions"; at face value, a function and a scalar product are different kinds of object.

concrete states and their wave functions

generalized quantum configuration spaces

There are mathematical relations between the accounts of different generalized quantum configuration spaces. Such transformations necessarily indicate different generalized quantum configuration spaces and correspondingly physically different dynamical states and wave functions.

Naturally, it would be nonsensical to try to superpose wave functions from different generalized quantum configuration spaces. "This, of course, is true only provided the two states that are superposed refer to the same beam of light, i.e. all that is known about the position and momentum of a photon in either of these states must be the same for each."

Nevertheless, the different generalized quantum configuration spaces can be related in terms of an abstract vector space. The wave functions can be derived by applications of operations to that abstract vector space.

The relations between two different generalized quantum configuration spaces include the important special cases in which the members of the pair have precisely conjugate dynamical variables, such as position and momentum. The abstract vector space is characterised by a more abstract form of Hamiltonian that treats the different generalized quantum configuration spaces as abstract coordinates. Physically, such an abstract Hamiltonian characterises a species, rather than a dynamical state. Each species can be prepared in many dynamical pure states, each state being characterised uniquely by its characteristic configuration space and a unique point in that space, with a corresponding wave function, unique up to coordinate changes within the configuration space. Physically, a species is characterised by chemical purity, and atoms or molecules of the species are supplied to the preparative quantum analyser as a beam emerging from an oven full of the vaporized species, or some such. The quantum analyser filters or selects the individual systems of the raw beam to produce a beam of individual systems each in some common pure state.

• Dirac, P.A.M. (1930). The Principles of Quantum Mechanics, 1st edition, Oxford University Press, Oxford UK.

pure state or pure case

By definition, each kind of quantum analyser prepares its characteristic set of pure states, which just constitutes its characteristic degree of freedom in its characteristic generalized quantum configuration space. In other words, a quantum analyser creates a characteristic dynamical variable, specified by the set of its output channels. With respect to a given analyser, a particular pure state is fully determined by the choice of just one of its output channels. To depict the whole configuration space, one would imagine that every output channel of every component one-variable sub-analyser would be open, forming a large branching array of channels.

Different pure states would be produced by using a different output channel of the same overall quantum analyser, or by using different component one-variable sub-analysers.

To each pure state, there is a range of admissible coordinate systems. Mathematical relations hold between the different coordinate systems, but these do not alter the pure state, which is physically invariant under coordinate transformations of generalized quantum configuration space. The wave function of the pure state is a function of points of the configuration space. Its explicit mathematical form, expressing it as a function of the coordinates, changes with coordinate changes, while as a function of points in generalized configuration space, it remains the same.

A given configuration space describes a range of possible pure states, which have respective wave functions that form a function space that has the structure of a vector space, of which the members are mutually superposable. It happens that such vector space is a Hilbert space, but it is not the Hilbert space known as (abstract) state space.

The pure states, defined with respect to any given configuration space, correspond one-to-one with a matching set of wave functions.

Pauli on the pure case

"The definition of the pure case as that ensemble for which one eigenvalue of the density matrix is 1 and all the others 0, is, therefore, equivalent to the statement that a pure case cannot be produced by mixing two different ensembles."

• Pauli, W. (1958/1980). General Principles of Quantum Mechanics, translated by P. Achuthan, P.K. Venkatesan, Springer, Berlin, ISBN 3-540-02298-9.

Langevin on "le cas pur"

Paul Langevin writes: "... et traduit très exactement le fait que certaines grandeurs, dites non commutables, ne peuvent être simultanément connues avec une entière certitude, de sorte que chaque système caractérise, par un certain nombre de grandeurs observables, différentes formes de connaissance maximum qui correspondent à ce qu'on appelle les cas purs." "It also expresses quite exactly the fact that certain quantities, called noncommutable, cannot be simultaneously known with complete certainty. It characterizes the system by a certain number of observable quantities, different forms of "maximum knowledge" corresponding to different so-called "pure cases.""

• London, F., Bauer, E. (1939). La Théorie de l'Observation dans la Mécanique Quantique, Hermann & Cie, Paris.
• Langevin, P. (1939). Préface de M. Langevin, pp. 3–4 of London, F., Bauer, E. (1939).
• Shimony, A., Wheeler, J.A., Zurek, W.H., McGrath, J., McGrath, S.M. (1983). Translation of London, F., Bauer, E. (1939) at pp. 217–259 in Wheeler, J.A., Zurek, W.H. (1983).
• Wheeler, J.A., Zurek, W.H., editors (1983). Quantum Theory and Measurement, Princeton University Press, Princeton NJ.

Dirac on states

Dirac's Principles of Quantum Mechanics changed in some respects as it passed through its four editions, of 1930, 1935, 1947, and 1958.

Dirac defines a state in various ways in various editions:

first edition, 1930

"We must first generalize the meaning of a 'state' so that it can apply to any atomic system. Corresponding to the case of the photon, which we say is in a given state of polarization when it has been passed through suitable polarizing apparatus, we say that any atomic system is in a given state when it has been prepared in a given way, which may be repeated arbitrarily at will. The method of preparation may then be taken as the specification of the state."

"... we may still regard it as being entirely in a certain single state. In fact it still satisfies the definition of having been prepared in a definite way which may be repeated at will."

    "The case of greatest interest of the compatibility of two observations is when they both refer to the same instant of time. The compatibility condition is now that if either is made a very short time before the other, the probability of any given result being obtained with the second shall be the same as if the first had not been made.

    It is often convenient to count two or more compatible observations, particularly when they are simultaneous, as a single observation, the result of such an observation being expressible by two or more numbers. We shall frequently have to consider the greatest possible number of independent compatible simultaneous observations being made on a system and shall, for brevity, call such a set of observations a maximum observation. When a maximum observation is made on a system, its subsequent state is completely determined by the result of the observation and is independent of its previous state. This may be considered as an axiom, or as a more precise definition of a state.

    The state of a system after a maximum observation has been made on it is such that there exists a maximum observation (namely, an immediate repetition of the maximum observation already made) which, when made on the system in this state, will for a certainty lead to one particular result (namely, the previous result over again). Any state can be specified only as the state ensuing after a given maximum observation has been made for which a given result was obtained, or in some equivalent way. We can therefore draw the conclusion that for any state there must exist one maximum observation which will for a certainty lead to one particular result, and conversely, if we consider any possible result of a maximum observation, there must exist a state of the system for which this result for the observation will be obtained with certainty."

second edition, 1935

"A state of motion is completely specified when one is given that it is associated with a certain beam."

"Corresponding to the description that we had in the case of the polarization, we must now describe the photon as going partly into each of the two components into which the incident beam is split. The photon is then, as we may say, in a state of motion given by the superposition of the two states of motion associated with the two components. We are thus led to a generalization of the term 'state of motion' applied to a photon. For a photon to be in a definite state of motion it need not be associated with one single beam of light, but may be associated with two or more beams of light which are the components into which the original one beam has been split."

I think there is a problem with Dirac's choice of the word 'partly'. It suggests division of the photon, but does not intend it. A better word would be 'jointly'. This word 'jointly' suggests a more abstract link than does 'partly', and the intended link is indeed more abstract. The notion of an 'enduring physical object', such as for example a classical material object, is, according to Whitehead, an abstraction, perhaps a little uncomfortably for some people. It is then perhaps not too fanciful to consider a quantal entity such as an electron also to be an abstract object, a step more abstract than a classical material object, because in some senses, quantum theory conceives the state of the electron to be jointly in two separate places.

Dirac's next paragraph reads

"Let us consider now what happens when we determine the energy in one of the components. The result of such a determination must be either the whole photon or nothing at all. Thus the photon must change suddenly from being partly in one beam and partly in the other to being entirely in one of the beams. This sudden change is due to the disturbance in the state of motion of the photon which the observation necessarily makes. It is impossible to predict in which of the two beams the photon will be found. Only the probability of either result can be calculated from the previous distribution of the photon over the two beams."

If in this paragraph, the word 'partly' is replaced by the word 'jointly', the feeling of conceptual incongruity (Dirac's phrase is "strange idea") is reduced.

third edition, 1947

"We know that each of them [photons in a beam] is located somewhere in the region of space through which the beam is passing, and has a momentum in the direction of the beam of magnitude given in terms of the frequency of the beam by Einstein's photo-electric law—momentum = frequency multiplied by a universal constant. When we have such information about the location and momentum of a photon, we shall say that it is in a definite translational state."

"A state of a system may be defined as an undisturbed motion that is restricted by as many conditions or data as are theoretically possible without mutual interference or contradiction. In practice, the conditions could be imposed by a suitable preparation of the system, consisting perhaps of passing it through various kinds of sorting apparatus, such as slits and polarimeters, the system being undisturbed after preparation. The word 'state' may be used to mean either the state at one particular time (after the preparation), or the state throughout the whole of the time after the preparation. To distinguish these two meanings, the latter will be called a 'state of motion' when there is liable to be ambiguity."

This definition passes intact from the 3rd to the 4th edition.

fourth edition, 1958

"A state of a system may be defined as an undisturbed motion that is restricted by as many conditions or data as are theoretically possible without mutual interference or contradiction. In practice, the conditions could be imposed by a suitable preparation of the system, consisting perhaps of passing it through various kinds of sorting apparatus, such as slits and polarimeters, the system being undisturbed after preparation. The word 'state' may be used to mean either the state at one particular time (after the preparation), or the state throughout the whole of the time after the preparation. To distinguish these two meanings, the latter will be called a 'state of motion' when there is liable to be ambiguity."

This definition passed intact from the 3rd to the 4th edition.

Bibliography for references

• Dirac, P.A.M. (1930). The Principles of Quantum Mechanics, 1st edition, Oxford University Press, Oxford UK.
• Dirac, P.A.M. (1935). The Principles of Quantum Mechanics, 2nd edition, Oxford University Press, Oxford UK.
• Dirac, P.A.M. (1947). The Principles of Quantum Mechanics, 3rd edition, Oxford University Press, Oxford UK.
• Dirac, P.A.M. (1958). The Principles of Quantum Mechanics, 4th edition, Oxford University Press, Oxford UK.

Dirac on bras and kets

"We now assume that each state of a dynamical system at a particular time corresponds to a ket vector, the correspondence being such that if a state results from a superposition of certain other states, its corresponding ket vector is expressible linearly in terms of the corresponding ket vectors of the other states, and conversely.

This means at least that there is a map from states to kets. It does not say whether the map is one-to-one or many-to one. It does not say whether or not the map is onto.

"All the states of a dynamical system are in one-to-one correspondence with all the possible directions for a ket vector, no distinction being made between the directions of the ket vectors |A〉 and −|A〉."

This settles the question, at least for the present.

It is a fact of algebra that, given a vector space, and given a basis for it, one has a unique corresponding basis in the dual space. In order to have phase for superposition, it is convenient, almost necessary, to make the kets vectors over the field of complex numbers. To get a basis, for the sake of having a dual space, it is convenient, almost physically necessary, to arbitrarily prescribe a quantum analyser. It is in the nature of a quantum analyser that it must be self-dual. This means that the input and output ends of a quantum analyser must be interchangeable, as for example in a Stern-Gerlach magnet, or in a prism, or in a half-silvered mirror.

So, in order to specify a quantum state with respect to a specific kinematic frame, one needs to presuppose a specific quantum analyser.

Following Bohr's doctrine of quantum phenomena as the categorical facts of quantum physics, one wants to describe quantum phenomena, and for this one needs initial and final conditions. If the initial and final conditions are allowed on occasion to be the same, it is convenient to describe them by practically congruent kinematic frames. For congruence, one uses a suitable quantum analyser, and then one has bras and kets as elements of isomorphic dual spaces.

• Dirac, P.A.M. (1958). The Principles of Quantum Mechanics, 4th edition, Oxford University Press, Oxford UK.

others on states

London & Bauer

"$\psi (q,t_{0})$ represents the "state" of the system at an instant $t_{0}$. Here we take this term in a sense completely analogous to that which it has in classical mechanics, where one says that the data $q_{1}(t_{0})\,...\,q_{f}(t_{0}),p_{1}(t_{0})\,...\,p_{f}(t_{0})$ "represent a state". The knowledge of the representative of the state at a given instant is necessary and sufficient for an unambiguous calculation, with the aid of the dynamic law, of the representative of the state at every subsequent moment. We cannot forgo a part of these data without losing the ability to calculate the future. Neither can we add to them supplementary data without introducing useless tautologies or contradictions of the data already in hand."

"But the Schrödinger equation has all the features of a causal connection."

"Heisenberg found the solution to this dilemma. He emphasized that it is the process of measurement itself which introduces the element of indeterminacy in the state of the object.

"Thus the statistical feature would not show up except on the occasion of a measurement."

Three comments.

1. The evolution of the representative of a state, and likewise of a state vector, is fully deterministic. Quantum evolution is not stochastic; it is deterministic.
2. The state (as here defined, and identifiable with the state vector) is not the same thing as the representative of the state. The state (as here defined), or state vector, is an abstraction that is useful for mathematical manipulations, but does not identify a concrete physical state in the ordinary physical sense of the word. The state vector does not contain all the information about the concrete physical state that is contained in the representative of the state. This is because the state vector omits the identity of the basis, which tells how the representative was prepared or how it was detected. The concrete physical state also evolves deterministically until it encounters an external system with which it interacts.
3. London & Bauer are here in effect stating that there are no "hidden variables", though they do not use this terminology.

Dirac remarks "... for some purposes it is more convenient to replace the abstract quantities [state vectors] by sets of numbers with analogous mathematical properties and to work in terms of these sets of numbers. ... The way in which the abstract quantities are to be replaced is not unique, there being many possible ways corresponding to the many systems of coordinates one can have in geometry. Each of these ways is called a representation and the set of numbers that replaces the abstract quantity is called a representative of that abstract quantity in the representation. ... When one has a particular problem to work out in quantum mechanics, one can minimize the labour by using a representation in which the representatives of the more important abstract quantities occurring in that problem are as simple as possible."

Comments:

1. Dirac does not mention here that a "representative" denotes none other than a concrete physical system defined by its preparation or detection as a pure state.
2. The word 'abstraction' by definition indicates that something has been taken out. What has been taken out to yield Dirac's abstract state vectors? The removed item is the specification in terms of the basis. Physically, this means that it is not stated as to how the system was prepared or detected, at very last step, by filtration through, for example, a Stern–Gerlach apparatus. It is not specified as a concrete physical system in a particular pure state. It is indicated that it refers to a pure state, but it is not indicated as to which one. In this ordinary language sense, a state vector is stated as a genus, not as a species (in this sentence, the words 'genus' and 'species' are used as in ordinary language). For example, a particular individual person is, in this ordinary language sense, a specimen or member of the human species, homo sapiens. And the human species is a member of the genus homo. In contrast, in the present physical context, in the usage of physics and chemistry, the term 'species' refers to a particular state vector (or genus in the ordinary language), not to a class of representatives, such as in ordinary language might be labeled a species.

• Dirac, P.A.M. (1958). The Principles of Quantum Mechanics, 4th edition, Oxford University Press, Oxford UK.
• London, F., Bauer, E. (1939). La Théorie de l'Observation dans la Mécanique Quantique, Hermann & Cie, Paris.
• Shimony, A., Wheeler, J.A., Zurek, W.H., McGrath, J., McGrath, S.M. (1983). Translation of London, F., Bauer, E. (1939) at pp. 217–259 in Wheeler, J.A., Zurek, W.H. (1983).
• Wheeler, J.A., Zurek, W.H., editors (1983). Quantum Theory and Measurement, Princeton University Press, Princeton NJ.

Heisenberg

"Quantum mechanics represents a physical system by a wavefunction in a configuration space whose number of dimensions is determined by the number of degrees of freedom of the system in question. The square of the absolute value of the wavefunction at a specific point of this space gives the probability that the anschaulich physical quantities denoted by the coordinates of the space take on the specific values corresponding to that point, if the system is observed with regard to these values. The formalism of quantum mechanics is thus based on the assumption that a physical system can be represented by classical-anschaulich variables, and that, as in classical theory, there can be an objective sense, independent of the processes of observation, in speaking of the actual value of a specific physical quantity, e.g. of the 'position of the electron'."

Also Heisenberg in his succinct textbook describes the preparation of atoms in a state: "The original beam will thus be divided into several, each containing only atoms in one state." This part of the text about the state of atoms does not refer to Dirac's abstract formulation.

• Heisenberg, W. (1930). The Physical Principles of the Quantum Theory, translated by C. Eckart, F.C. Hoyt, University of Chicago Press, Chicago IL.
• Heisenberg, W. (1935/2011). Is a deterministic completion of quantum mechanics possible? translation at pp. 6–19 of Crull, E., Bacciagaluppi, G. (2011).
• Crull, E., Bacciagaluppi, G. (2011). Translation of: W. Heisenberg, 'Ist eine deterministsche Ergänzung der Quantenmechanik möglich?'

Newton

According to Newton, "... a measurement is, in effect, identical to the preparation of a state; by measuring the variable A and finding the value A, we have prepared the system in an eigenstate of the operator A. ... To prepare a system in a precisely specified state implies that all the dynamical variables needed for its unique determination have been measured, and vice versa."

• Newton, R.G. (2002). Quantum Physics:a Text for Graduate students, Springer, New York, ISBN 0-387-95473-2.

The certain other states can be found by analysing the pure state with a quantum analyser other than the one that defines it as pure, or, mathematically, operating on the ket with an observable other than the one that defines it as pure.

Feynman

"As long as we deal only with “pure states”—that is, we have only one channel open—there are nine such amplitudes ..."

"The important point is this: if the T filter passes only one beam, the fraction that gets through the second S filter depends only on the setup of the T filter, and is completely independent of what precedes it. The fact that the same atoms were once filtered by an S filter has no influence whatever on what they will do once they have been sorted again into a pure beam by a T apparatus. From then on, the probability for getting into different states is the same no matter what happened before they got into the T apparatus."

"We should be careful to say that we are considering good filters which do indeed produce “pure” beams. If, for instance, our Stern-Gerlach apparatus didn't produce a good separation of the three beams so that we could not separate them cleanly by our masks, then we could not make a complete separation into base states. We can tell if we have pure base states by seeing whether or not the beams can be split again by another filter of the same kind. ... Our statement about base states means that it is possible to filter to some pure state, so that no further filtering by an identical apparatus is possible."

• Feynman, R.P., Leighton, R.B., Sands, M. (1963). The Feynman Lectures on Physics, Volume 3, Addison-Wesley, Reading MA, available here.

Messiah

"When the system is in a state represented by a wave of type II.34 [Ψ = ψe^−iEt/ħ], it is said to be in a stationary state of energy E; the time-independent wave function ψ is usually called the wave function of the stationary state, although the true wave function differs from the latter by a phase factor exp (−iEt/ħ)."

"A precise measurement carried out on a complete set of compatible variables of a system represents the maximum information one can obtain on that system. It therefore defines the dynamical state completely, and a definite wave function corresponds to it. ... During the process of observation, the measured system can no longer be considered separately and the very notion of a dynamical state defined by the simpler wave function ψ(t) loses its meaning."

"The dynamical state of [an isolated] system is represented at a given instant of time by its wave function at that instant."

"In contrast to what happens during the process of measurement, the system, once the measurement is completed, can again be treated as an entity completely separated from the measuring apparatus. It becomes possible again to describe it by means of a wave function involving only its own dynamical variables. This wave function of the system after measurement is certainly different from the wave function immediately before measurement, except perhaps if the latter is an eigenfunction of the observable A associated with the measured quantity. We shall call this (non-causal) change of the wave function by the measurement process the filtering of the wave packet."

"The time sequence of an experiment in physics can be viewed in the following manner. At the initial time t₀ one prepares the system by performing on it the simultaneous measurement of a complete set of compatible variables. Its dynamical state is thus determined at time t₀. ... In practice, a complete "preparation" of the system like the one contemplated above, is rarely achieved. Most frequently, the dynamical variables measured during the preparation do not constitute a complete set. As a consequence, the dynamical state of the system is known incompletely and one must have recourse to the methods of statistics. Instead of a given dynamical state, one one is dealing with a statistical mixture of states; instead of a signing to the system a well-defined wave function, one assigns to it a statistical mixture of wave functions each having a suitable statistical weight. ... When the preparation is complete, and consequently the dynamical state of the system is completely known, one says that one is dealing with a pure state, in contrast to the statistical mixtures which characterize incomplete preparations."

"One assumes, however, that one can always add to every dynamical variable a certain number of others and thus form a complete set of compatible variables; by definition, all the variables of such a set are compatible with each other and there exist no other variables compatible with each of them, aside from functions of these variables themselves. The precise determination of the variables of a complete set constitutes the largest possible amount of information one can obtain on the dynamical state of a quantum system. Consequently, the dynamical state of a quantum system is no longer defined, as in Classical Theory, by the precise specification of all the dynamical variables associated with the system, but rather by the specification of those which occur in one of the various complete sets of compatible variables one can construct."

"In a given dynamical state, a definite statistical distribution of values is associated with each dynamical variable of the system."

"The dynamical state of a quantum system is defined by a collection of precisely defined quantities, namely the particular values taken by the dynamical variables of a complete set of compatible variables. By carrying out the precise measurement of the variables of such a set, one defines unambiguously the state of the system at time t when the measurement was performed."

"... all the elements for the description of a quantum system are present when one has defined its fundamental dynamical variables, the commutation relations obeyed by the representative observables, and the explicit expression as a function of those observables, of the Hamiltonian which governs the motion of the system."

It may be remarked that Messiah's use of the customary term 'simultaneous' is potentially confusing. The component 'measurements' are in general performed in the same experiment as aspects of one and the same quantum process or phenomenon, and are in that sense "simultaneous". But that process is physically determined by the structure of the 'measuring' apparatus. Physically, that apparatus is a spatial sequence of quantum analysers. It it natural to imagine that the quantal entity that is the subject of the 'measurement' passes successively in time through the sequence of analysers, which therefore are imagined to act non-simultaneously in the ordinary language sense. This reasoning allows that the passage of the quantal entity through the 'measuring' apparatus could be in either of two senses, according to a kind of reciprocity principle, indicated by the Hermiticity of the operators for the several commuting dynamical variables, in close analogy with the Helmholtz reciprocity principle. Howsoever, in quantum mechanics the term 'simultaneous' is customary. Perhaps a better term would be 'co-existent'.

It may be further remarked that the term 'measure' used by Messiah here, though regrettably customary, is also potentially confusing. The "measurement" intended here is not a true measurement, because the latter includes a detection that is absent here, necessarily because the detection would destroy the state instead of supplying it as required at the output of the preparative device. An alternative to Messiah's present slightly confusing use of the word 'measurement' would be a term perhaps such as 'quantum analysis'. That is to say, the preparation of a pure state is completed by a complete quantum analyser, the output of which, instead of being sent to a detector, is supplied as output from the preparative device of a system in a pure state. Dirac refers to the quantum analyser in his words on page 11 "consisting perhaps in passing it through various kinds of sorting apparatus, such as slits and polarimeters, the system being left undisturbed after preparation.". There are other usages of terms such as by Aspect et al. of polarization analyser. The regrettable lack of a customary term for the general concept of a quantum analyser is a cause of much confusion in the literature.

It is important to be aware that later in his text, Messiah changes the sense of the word state, from the present sense of an actually preparable or detectable state, and makes it refer to the abstract "state" sense of Dirac, that comprises many ordinary states.

"Thus we postulate that the wave function completely defines the dynamical state of the system under consideration."

Here Messiah is setting things up for problems, as will soon emerge. Previously, the state specification was defined to include the configuration space of relevance (implying inclusion of the chosen measurement or preparatory filter), but soon it will emerge that the "wave function" and the "state" are not tied to the configuration space of relevance. This is means a change of definition of "state".

• Messiah, A. (1961). Quantum Mechanics, volume 1, translated by G.M. Temmer from the French Mécanique Quantique, North-Holland, Amsterdam.

Jauch

"Definition: A state is the result of a series of physical manipulations on the system which constitute the preparation of the state. Two states are identical if the relevant conditions in the preparation of the state are identical."

"Thus, a state of a quantum system can only be measured if the system can be prepared an unlimited number of times in the same state."

• Jauch, J.M. (1968). Foundations of Quantum Mechanics, Addison–Wesley, Reading MA.

Ballentine

"... a state ... is identified with a specification of a probability distribution for each observable. (An observable is a dynamical variable that can in principle be measured.)

"Any repeatable process that yields well-defined probabilities for all observables may be termed a state preparation procedure."

• Ballentine, L.E. (1998). Quantum Mechanics: a Modern Development, World Scientific, Singapore.

kets and bras as input and output state specifiers

Landau Lifshitz (translated into English)

The first (1958) and second (1965) editions of Landau and Lifshitz on non-relativistic quantum mechanics do not mention Dirac's kets and bras. They are introduced in the third edition.

In the first edition, L&L discuss the wave function and measurements in §7. They work with wave functions, not kets and bras. On page 23, they write:

"All these properties of the functions Ψ_n(q) show that they are the eigenfunctions of some physical quantity (denoted by f) which characterises the electron, and the measurement concerned can be spoken of as a measurement of this quantity.

"It is very important to notice that the functions Ψ_n(q) do not, in general, coincide with the functions φ_n(q); the latter are in general not even mutually orthogonal, and do not form a set of eigenfunctions of any operator. This expresses the fact that the results of measurements in quantum mechanics cannot be reproduced."

This account continues and comes to the integral $\int \phi _{n}(q,t){\Psi _{m}}^{*}(q){\text{d}}q$. This corresponds with Dirac's first and second editions' $\phi$ versus $\psi$ distinction, which in the third and fourth editions he expressed in the bra versus ket distinction.

In the third (1977) edition of L&L, kets and bras are described in §11 on pages 34-35:

"A widely used notation (introduced by Dirac) in recent literature is that which denotes the matrix elements f_nm by†

⟨n| f |m⟩.                                                                                    (11.17)

This symbol is written so that it may be regarded as "consisting" of the quantity f and the symbols |m⟩ and ⟨n| which respectively stand for the initial and final states as such (independently of the representation of the wave functions of the states). With the same symbols we can construct notations for the expansion coefficients of wave functions: if there is a complete set of wave functions corresponding to the states |n₁⟩, |n₂⟩, ... , the coefficients in the expansion in terms of these of the wave function of a state |m⟩, are denoted by

⟨n_i|m⟩ $=\int {\psi _{n_{i}}}^{*}\psi _{m}{\text{d}}q$.                                                                (11.18)

† Both notations are used in the present book. The form (11.17) is especially convenient when each suffix has to be written as several letters."

Auletta Fortunato Parisi

"In this context, we see that kets may be thought of as input states, whereas bras as output states of a certain physical evolution or process."

Though these authors do not say so at this point, one may recall Dirac's dictum that the theory of quantum mechanics is completely symmetrical between bras and kets. Consequently, one could also say that bras may be thought of as input states, with kets as output states of a certain physical evolution or process. This symmetry is the reason why causality is not exhibited in an obvious way in quantum mechanics. For causality is unsymmetrical precisely in this regard. The elision of causality is the result of quantum mechanics's formalism that cuts the quantum state from phase space to generalized quantum configuration space, and brings in the uncertainty principle.

• Auletta, G., Fortunato, M., Parisi, G. (2009). Quantum Mechanics, Cambridge University Press, Cambridge UK, ISBN 978-0-521-86963-8.

Dirac

Dirac does not explicitly say so. But his presentation is built on it.

Representations

Dirac

"In the two preceding chapters we dealt with certain abstract symbols, denoting states and observables, whose exact nature was not specified, but which were assumed to obey certain definite laws. In the present chapter we shall consider representations of these abstract symbols, i.e. sets of numbers having properties that correspond completely to those of the symbols they represent. When once one has found such a representation and understood the nature of the correspondence, one can obtain all the properties of the abstract symbols that one wants by dealing entirely with their representatives, to which, since they are just sets of numbers, ordinary mathematical methods apply. One cannot, of course, obtain in this way any relation between the abstract symbols that one could not obtain directly from the algebra of the abstract symbols without the help of a representation. One can, however, often obtain results much more easily and conveniently with the help of a representation than without it, and further the numbers occurring in a representation have often a very direct physical interpretation, so that representations are of great use in applications of the theory."

"The preceding chapter dealt with the fundamental laws governing states and observables in quantum mechanics and included all the axioms of the underlying mathematical formalism as well as the assumptions for the physical interpretation of the mathematics. The present chapter and the following one will be concerned, not with making new laws and assumptions, but with systematizing and developing ideas already introduced, and generally with arranging the theory in a form fitted for subsequent applications. One matter that we must deal with is the setting up of a suitable notation for coordinates—a notation which can be consistently followed all through the future very extensive use of coordinates and is at the same time as simple and as easily remembered as possible."

"A system of coordinates for ψ- and φ- vectors and linear operators will be called in future a representation. The coordinates of any ψ- or φ- vector or linear operator will be called the representative of that quantity and will be said to represent that quantity. They may also be called the representative of the corresponding state or observable."

"14. Basic vectors

"In the preceding chapters we set up an algebraic scheme involving certain abstract quantities of three kinds, namely bra vectors, ket vectors, and linear operators, and we expressed some of the fundamental laws of quantum mechanics in terms of them. It would be possible to continue to develop the theory in terms of the abstract quantities and to use them for applications to particular problems. However, for some purposes it is more convenient to replace the abstract quantities by sets of numbers with analogous mathematical properties and to work in terms of these sets of numbers. The procedure is similar to using coordinates in geometry, and has the advantage of giving one greater mathematical power for the solving of particular problems.

     "The way in which the abstract quantities are to be replaced by numbers is not unique, there being many possible ways corresponding to the many systems of coordinates one can have in geometry. Each of these ways is called a representation and the set of numbers that replace an abstract quantity is called a representative of that abstract quantity in the representation. Thus the representative of an abstract quantity corresponds to the coordinates of a geometrical object. When one has a particular problem to work out in quantum mechanics, one can minimize the labour by using a representation in which the representatives of the more important abstract quantities occurring in that problem are as simple as possible.

     "To set up a representation in a general way, we take a complete set of bra vectors, i.e. a set such that any bra can be expressed linearly in terms of them (as a sum or an integral or possibly an integral plus a sum). These bras we call the basic bras of the representation. They are sufficient, as we shall see, to fix the representation completely.

     "Take any ket  | a〉 and form its scalar product with each of the basic bras. The numbers so obtained constitute the representative of  | a〉. They are sufficient to determine the ket  | a〉 completely, since if there ins a second ket,  | a₁〉 say, for which these numbers are the same, the difference  | a〉 −  | a₁〉 will have its scalar product with any basic bra vanishing, and hence its scalar product with any bra whatsoever will vanish and  | a〉 −  | a₁〉 itself will vanish.

     "We may suppose the basic bras to be labelled by one or more parameters λ₁, λ₁, λ₂, ..., λ_u, each of which will take on certain numerical values. The basic bras will then be written 〈λ₁ λ₁ λ₂ ... λ_u |  and the representative of  | a〉 will be written 〈 λ₁ λ₁ λ₂ ... λ_u | a〉. This representative will now consist of a set of numbers, one for each set of values that λ₁, λ₁, λ₂, ..., λ_u may have in their respective domains. Such a set of numbers just forms a function of the variables λ₁, λ₁, λ₂, ..., λ_u. Thus the representative of a ket may be looked upon either as a set of numbers or as a function of the variables used to label the basic bras."

The construction of the representative is given a notational appearance as if it were an 'inner product' of a bra with a ket. Indeed, as noted below, Weinberg on his page 59 actually calls it an "inner product". This is an abbreviated notation for an object resulting from a "product", of a string or 'vector' of bras, with the ket. The string of bras might be construed as belonging to a species of linear operator that acts on kets to produce sets of numbers that indicate functions. Its value is not of the customary kind for an 'inner product', just a single number. The resultant object, considered as a function, is called a wave function.

The physical meaning of this is that a state can be expressed as a superposition of other states, by sending it through a quantum analyzer. The original state is pre-probabilistically split into a complex-number-weighted sum of states, which form a coherent superposition and can in principle be re-assembled unless they are interrupted by detection or other disruption before re-assembly.

There are many kinds of analyzer to choose from for such analyses. Amongst these, one is distinguished kind of analyzer, namely the one that that was used to prepare the original pure state. It has the property that the split state that emerges from it consists of just precisely one distinguished sub-beam. The probability that a system in the original beam will pass entirely into this distinguished sub-beam is 1. The is the criterion of purity of state. The distinguished analyzer is unique or nearly unique. The two distinguishing objects, analyzer and sub-beam, determine the statistical operator as idempotent, and pick out the relevant row and column of the density matrix.

• Dirac, P.A.M. (1930). The Principles of Quantum Mechanics, 1st edition, Oxford University Press, Oxford UK.
• Dirac, P.A.M. (1935). The Principles of Quantum Mechanics, 2nd edition, Oxford University Press, Oxford UK.
• Dirac, P.A.M. (1947). The Principles of Quantum Mechanics, 3rd edition, Oxford University Press, Oxford UK.
• Dirac, P.A.M. (1958). The Principles of Quantum Mechanics, 4th edition, Oxford University Press, Oxford UK.

London Bauer

    "This decomposition into components is quite analogous to the resolution of a vector in ordinary space into projections. We can consider the set of ψ_ρ as equivalent to the function ψ(q) itself; it is one particular decomposition of the vector ψ into orthogonal components. The coefficients

$\psi _{k}=(u_{k},\psi )$

give an analogous decomposition of the same vector with respect to another system of orthogonal axes u₁, u₂ ... u_k, ....

    "The representation of ψ by itself, that is the function ψ(q), may be regarded as a special case of representation in terms of orthogonal components—specifically, the orthogonal system composed of the eigenfunctions of the particular operator F = q. The eigenvalue problem for this operator has the form,

   $q\,v_{\alpha }(q)=q_{\alpha }v_{\alpha }(q),$

or

$(q-q_{\alpha })v_{\alpha }(q)=0.$

The solutions are the "limiting" or symbolic functions of Dirac

$v_{\alpha }(q)=\delta (q-q_{\alpha }).$

...

    "In terms of these special eigenfunctions one obtains for [the function] ψ(q) the trivial expansion

 $\psi (q)=\sum _{\alpha }\psi (q_{\alpha })\delta (q-q_{\alpha }),$

where the [numbers]

$\psi (q_{\alpha })=\int \psi (q)\delta (q-q_{\alpha })\mathrm {d} q$

are the coefficients in the expansion.

• London, F., Bauer, E. (1939). La Théorie de l'Observation dans la Mécanique Quantique, Hermann & Cie, Paris.
• Shimony, A., Wheeler, J.A., Zurek, W.H., McGrath, J., McGrath, S.M. (1983). Translation of London, F., Bauer, E. (1939) at pp. 217–259 in Wheeler, J.A., Zurek, W.H. (1983).
• Wheeler, J.A., Zurek, W.H., editors (1983). Quantum Theory and Measurement, Princeton University Press, Princeton NJ.

Auletta Fortunato Parisi

"It is a historical contingency depending on the development of quantum mechanics that, when the first wave function was introduced, it was written ψ(x). If it had been written ψ_x(x), one would have written ψ_p(p_x) for indicating the wave function of the momentum in the momentum representation. Since it did not happen we are obliged to choose forms like ${\tilde {\psi }}(p_{x})$ in order to indicate both the different dependence and the different representation."

"Using Dirac formalism ... "

${\tilde {\psi }}(p_{x})=\langle p_{x}|\psi \rangle .$

• Auletta, G., Fortunato, M., Parisi, G. (2009). Quantum Mechanics, Cambridge University Press, Cambridge UK, ISBN 9781107665897.

Weinberg

Weinberg writes on page xvi:

"The viewpoint of this book is that physical states are represented by vectors in Hilbert space, with the wave functions of Schrödinger just the scalar products of these states with the basis states of definite position."

Narrowly speaking, there is no scalar product of states, because they are physical, not mathematical, objects. Nevertheless, in quantum mechanics it is customary to speak of orthogonal states in the obvious sense. (See Dirac.) There are scalar products between vectors and dual vectors, which are mathematical objects. Weinberg's Section 3.7 on "Interpretation of Quantum Mechanics", on page 95, reads: "We can live with the idea that the state of a physical system is described by a vector in Hilbert space, rather than by numerical values of the positions and momenta of all the particles in the system, but it is hard to live with no description of physical states at all, only an algorithm for calculating probabilities."

Nevertheless, Weinberg distinguishes clearly between wave functions and state vectors. He writes on page 52:

"... wave mechanics has several limitations. It describes physical states by means of wave functions, which are functions of the positions of the particles of the system, but why should we single out position as the fundamental physical variable? For instance, we might want to describe the states in terms of probability amplitudes for particles to have certain values of the momentum or energy rather than position. ... The first postulate of quantum mechanics is that physical states can be represented as vectors in a sort of abstract space known as Hilbert space."

Weinberg's distinction between wave functions and his own state vectors is clear. He writes on page 59:

"... a wave function of Schrödinger's wave mechanics is nothing but the scalar product

ψ(x) = (Φ_x,Ψ)."

Weinberg writes on page 57:

"In Dirac's notation, a state vector Ψ is denoted  |Ψ〉. ... In the special cases where  Ψ is identified as a state with a definite value a for some observable  A, the ket in Dirac's notation is frequently written  |a〉."

Thus it appears that  |Ψ〉 denotes a state that is not identified with a definite observable that would make it a particular actual state. This indicates the abstract character of Weinberg's Hilbert space vectors, which comprise all possible relevant particular states of the species of system.

Cohen-Tannoudji et al. Volume 1, 2nd edition

<page 4> "This avoids the useless repetition which results from presenting the more general bra-ket formalism only after having developed wave mechanics uniquely in terms of wave functions."

<page 19> "The quantum state of a particle such as the [spinless] electron is characterised by a wave function ψ(r,t), which contains all the information it is possible to obtain about the particle."

<page 109>      "$\psi (\mathbf {r} )\leftrightarrow |\psi \rangle$

<page 147>     "$\langle \mathbf {r} |\psi \rangle =\psi (\mathbf {r} )$

$\langle \mathbf {p} |\psi \rangle ={\bar {\psi }}(\mathbf {p} )$."

Evidently, there is a set of correspondences of a kind between wave functions and kets as defined by Cohen-Tannoudji, but wave functions and kets are different things. Each such correspondence is a further one-to-one correspondence between points in generalized quantum configuration space. Many wave functions correspond to one ket as defined by Cohen-Tannoudji et al.. A ket of Cohen-Tannoudji describes a class of physical states proper for a species of particle, while a wave function defines a particular physical state for the species.

It is here evident that there are two radically different kinds of transformation in this. One kind is a transformation from one quantum configuration space to another (e.g. |ψ〉 → ${\bar {\psi }}$   and   ${\bar {\psi }}$ → |ψ〉, Fourier transforms of functions); the other is a coordinate transform within a fixed quantum configuration space, a geometrical transform, say $\mathbf {p}$ → $\mathbf {p} ^{\prime }$. The transform of quantum configuration spaces represents a change of experimental instrumentation in the laboratory. The transform of coordinates occurs in the physicist's notebook.

Sakurai & Napolitano

"The reader familiar with wave mechanics may have recognized by this time that 〈x′|α〉 is the wave function for the physical state represented by |α〉. We will say more about this identification of the expansion coefficient with the x-representation in Section 1.7."

That Section 1.7 is a detailed discussion, too long to quote in full here.

Messiah

"Just as the wave functions Φ and Ψ are equivalent representations of the same dynamical state, the operators B and A are equivalent representations of one and the same physical entity. The calculation of physically measurable quantities such as the mean values considered here may be carried out in a formally identical manner in either of these representations. This suggests that Quantum Theory might be formulated in a general way, independent of any representation. This general formulation will be given in Chapters VII and VIII."

It is evident in this passage that the writer is committing errors of anaphora, or some related errors. The word 'representation' is used in two different senses, the one referring to choice of generalized configuration space, the other to the semantic link between a mathematical operator and a physical quantity. The use of the word 'entity' is problematic here. It means 'numerically specified quantity', though formally it might be read as meaning 'enduring physical object', such as would be regarded as substantial by Aristotle. A clue to the problem is the writer's recourse to the phrase "formally identical manner". Why only "formally"? The problem is that the writer seems to have forgotten Dirac's dictum that "for any state there must exist one maximum observation which will for a certainty lead to one particular result, and conversely, if we consider any possible result of a maximum observation, there must exist a state of the system for which this result for the observation will be obtained with certainty." Messiah himself says something similar: "A precise measurement carried out on a complete set of compatible variables of a system represents the maximum information one can obtain on that system. It therefore defines the dynamical state completely, and a definite wave function corresponds to it. ... During the process of observation, the measured system can no longer be considered separately and the very notion of a dynamical state defined by the simpler wave function ψ(t) loses its meaning."

The wave function representations Φ and Ψ refer to the same physical system as defined by its component particles, but not the same dynamical state as defined in the quote from Dirac, or in the additional quote from Messiah himself. Position and momentum are incompatible in quantum mechanics, and cannot be "simultaneously" measured on a system in a definite state. The writer is in error at this point.

"Consider, for instance, the measurement of the position coordinate of a particle in a quantized system whose dynamical state is represented by the wave function Ψ (we shall henceforth say, more briefly, that the system is in the state Ψ). We assume that the measuring device at our disposal is infinitely precise. Thus the impossibility to predict with certainty the result of the observation is not due to any imperfection of the measuring device; the state Ψ does not in general correspond to a precise value of x; rather it is a superposition of dynamical states, each corresponding to a given value of x. Immediately after the operation of measurement, we can assert that the system is in a dynamical state where the x coordinate has the precise value x′ indicated by the measuring apparatus. Such a state can definitely not be represented by the wave function Ψ: the intervention of the measuring device has modified the dynamical state of the measured system."

In effect, 'measurement' with registration is done by analysis of the beam into superposed components, followed by discarding of all the superposed components except one, which is detected. The reassembly of the beam, that can show the nature of superposition, is rejected. The original beam's particle flux is cut down to the flux in the one retained and detected sub-beam. In a sense, this discarding is the collapse of the wave function. The original or source beam is in an initial condition, while the detected or destination beam is in a final condition. The two are in different states. The "collapse" of the wave function is thus a change of logical definition of the "beam".

If the 'measuring' analyser is of the same dynamical variable as was used to prepare the source beam, then the result of the 'measurement' is certain: it will be the same eigenstate as that of the original beam. If it is of another compatible dynamical variable, it will again be the eigenstate of that dynamical variable that was used to prepare the source beam, supposing that beam to be pure. If it is of an incompatible dynamical variable, the original beam will be really split probabilistically into several sub-beams, each of which, if sent to a detector, will be in a "collapsed" state; otherwise, if the sub-beams are reassembled by passage through a reversed copy of the 'measuring' analyser, superposition will be exhibited. Consequently, the actual physical state and the representation are indissolubly linked, and the physical state cannot retain its specificity by mere association with an abstract object that is not marked with a particular representation. In other words, the representationless abstraction belongs not to a unique physical pure state; it belongs to a plurality of physical pure states with their respectively corresponding plurality of wave functions.

• Dirac, P.A.M. (1930). The Principles of Quantum Mechanics, 1st edition, Oxford University Press, Oxford UK.
• Messiah, A. (1961). Quantum Mechanics, volume 1, translated by G.M. Temmer from the French Mécanique Quantique, North-Holland, Amsterdam.

general account

A basis is a list x′ = (x₁, x₂, ..., x_n) of vectors that just span a vector space. For Dirac's purpose, one can consider such a list as a list of vectors each belonging to a dual space 〈x′| of the space |Ψ〉of kets for a system. One could even name a space 〈X| of basis lists. The space〈X| is defined to be dual to the space of kets, and consists of the bras that are dual to the kets.

A wave function is a list (of scalar products such as x_i •|ψ〉), written rather cheekily〈x′|ψ〉.

The abstractive step

"The same considerations become clearer if one imagines the wave functions to represent the same vector of a space with infinitely many dimensions. ... One is thus led to build up the entire Quantum Theory by starting directly from the vector concept, without reference to the particular representation which can be made thereof."

Here Messiah is setting up an abstraction by which the distinct wave functions Φ and Ψ are stripped of their links to their respective distinct configuration spaces, and identified with a vector in an abstract vector space.

• Messiah, A. (1961). Quantum Mechanics, volume 1, translated by G.M. Temmer from the French Mécanique Quantique, North-Holland, Amsterdam.

abstract states or kets

Matrix mechanics

According to Beller, Heisenberg's matrix mechanics does not exhibit the stationary states of an atom, while wave mechanics does it clearly.

"The matrix theoreticians had realised that the theory's inability to incorporate the concept of the state of the system was a serious drawback. It was clear that an atomic system can exist in certain states for definite amounts of time; therefore a theoretical description of such states was needed. The ability of Schrödinger's theory to give a straightforward definition of stationary states through a wave function was one of its advantages, as Lorentz pointed out in a discussion of the comparative merits of the two theories. That superiority originally disposed Born to regard it as physically more significant than the matrix approach."

"But even then one could not say what this discrete stationary state was, and therefore now I am coming to the second part of my talk — the concept of a 'state'. In 1925, one did have a method for calculating the discrete energy values of the atom. One also had at least in principle a method for calculating the transition probabilities. But what was this state of the atom? How could it be described? It could not be described by referring to an electronic orbit. So far it could be described only by stating an energy and transition probabilities; but there was no picture of the atom. Furthermore it was clear that sometimes there are non-stationary states. The simplest example of a non-stationary state was an electron moving through a cloud chamber. So the question really was how to handle such a state which can occur in nature. Can such a phenomenon as the path of an electron through a cloud chamber be described in the abstract language of matrix mechanics?

Fortunately at that time wave mechanics had been developed by Schrödinger. And in wave mechanics, things looked very different.There one could define a wave function for the discrete stationary state. ... But even when we knew this and accepted the quantum jumps, we did not know what the word state could mean. ...

Then there came the transformation theory by Dirac and Jordan. In this scheme one could transform for instance from ψ(q) to ψ(p), and it was natural to assume that the square |ψ(p)|² would be the probability to find the electron with momentum p. ...

"... So finally one knew how to represent such a phenomenon as the path of the electron, but again at a very high price. Namely, this interpretation meant that the wave packet representing the electron is changed at every point of observation, that is at every water droplet in the cloud chamber. At every point we get new information about the state of the electron; therefore we have to replace the original wave packet by a new one, representing the new information.

"The state of the electron thus represented does not allow to ascribe to the electron in its orbit definite properties like coordinates, momentum and so on. What we can do is only to speak about the probability to find, under suitable experimental conditions, the electron at a certain point, or to find a certain value for its velocity. So finally we have come to the definition of a state which is much more abstract than the original electronic orbit. Mathematically we describe it by a vector in Hilbert space, and this vector determines probabilities for the result of any kind of experiments which can be carried out on this state. The state may change by every new information.

"This definition of state made a very big change, or as Dirac has said, a big jump in the description of natural phenomena ..."

• Beller, M. (1983). Matrix theory before Schrödinger: philosophy, problems, consequences, Isis 74: 469–491.
• Beller, M. (1992). The genesis of Bohr's complementarity principle and the Bohr–Heisenberg dialogue, pp. 273–293 in Ullman-Margalit, E. (1992).
• Heisenberg, W. (1973). Development of concepts in quantum theory, pp. 264–275 of Mehra (1973).
• Mehra, J. editor (1973). The Physicist's Conception of Nature, Reidel, Dordrecht, ISBN 90-277-0345-0.
• Ullman-Margalit, E., editor (1992). The Scientific Enterprise, Kluwer, Dordrecht, ISBN 978-94-010-5190-3.

Heisenberg

" ... the Copenhagen interpretation regards things and processes which are describable in terms of classical concepts, i.e. the actual, as the foundation of any physical interpretation."

"Since Schrödinger had recognized that wave functions were the elements of the transformation matrices for the transition from energy states to position states, Born's hypothesis formed a particular case of this more general assumption, which fitted naturally into the scheme of quantum mechanics.

"Even then, however, a complete interpretation of the quantum theory had not been achieved, for the question remained, how to define the word "state" within the theory. A hydrogen atom in its normal state could be represented in the mathematical scheme of the theory. There were, however, entirely different states. For example, the track of an electron was seen in the cloud chamber. How should one represent in the theory an electron which is moving at a definite point with a definite velocity?

"... I, for my part, attempted to extend the physical significance of the transformation matrices in such a way that a complete interpretation was obtained which would take account of all possible experiments.

"The clarification ... took place in the early part of 1927. ... it was now assumed in quantum mechanics that real states can always be represented as vectors in Hilbert space (or as "mixtures" of such vectors). The uncertainty principle [W. Heisenberg; Z. Phys. 43, 172, 1927.] was the simple expression for this assumption."

At this point in this 1955 essay, Heisenberg retreats from a physical account to a merely mathematical one, and does not specify to which Hilbert space he is referring. Heisenberg here refers as if definitive to the 1927 Born and Heisenberg report to the Fifth Solvay Conference.

"From the point of view of Bohr's theory a system can always be in only one quantum state. To each of these belongs an eigensolution φ_n⁰ of the unperturbed system. ... According to Bohr's principles, it makes no sense to say that a system is simultaneously in several states. The only possible solution seems to be statistical: the superposition of several eigensolutions expresses that through the perturbation, the system can go over into any other quantum state, and it is clear that as measure for the transition probability one has to take the quantity

Ψ_nk = |S_nk(t)|² ;"

This autumn 1927 statement by Born and Heisenberg brings the discussion back to physics. A physical state here is a pure state, represented by an eigenfunction of the operator that describes the effect of the physical filter that supplied the unperturbed pure state. Heisenberg is working according to the principle that a pure state has a parent operator, an observable indeed, that supplies it, and of which it is an eigenstate.

"Already Einstein [A. Einstein, Quantentheorie des einatomigen idealen gases, II, Berl. Ber., (1925), [3. Both the typescript and the French edition read p. 5.].], when he deduced from de Broglie's daring theory the possibility of 'diffraction' of material particles,^a tacitly assumed that it is the particle number that is determined by the intensity of the waves."

• Bacciagaluppi, G., Valentini, A. (2009), Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference, Cambridge University Press, Cambridge UK, ISBN 978-0-521-81421-8.
• Born, M., Heisenberg, W. (1928). Quantum mechanics, pp. 143–181 of Électrons et Photons: Rapports et Discussions du Cinquième Conseil de Physique, tenu à Bruxelles du 24 au 29 Octobre 1927, sous les Auspices de l'Institut International de Physique Solvay, Gauthier-Villars, Paris.
• Heisenberg, W. (1955). Niels Bohr and the development of physics. Essays dedicated to Niels Bohr on the occasion of his seventieth birthday. Edited by W. Pauli with the assistance of L. Rosenfeld and V. Weisskopf, Pergamon Press, London, pp. 12–29.

Dirac

        "The general interpretation of quantum mechanics was very much helped by another development, which is due mainly to Schrödinger, working from the ideas of de Broglie. This involved bringing into physics a new concept, the concept of a state in atomic theory.

        "We already had the idea of states in classical mechanics. When we think about a given classical system, there will be various possible states of motion coming from the various solutions of the equations in motion. But the peculiar thing is that a quantum state does not just correspond to a classical state. It corresponds to a whole set of classical states, what one may call a family of classical states, which are related to one another in a special mathematical way, which was discovered by Hamilton a hundred years before. Hamilton discovered this special relationship between classical states just by considerations of mathematical beauty, trying to get a powerful formulation of the equations. And this work of Hamilton's turns out to be just what is needed as a preparation for our understanding of quantum states. Each quantum state corresponds to one of Hamilton's families of states.

        "Then the surprising thing turns up that the quantum states have superposition relations between them. That means they are to be pictured as a kind of vector. They are something which can be added to quantities of the same nature to produce sums of the same nature again."

Dirac writes: "We now assume that each state of a dynamical system at a particular time corresponds to a ket vector, the correspondence being such that if a state results from the superposition of certain other states, its corresponding ket vector is expressible linearly in terms of the corresponding ket vectors of the other states, and conversely."

• Dirac, P.A.M. (1958). The Principles of Quantum Mechanics, 4th edition, Oxford University Press, Oxford UK.
• Dirac, P.A.M. (1973). Development of the physicist's conception of nature, pp. 1–55 in The Physicist's Conception of Nature, edited by J. Mehra, D. Reidel, Dordrecht, ISBN 90-277-0345-0.

Dirac's kets and Heisenberg's "states" are not states in the ordinary sense of the word. No, they are vectors in an abstract state space, which is a set, each element or ket of which is not itself a state in the ordinary sense of the word, but is instead an encoding of many states in the ordinary sense of the word. The several ordinary-sense states that comprise a ket differ from one another by referring respectively to their own generalized quantum configuration spaces, or, in another view, to their own complete sets of commuting variables. The abstract "state" or ket belongs to the species of particle, not just to a particular state of it in the ordinary sense, in which it may be actually prepared or detected. For example, two generalized quantum configuration states might be the position configuration and the momentum configuration state of a species of particle. These two are mathematically related by a Fourier transform. Such a relation is analogous to a canonical transformation in Hamiltonian classical mechanics, which may be viewed as a change of coordinates in phase space. This is the reason why the Dirac version of the theory is called the transformation theory.

It is odd that Dirac finds superposition "surprising", and that Feyman finds it "mysterious". It is no more than the additivity of statistical tables of particle counts add, exhibited in nature only when a natural object has been analyzed into superposable components by a quantum analyzer, and then reassembled by another such.

The key here, pointed out by Max Born, is that the 'wave' aspect of particles is a virtual wave, in a statistical counting sense, not in an ordinary continuously distributed material sense.

Messiah

"The entire interpretation of wave mechanics developed in Chapters IV and V had as its starting point the definition of the probability densities of position and momentum by means of the wave functions Ψ and Φ referring to configuration and momentum space respectively."

"... just as the functions Φ and Ψ are equivalent representations of one and the same dynamical state ..."

"... we associate with every dynamical state a certain type of vector, which we call, following Dirac, ket vector or ket and which we represent by the symbol  |〉."

The immediately foregoing sentence suffers from a grammatical error, which makes it perhaps ambiguous. Probably the writer intended "we associate with every dynamical state a vector of a certain type".

• Messiah, A. (1961). Quantum Mechanics, volume 1, translated by G.M. Temmer from the French Mécanique Quantique, North-Holland, Amsterdam.

Dual spaces

Dirac 1930

"The sum of a φ and a ψ has no meaning and will never appear in the analysis."

"We must therefore remember, when using the vector picture, that, in so far as it would allow one to add together two vectors representing a ψ and a φ respectively, it is imperfect and gives the ψ's and φ's more properties than quantum mechanics requires or allows."

"There are, however, two cases when we are obliged in general to consider the disturbance as causing a change in the state of the system, namely when the disturbance is an observation and when it consists in preparing the system so as to be in a given state."

Dirac 1935

"There will be no meaning for the sum of one of the vectors in the new vector space with one of the ψ's in the original vector space. ... We shall call the vectors in the new vector space φ's."

Dirac 1947, 1958

"Whenever we have a set of vectors in any mathematical theory, we can always set up a second set of vectors, which mathematicians call the dual vectors. The procedure will be described for the case when the original vectors are our ket vectors."

"The new vectors are, of course, defined only to the extent that their scalar products with the original ket vectors are given numbers, but this is sufficient for one to be able to build up a mathematical theory about them.

    "We shall call the new vectors bra vectors, or simply bras, and denote a general one of them by the symbol $\langle |$ , the mirror image of the symbol for a ket vector."

"... a bra and a ket vector are of different natures and cannot be added together."

• Dirac, P.A.M. (1930). The Principles of Quantum Mechanics, 1st edition, Oxford University Press, Oxford UK.
• Dirac, P.A.M. (1935). The Principles of Quantum Mechanics, 2nd edition, Oxford University Press, Oxford UK.
• Dirac, P.A.M. (1947). The Principles of Quantum Mechanics, 3rd edition, Oxford University Press, Oxford UK.
• Dirac, P.A.M. (1958). The Principles of Quantum Mechanics, 4th edition, Oxford University Press, Oxford UK.

Messiah

"In order to introduce a metric in the vector space we have just defined, we make the hypothesis that there exists a one-to-one correspondence between the vectors of this space and those of the dual space. ... Thus the correspondence between kets and bras is analogous to the correspondence between the wave functions of wave mechanics and their complex conjugates."

• Messiah, A. (1961). Quantum Mechanics, volume 1, translated by G.M. Temmer from the French Mécanique Quantique, North-Holland, Amsterdam.

Gottfried

"Although we shall frequently say that  $|\alpha \rangle$ and  $|\beta \rangle$ are orthogonal if  $\langle \beta |\alpha \rangle \,=\,0$ , it should be remembered that the scalar product is only defined between a vector and a dual vector."

"It is necessary, first, to associate a dual vector to every ket in a one-to-one manner, called a bra, denoted by the symbol  $\langle \,|$ ; and second, to define the scalar products as being between bras and kets."

Weinberg

Weinberg, in his Lectures on Quantum Mechanics, does not use the term dual space, and says he has no occasion in that book to use the dagger notation for the related operator. He writes parenthetically:

"(For any given state vector $\Omega$ we can if we like introduce an operator $\Omega ^{\dagger }$ , which, operating on any state vector $\Phi$ yields the number $(\Omega ,\Phi )$ , but in this book we will not have occasion to employ the symbol $\Omega ^{\dagger }$ except as an ingredient for for dyads like ${\begin{bmatrix}\Psi \Omega ^{\dagger }\end{bmatrix}}$ .)"

At first glance, this doesn't look as if Weinberg thinks that it is wrong to talk in terms of dual vectors and dual spaces. It just looks as if he doesn't think it necessary for all purposes.

• Weinberg, S. (2013). Lectures on Quantum Mechanics, Cambridge University Press, Cambridge UK, ISBN 978-1-107-02872-2.

ψ- and φ- symmetry

"The theory will throughout be symmetrical between φ's and ψ's."

"There is, in fact, perfect symmetry between the φ's and ψ's, which symmetry will survive all through the theory."

"In fact the whole theory will be symmetrical in its essentials between bras and kets."

• Dirac, P.A.M. (1930). The Principles of Quantum Mechanics, 1st edition, Oxford University Press, Oxford UK.
• Dirac, P.A.M. (1935). The Principles of Quantum Mechanics, 2nd edition, Oxford University Press, Oxford UK.
• Dirac, P.A.M. (1947). The Principles of Quantum Mechanics, 3rd edition, Oxford University Press, Oxford UK.
• Dirac, P.A.M. (1958). The Principles of Quantum Mechanics, 4th edition, Oxford University Press, Oxford UK.

complete set of commuting observables

"Let us define a complete set of commuting observables to be a set of observables which all commute with one another and for which there is only one simultaneous eigenstate belonging to any set of eigenvalues."

• Dirac, P.A.M. (1958). The Principles of Quantum Mechanics, 4th edition, Oxford University Press, Oxford UK.

"For every quantum-theoretical quantity there exists a coordinate system in which the statistical error for this quantity is zero. Therefore a definite experiment can never give exact information on all quantum-theoretical quantities. Rather, it divides physical quantities into "known" and "unknown" (or more and less accurately known quantities) in a way characteristic of the experiment in question."

For a system in a pure state, there is a unique maximal set of compatible dynamical variables, with the state being their common eigenstate. This is the set of dynamical variables by which the state is prepared or observed, depending on whether the state is a ket state or a bra state.

Constructing a description

How to construct a description of atomic phenomena? Physically, a phenomenon is an observable process, marked both by preparation and by detection devices.

Bohr's metaphysics is of actuality as evolution through discrete stages of state occupation demarcated by quantal transitions. In general, actual particular quantal transitions are not observed. Rather, a phenomenon reveals an overall outcome of indefinitely many virtual or unobserved quantal transitions. An observable quantity thus needs a matrix representation, to account for multiple potential contributory transition pathways. The actually observed overall transition in a phenomenon is the probabilistic sum of the potential contributory transitions. The quantity itself is not a matrix, but its changes are accounted for by a matrix in the obvious way.

The quantum mechanical formalism represents the potential stages of state occupation by removing their internal causal progression with time. Instead, they have probabilistic and non-causal internal cycling. The formalism does this by making the preparation and detection endpoints interchangeable, the bra-ket symmetry. Such a phenomenon must have a time-independent Hamiltonian. It must be observed through devices (for example prisms and Stern-Gerlach magnets) that obey a kind of Helmholtz reciprocity principle, and in a given experiment are commutative. This is expressed through Hermitian operators, in which time is in a sense reversible. In general, non-commutative devices in an experiment do not admit interchangeability of preparation and detection. With a time-independent Hamiltonian, causality has no representation or meaning in the quantum mechanical formalism.

To represent causality, the quantum mechanical formalism requires the use of a time-dependent Hamiltonian.

The formalism should allow counting of detections of particles, with a probability interpretation.

It should represent conservation in some way. In the first quantization, material particles are conserved, being neither created nor annihilated, but photons are not considered as such.

causality

    "A consequence of the preceding discussion is that we must revise our ideas of causality. Causality applies only to a system which is left undisturbed. If a system is small, we cannot observe it without producing a serious disturbance and hence we cannot expect to find any causal connection between the results of our observations. There is thus an essential indeterminacy in the quantum theory, of a kind that has no analogue in the classical theory, where causality reigns supreme. The quantum theory does not enable us, in general, to calculate the result of an observation, but only the probability of our obtaining a particular result when we make the observation.

    "The lack of determinacy in the quantum theory should not be considered as a thing to be regretted. It is necessary for a rational theory of the ultimate structure of matter. One of the most satisfactory features of the present quantum theory is that the differential equations that express the causality of classical mechanics do not get lost, but are all retained in symbolic form, and indeterminacy appears only in the application of these equations to the results of observations."

• Dirac, P.A.M. (1930). The Principles of Quantum Mechanics, 1st edition, Oxford University Press, Oxford UK.
• Dirac, P.A.M. (1935). The Principles of Quantum Mechanics, 2nd edition, Oxford University Press, Oxford UK.
• Dirac, P.A.M. (1947). The Principles of Quantum Mechanics, 3rd edition, Oxford University Press, Oxford UK.
• Dirac, P.A.M. (1958). The Principles of Quantum Mechanics, 4th edition, Oxford University Press, Oxford UK.

energy

"..., in our present formalism energy does not play an exceptional part except for the time-evolution of the state as represented by its ψ function."

• London, F., Bauer, E. (1939). La Théorie de l'Observation dans la Mécanique Quantique, Hermann & Cie, Paris.
• Shimony, A., Wheeler, J.A., Zurek, W.H., McGrath, J., McGrath, S.M. (1983). Translation of London, F., Bauer, E. (1939) at pp. 217–259 in Wheeler, J.A., Zurek, W.H. (1983).
• Wheeler, J.A., Zurek, W.H., editors (1983). Quantum Theory and Measurement, Princeton University Press, Princeton NJ.

time of observation

"... an observation is not specified unless the time when it is made is given. In special cases it may so happen that the result of the observation, or the probability of any particular result being obtained, is independent of this time. If the state of the system is such that this is so for every observation that could be made on the system, then the state is such that the state is said to be a stationary state and we should picture it as one in which the conditions are not varying."

• Dirac, P.A.M. (1930). The Principles of Quantum Mechanics, 1st edition, Oxford University Press, Oxford UK.

Inverse problem

The forward problem for a quantum state may be regarded as the prediction of physically measured results given the mathematically defined state. The inverse problem would then be the identification of the mathematically defined state given an array of physical measurements.

For spin states, non-commutativity arises from classical considerations. Classically the three spin components do not commute. Commutative variables measured simultaneously can be chosen to solve the inverse problem for spin states.

"The density matrix of a two-level system (spin, atom) is usually determined by measuring the three noncommuting components of the Pauli vector. This density matrix can also be obtained via the measurement data of two commuting variables, using a single apparatus. This is done by coupling the two-level system to a mode of radiation field, where the atom-field interaction is described with the Jaynes-Cummings model. The mode starts its evolution from a known coherent state. The unknown initial state of the atom is found by measuring two commuting observables: the population difference of the atom and the photon number of the field. We discuss the advantages of this setup and its possible applications."

Quantum system

A quantum system is a physical object as it is described by quantum mechanics, with a conceptual physical existence that really endures for some time.

Its existence is conceptual in the sense that it cannot be directly and actually sensed or directly and specifically manipulated by a human observer. Manipulation or observation of it requires the mediation of suitable specifically designed and constructed apparatus, consisting of a preparative and of a registrative device. It is supposed that the preparative device generates or supplies many independent copies or instances of the quantum system, and provides them to the registrative device. The quantum system is conceived of a a physical object that transmits causal efficacy from the preparative to the registrative apparatus. The passage of a single copy or instance of the quantum system is regarded as a quantum physical phenomenon. Quantum mechanics does not immediately and directly describe individual phenomena, though it refers to them through probabilities, and regards them as really existing physical objects. Statements about actual physical fact by quantum mechanics are always and only probabilistic.

In general, the boundary between the preparative and registrative device can be chosen in a diversity of ways. Some intermediate device parts are so placed that they can be assigned as parts of either the preparative or the registrative device.

The ontological distinction trumpeted by Bohr, however, is not that between preparative and registrative device, for they are both classical objects. It is between the system and the conjunction of the devices. The devices are considered fully described by ordinary language supplemented with terms of classical physics. The system is a strictly quantal entity, uniquely and characteristically defined by quantum mechanics. This is an ontological taxonomic distinction.

Quantum superposition

Dirac

Classical superposition refers only to waves. Quantum superposition refers indifferently to waves or particles with wave- or particle-like behaviour. In classical thinking, the terms 'waves', 'particles', 'with wave-' or 'particle-like behaviour', each seem distinct in meaning. All are needed to account, in classical terms, for quantum phenomena. In quantum thinking, they are all conceptually identical, so that, in a sense, in quantum thinking, there is no distinction between particle and wave. In a sense, then, the locution "complementarity between wave and particle pictures" belongs to classical thinking, not to quantal thinking sensu strictu.

The pure wave picture of superposition is in a sense shared by classical and quantum thinking. But it belongs only to quantum superposition that it can be pictured particle-by-particle; this makes the quantum view utterly incompatible with the classical view.

The last previous sentence is often interpreted in classical terms by saying that the particle "goes through both slits", or some such. In quantum thinking, such a locution has no meaning. In quantum thinking, the quantal entity is manifest in a phenomenon, a process in which it passes from source to destination through the apparatus. The initial and final states are well-defined, but the intermediate "trajectory" is undefined. In Dirac's terms, the initial state is expressed in a ket, |〉, the final state in a bra,〈|.

If source and destination are physically congruently contiguous with nothing between them, and are mutually dual, then together they exhibit a stationary state of the quantal entity. This is written mathematically in Dirac notation, omitting an intermediate symbol |, as

$\langle |\rangle =1$.

This is expressed by Paul Dirac thus: "Each photon then interferes only with itself. Interference between different photons never occurs."

"Any state may be considered as the result of a superposition of two or more other states, and indeed in an infinite number of ways. Conversely any two or more states may be superposed to give a new state. The procedure of expressing a state as the result of superposition of a number of other states is a mathematical procedure that is always permissible, independent of any reference to physical conditions, like the procedure of resolving a wave into Fourier components. Whether it is useful in any particular case, though, depends on the special physical conditions of the problem under consideration."

"This, of course, is true only provided the two states that are superposed refer to the same beam of light, i.e. all that is known about the position and momentum of a photon in either of these states must be the same for each."

Dirac is mostly referring here to a situation specified in the above quote from the 1st edition page 8 about "the same beam of light". Evidently, by "useful", he mainly intends to mean 'physically meaningful'.

"We must first generalize the meaning of a 'state' so that it can apply to any atomic system. Corresponding to the case of the photon, which we say is in a given state of polarization when it has been passed through suitable polarizing apparatus, we say that any atomic system is in a given state when it has been prepared in a given way, which may be repeated arbitrarily at will. The method of preparation may then be taken as the specification of the state. The state of a system in the general case includes any information that may be known about its position in space from the way in which it was prepared, as well as any information about its internal condition."

"We are thus led to a generalization of the term 'state of motion' applied to a photon. For a photon to be in a definite state of motion it need not be associated with one single beam of light, but may be associated with two or more beams of light which are the components into which the original one beam has been split."

"We are thus led to a generalization of the term 'translational state' applied to a photon. For a photon to be in a definite translational state it need not be associated with one single beam of light, but may be associated with two or more beams of light which are the components into which one original beam has been split."

• Dirac, P.A.M. (1930). The Principles of Quantum Mechanics, 1st edition, Oxford University Press, Oxford UK.
• Dirac, P.A.M. (1935). The Principles of Quantum Mechanics, 2nd edition, Oxford University Press, Oxford UK.
• Dirac, P.A.M. (1947). The Principles of Quantum Mechanics, 3rd edition, Oxford University Press, Oxford UK.
• Dirac, P.A.M. (1958). The Principles of Quantum Mechanics, 4th edition, Oxford University Press, Oxford UK.

Landau & Lifshitz

    "Suppose that, in a state with a wave function $\Psi _{1}(q)$ , some measurement leads with certainty to a definite result (result 1), while in a state with $\Psi _{2}(q)$ , it leads to result 2. Then it is assumed that every linear combination of $\Psi _{1}$ and $\Psi _{2}$ , i.e. every function of the form $c_{2}\Psi _{1}+c_{2}\Psi _{2}$ (where $c_{1}$ and $c_{1}$ are constants), gives a state in which that measurement leads to either result 1 or result 2. Moreover, we can assert that, if we know the time dependence of the states, which for the one case is given by $\Psi _{1}(q,t)$ , and for the other by $\Psi _{2}(q,t)$ , then any linear combination also gives a possible dependence of a state on time. These propositions can be immediately generalized to any number of different states.

    "The above assertions regarding wave functions constitute what is called the principle of superposition of states, the chief positive principle of quantum mechanics."

• Landau, L.D., Lifshitz, E.M. (1958/1965). Quantum Mechanics: Non-Relativistic Theory, second edition, (first published in English in 1958), translated from the Russian by J.B. Sykes, J.S. Bell, Pergamon Press, Oxford UK.

Merzbacher

"If $\psi _{1}(x,y,z,t)$ and $\psi _{2}(x,y,z,t)$ describe two waves, their sum $\psi (x,y,z,t)=\psi _{1}+\psi _{2}$ also describes a possible physical situation. This assumption is known as the principle of superposition."

"Mathematically, the superposition appears as a linear combination of the appropriate eigenfunctions."

• Merzbacher, E. (1961/1970). Quantum Mechanics, second edition, Wiley, New York.

Isham

"A crucial idea in quantum theory is that any pair of wave functions $\psi _{1}$ and $\psi _{2}$ can be superimposed with arbitrary complex coefficients $\alpha _{1}$ and $\alpha _{2}$ to give new wave function $\alpha _{1}\psi _{1}(x)+\alpha _{2}\psi _{2}(x)$ (provided that $\alpha _{1}$ and $\alpha _{2}$ are chosen such that this new function is normalized to one)."

"... it is usually not meaningful to spesk of an observable 'having' a particuoar value. Rather, a typical state will be a linear superposition of eigenstates of the associated operator."

"The typical quantum-mechanical situation in which a non-trivial superposition of eigenvectors leads to a proposition being neither true nor false would then be assigned to the indeterminate category."

• Isham, C.J. (1995). Lectures on Quantum Theory: Mathematical and Structural Foundations, Imperial College Press, London, ISBN 1860940005.

De Muynck

"... the superposition principle, stating that a normalized linear combination $|\psi \rangle =\alpha _{1}|\psi _{1}\rangle +\alpha _{2}|\psi _{2}\rangle$ of two (or more) state vectors is also a possible state vector. In the superposition it is not necessary that $|\psi _{1}\rangle$ and $|\psi _{2}\rangle$ be orthogonal."

• De Muynck, W. (2002). Foundations of Quantum Mechanics: an Empiricist Approach, Kluwer Academic Publishers, Dordrecht, ISBN 1-2040-0932-1.

Gustafson & Segal

"1.2  Wave functions

"In quantum mechanics, the state of a particle is described by a complex-valued function of position and time, $\psi (x,t)$, $x\in \mathbb {R} ^{3}$, $t\in \mathbb {R}$. This is called a wave function (or state vector).

...

"1.4  The Schrödinger equation

"... the evolution of a particle's wave function. ... An evolving state at time $t$ is denoted by $\psi (x,t)$ with the notation $\psi (t)(x)\equiv \psi (x,t)$.

"Superposition principle: If $\psi (t)$ and $\phi (t)$ are evolutions of states, then $\alpha \psi (t)+\beta \phi (t)$   ($\alpha$, $\beta$ constants) should also describe the evolution of a state. "

• Gustafson, S.J., Segal, I.M. (2011). Mathematical Concepts of Quantum Mechanics, second edition, Springer, Heidelberg, ISBN 978-3-642-21865-1.

Evidently, these authors are not fussed about a putative difference between wave functions and state vectors! One may observe that the configuration space domains of $\psi (t)$ and $\phi (t)$ are not made explicitly the same, but they are not noted to be different, as for example they might be indicated by $x$ and $p$ if they were intended to be different, and one might reasonably suppose that they are implicitly and tacitly intended to be the same, indicated by $x$.

Comparison of classical with quantum superposition

Dirac

"The nature of the relationships which the superposition principle requires to exist between the states of any system is of a kind that cannot be explained in terms of familiar physical concepts. One cannot in the classical sense picture a system being partly in each of two states and see the equivalence of this to the system being completely in some other state. There is an entirely new idea involved, to which one must get accustomed and in terms of which one must proceed to build up an exact mathematical theory, without having any detailed classical picture."

"It is important to remember, however, that the superposition that occurs in quantum mechanics is of an essentially different nature from any occurring in the classical theory, as is shown by the fact that the quantum superposition principle demands indeterminacy in the results of observations in order to be capable of a sensible physical interpretation. The analogies are thus liable to be misleading."

"The assumption just made shows up very clearly the fundamental difference between the superposition of the quantum theory and any kind of classical superposition. In the case of a classical system for which a superposition principle holds, for instance a vibrating membrane, when one superposes a state with itself the result is a different state, with a different magnitude of the oscillations. There is no physical characteristic of a quantum state corresponding to the magnitude of the classical oscillations, as distinct from their quality, described by the ratios of the amplitudes at different points of the membrane. Again, while there exists a classical state with zero amplitude of oscillation everywhere, namely the state of rest, there does not exist any corresponding state for a quantum system, the zero ket vector corresponding to no state at all."

• Dirac, P.A.M. (1958). The Principles of Quantum Mechanics, 4th edition, Oxford University Press, Oxford UK.

Beltrametti & Cassinelli

"The appearance of these superpositions of pure states is a distinguishing feature of quantum mechanics: it has no analogue in classical mechanics. True, a notion of superposition is also present in classical mechanics (e.g., in classical wave phenomena), but what is superposed in classical examples is quite different from what is superposed in quantum mechanics."

• Beltrametti, E.G., Cassinelli, G. (1981). The Logic of Quantum Mechanics, volume 15 of Encyclopedia of Mathematics and its Applications, edited by G.-C. Rota, Addison—Wesley, Reading MA, ISBN 0-201-13514-0.

Relativistic invariance of state and observation

"It is convenient at this stage to modify slightly the meaning of the word 'state' and to make it more precise. We must regard the state of a system as referring to its condition throughout an indefinite period of time and not to its condition at a particular time, which would make the state a function of the time. Thus a state refers to a region of 4-dimensional space-time and not to a region of 3-dimensional space. A system, when once prepared in a given state, remains in that state so long as it remains undisturbed. This does not, of course, imply that it is not undergoing changes which could be revealed by experiment. In general it will be following out a definite course of changes, predictable by the quantum theory, belonging to that state."

• Dirac, P.A.M. (1930). The Principles of Quantum Mechanics, 1st edition, Oxford University Press, Oxford UK.

waves and particles

"The waves and particles should be regarded as two abstractions which are useful for describing the same reality."

• Dirac, P.A.M. (1930). The Principles of Quantum Mechanics, 1st edition, Oxford University Press, Oxford UK.

Physics, mathematics, and metaphysics

London & Bauer on questions of principle

"Physicists are to some extent sleepwalkers, who try to avoid such issues and are accustomed to concentrate on concrete problems. But it is exactly these questions of principle which nevertheless interest nonphysicists and all who wish to understand what modern physics says about the analysis of the act of measurement itself."

• London, F., Bauer, E. (1939). La Théorie de l'Observation dans la Mécanique Quantique, Hermann & Cie, Paris.
• Shimony, A., Wheeler, J.A., Zurek, W.H., McGrath, J., McGrath, S.M. (1983). Translation of London, F., Bauer, E. (1939) at pp. 217–259 in Wheeler, J.A., Zurek, W.H. (1983).
• Wheeler, J.A., Zurek, W.H., editors (1983). Quantum Theory and Measurement, Princeton University Press, Princeton NJ.

Dirac on physics and mathematics

"Mathematics is the tool specially suited for dealing with abstract concepts of any kind and there is no limit to its power in this field. For this reason a book on the new physics, if not purely descriptive of experimental work, must be essentially mathematical. All the same the mathematics is only a tool and one should learn to hold the physical ideas in one's mind without reference to the mathematical form. In this book I have tried to keep the physics to the forefront, by beginning with an entirely physical chapter and in the later work examining the physical meaning underlying the formalism wherever possible."

Dirac on interpretation

"The axioms and assumptions that we have made about observables are so far purely mathematical and have no physical implications. The physical connections, which cause these axioms and assumptions to become physical laws, will now be given."

"We have made a number of assumptions about the way in which states and dynamical variables are to be represented mathematically in the theory. These assumptions are not, by themselves, laws of nature, but become laws of nature when we make some further assumptions that provide a physical interpretation of the theory. Such further assumptions must take the form of establishing connections between the results of observations, on the one hand, and the equations of the mathematical formalism, on the other."

Dirac references

• Dirac, P.A.M. (1930). The Principles of Quantum Mechanics, 1st edition, Oxford University Press, Oxford UK.
• Dirac, P.A.M. (1935). The Principles of Quantum Mechanics, 2nd edition, Oxford University Press, Oxford UK.
• Dirac, P.A.M. (1947). The Principles of Quantum Mechanics, 3rd edition, Oxford University Press, Oxford UK.
• Dirac, P.A.M. (1958). The Principles of Quantum Mechanics, 4th edition, Oxford University Press, Oxford UK.

Heisenberg on mathematics and ordinary language for atomic phenomena

    "... In atomic physics, we make use of a highly sophisticated mathematical language which satisfies any criterion of clarity and precision. At the same time we realize that we cannot describe atomic phenomena unambiguously in any ordinary language; for instance we cannot speak unambiguously about the behaviour of an electron inside an atom. It would be a rash conclusion if we should seek to avoid the difficulty by restricting ourselves to the use of mathematical language. This is no way out since we do not know how far this mathematical language applies to the phenomena. Finally, science must rely on the ordinary language as well, because this is the only language in which we can be certain to grasp the phenomena.

    "This situation throws some light on the tension between the scientific method on the one hand and the relation of society to the 'one', to the fundamental principle behind the phenomena on the other. It seems obvious that this latter relation cannot and should not be described in a highly sophisticated and precise language, the application of which to reality may be very limited. For this purpose, only the natural language will do, which can be understood by everybody. Reliable results in science, however, require unambiguous statements, there we cannot do without the precision and clarity of an abstract mathematical language."

Heisenberg on understanding

    "... ″Understanding″ then means: adaptation of our conceptual thinking to the totality of the new phenomena; or: discovering in the wealth of phenomena some underlying structures, which correspond to fundamental innate structures in our conceptual equipment and which therefore enable us to form concepts.

    "It is evident from this discussion that narrow specialization is a hindrance for understanding. It is only by looking at the whole field of new phenomena that the correct concepts can be found. Even in a very special problem, ″understanding″ can frequently be obtained by referring to a similar problem and its solution in a different field of physics.

    "Mathematical analysis can be an important help after the correct concepts have been found, since it may then enable the physicists to draw precise conclusions and to compare them with the facts. Before the correct concepts have been found it is only of little use. Because then it can only establish a precise connection between assumptions, expressed in the old concepts, and their consequences. But the assumptions are probably incorrect and therefore their consequences need not represent the phenomena. Hence mathematical analysis is usually not the direct way toward understanding; mathematical physics and theoretical physics are very different sciences."

Heisenberg on Bohr's attitude to mathematics and ordinary language

"But I noticed that mathematical clarity had in itself no virtue for Bohr. He feared that the formal mathematical structure would obscure the physical core of the problem, and in any case he was convinced that a complete physical explanation should absolutely precede the mathematical formulation."

Heisenberg on what can be described in the quantum mechanical formalism

"By turning round, I had to investigate what can be described in this formalism; and then it was very easily seen, especially when one used the new mathematical discoveries of Dirac and Jordan about transformation theory, that one could not describe at the same time the exact position and the exact velocity of an electron; one had these uncertainty relations. In this way things became clear. When Bohr returned to Copenhagen, he had found an equivalent interpretation with his concept of complementarity, so finally we all agreed that we now had understood quantum theory."

Whitehead on metaphysics

(From the Wikipedia article on Alfred North Whitehead)

"However, interest in metaphysics – the philosophical investigation of the nature of the universe and existence – had become unfashionable by the time Whitehead began writing in earnest about it in the 1920s. The ever-more impressive accomplishments of empirical science had led to a general consensus in academia that the development of comprehensive metaphysical systems was a waste of time because they were not subject to empirical testing.

"Whitehead was unimpressed by this objection. In the notes of one his students for a 1927 class, Whitehead was quoted as saying: "Every scientific man in order to preserve his reputation has to say he dislikes metaphysics. What he means is he dislikes having his metaphysics criticized." In Whitehead's view, scientists and philosophers make metaphysical assumptions about how the universe works all the time, but such assumptions are not easily seen precisely because they remain unexamined and unquestioned. While Whitehead acknowledged that "philosophers can never hope finally to formulate these metaphysical first principles," he argued that people need to continually re-imagine their basic assumptions about how the universe works if philosophy and science are to make any real progress, even if that progress remains permanently asymptotic. For this reason Whitehead regarded metaphysical investigations as essential to both good science and good philosophy."

phase of the wave function

Heisenberg

"Kramers was guided by the idea of virtual harmonic oscillators in the atom corresponding to the harmonics, in writing down a dispersion formula. ... one had to give phases to these amplitudes ... Only when we did that did we get reasonable formulas for the dispersion."

• Heisenberg, W. (1973). Development of concepts in quantum theory, pp. 264–275 of Mehra (1973).
• Mehra, J. editor (1973). The Physicist's Conception of Nature, Reidel, Dordrecht, ISBN 90-277-0345-0.

Messiah

"Moreover, the density operator representing the state of a system is defined in a unique manner, while the vector representing a pure state is at best defined only to within a phase factor."

• Messiah, A. (1961). Quantum Mechanics, volume 1, translated by G.M. Temmer from the French Mécanique Quantique, North-Holland, Amsterdam.

London & Bauer

"Given that the ψ_k are normally complex quantities, we can write them in the form

$\psi _{k}={\sqrt {p_{k}}}\,\exp {(i\alpha _{k})},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)$

where the the phases α_k are still indeterminate. One easily verifies that the difference between cases II and I arises from this ignorance about the phases α_k.

• London, F., Bauer, E. (1939). La Théorie de l'Observation dans la Mécanique Quantique, issue 775 of Actualités Scientifiques et Industrielles, section Exposés de Physique Générale, directed by Paul Langevin, Hermann & Cie, Paris.
• Shimony, A., Wheeler, J.A., Zurek, W.H., McGrath, J., McGrath, S.M. (1983). Translation of London, F., Bauer, E. (1939) at pp. 217–259 in Wheeler, J.A., Zurek, W.H. (1983).
• Wheeler, J.A., Zurek, W.H., editors (1983). Quantum Theory and Measurement, Princeton University Press, Princeton NJ.

Born rule

Einstein's 1916/1917 paper on the A and B coefficients developed the idea of time rates of probabilistic jumps between quantum states in a more or less classical context, with Bohr's 1913 quantum states.

Heisenberg in 1925 systematized the As and Bs into what Born recognized as square matrices of time rates of probabilistic jumps between quantum states.

In 1936 Einstein wrote a paper, in German, in which, amongst other matters, he considered quantum mechanics in general conspectus.

He asked "How far does the ψ-function describe a real state of a mechanical system?"

For the present purpose, it is convenient to use a notation for discussing statistical sample spaces. The term "ensemble" that is customary in this area is here treated as synonymous with 'statistical sample space'. The statistical sample space is the conceptual or fictive set of possible specimens ψ(ω) of the species ψ, where ω∈Ω denotes a variable index that uniquely identifies the possible specimens, one by one.

Scalar product vs. inner product (lifted from Talk:Quantum state)

To editor Chjoaygame: (In response to diff/697869911) I think that inner product is a more appropriate term, because it generalizes the scalar product to abstract vector spaces over (possibly complex) fields. The scalar product applies only to euclidean spaces. Petr Matas 14:24, 2 January 2016 (UTC)

As I read it, Wikipedia posts what reliable sources say, in context. The sub-section is headed Bra–ket notation. Dirac would have a fair chance of being a reliable source on this topic. As cited, he says "scalar product". So do Gottfried and Yan (2003).

Weinberg (2013) also speaks of the "scalar product", as does Messiah (1961). Also, mostly Auletta, Fortunato, and Parisi (2009). Ballentine (1998) sees 'inner' and 'scalar' as alternatives. Beltrametti and Cassinelli (1982) speak of the "scalar" product. As do Cohen-Tannoudji, Diu, and Laloë, F. (1973/1977). And Jauch (1968). And Zettili (2009).

Robinett (2006) mixes Dirac notation with the ψ(x, t) notation, and uses "inner product".

Some authors who do not use the Dirac bra–ket notation, such as Von Neumann (1932/1955) and Schiff (1949), though not Weinberg, use "inner product".

Perhaps a further survey of sources may be needed?Chjoaygame (talk) 17:41, 2 January 2016 (UTC)

D   I don't think so, your sourcing is clear. But that means that the "scalar product" from all these citations is actually described in the article inner product. Shouldn't it be merged with scalar product then?Petr Matas 18:52, 2 January 2016 (UTC)

It is outside my scope to consider such a merge. There is a difference between an inner product and a 'scalar product' as the term is used in the present context. An inner product is between two vectors of the same space. The present 'scalar product' is between vectors from a space and its dual space; these two spaces in the present case are isomorphic.Chjoaygame (talk) 19:05, 2 January 2016 (UTC)

Shouldn't $\langle \phi |\psi \rangle$ be read either as a scalar product between two vectors of the same space, or as a functional acting on a vector, but never as a scalar product between a functional and a vector? Petr Matas 21:41, 2 January 2016 (UTC)

By the mathematics books, you have to be right. But the context is physics as found in sources.Chjoaygame (talk) 22:11, 2 January 2016 (UTC)}}

Surveying a little more:

Bransden & Joachain's Physics of Atoms and Molecules (1983/1990) routinely uses 'scalar product', though it once mentions (in parentheses) 'inner product' as an alternative. Their Quantum Mechanics (2nd edition 2000) uses only 'scalar product'.

Busch, Lahti & Mittelsteadt (The Quantum Theory of Measurement, 2nd edition 1991/1996) uses the Dirac notation and 'inner product'.

David (2015) uses 'scalar product'.

Davydov (1965) uses 'scalar product'.

De Muynck (Foundations of Quantum Mechanics, an Empiricist Approach, 2004) uses 'inner product'.

D.J. Griffiths (1995) uses Dirac notation and 'inner product'.

R.B. Griffiths (2002) uses Dirac notation and 'inner product'.

Duarte on Ward

In the opinion of F. J. Duarte, expressed in his textbook, Ward appears to have followed what Duarte views as Dirac's doctrine, in that he was never bothered by issues of interpretation in quantum mechanics. It is not clear how much this derives from Ward's personal character, and how much from physics, and how much from Duarte's personal viewpoint. Duarte's own perspective may be partly inferred from his having written about his own work: "In other words, we postulate that the most efficient and practical interpretation of quantum mechanics is... no interpretation at all."

In his admirable text Tunable Laser Optics, Duarte writes: "In the past this concept has been the source of some controversy due to a misunderstanding of the Dirac interpretation that implies that indistinguishable photons, regardless of source of origin, are the same photon.". Duarte knows that Dirac is an interpreter of quantum mechanics. Dirac's Chapter 1 is for me a mighty source of education. But it is interpretive. At its extreme, this interpretation gives help and sustenance to one of the worst mysterian elements of Copenhagenism, that may perhaps have helped sway Feynman into his odd belief that diffraction of particles is mysterious: "It is all quite mysterious. And the more you look at it the more mysterious it seems." Dirac in Chapter 1 writes of a photon being partly in one and partly in another of the sub-beams into which an original beam is split. Fortunately, Dirac's bizarre interpretation did not lead Feynman to believe in a simple way that an electron could be partly in several beams at once, whatever that might mean! He wrote: ""They split in half and . . . ” But no! They cannot, they always arrive in lumps." I don't know exactly what a good Copenhagenist will believe in these circumstances.

Bohr

Bohr (1961, The Unity of Human Knowledge) p. 63:

     We admire the Greek mathematicians who in many respects laid the firm foundation on which later generations have built. It is important to realize that the definition of mathematical symbols and operations is based on simple logical use of common language. Mathematics is therefore not to be regarded as a special branch of knowledge based on the accumulation of experience but rather as a refinement of general language, supplementing it with appropriate tools to represent relations for which ordinary verbal expression is imprecise or too cumbersome.

Ok, but p. 66 is windbagism:

     The flexibility of the subject-object separation in the account of conscious life corresponds to a richness of experience so multifarious that it involves a variety of approaches. As regards our knowledge of fellow-beings, we witness of course only their behaviour, but we must realize that the word consciousness is unavoidable when such behaviour is so complex that its account in common language entails reference to self-awareness. It is evident, however, that all search for an ultimate subject is at variance with the aim of objective description which demands the contraposition of subject and object.

Whitehead's ontology circumvents that kind of chatter.

Pauli

Pauli, W. (12 January 1958/1994), 'Albert Einstein and the development of physics', Neue Zürcher Zeitung, reprinted on pp. 117–123 of Writings on Physics and Philosophy, edited by C.P. Enz and K. von Meyenn, translated by R. Schlapp, Springer, Berlin, ISBN 978-3-642-08163-7, p. 122:

... the so-called "Copenhagen interpretation" of quantum mechanics, founded by Bohr, which I also follow, in common with most theoretical physicists. Einstein's opposition to it is again reflected in the papers which he published, at first in collaboration with Rosen and Podolsky, and later alone, as a critique of the concept of reality in quantum mechanics.

Ibid.: (1957/1994), pp. 127–135, 'Phenomenon and Physical Reality', Dialectica, 11: 35-48, p. 132:

Other investigators, especially Bohr, Heisenberg and Born, with whom I am in full agreement, do not share these misgivings, and regard as definitive just this step, which was made in quantum mechanics, of including the observer and the conditions of experiment in a more fundamental way in the physical explanation of nature.

Margenau

Margenau, H. (1929/1978). 'The problem of physical explanation', The Monist, 39: 321, reprinted as pp. 3–20, Chapter 1 of Physics and Philosophy: Selected Essays, D. Reidel, Dordrecht, ISBN 978-94-009-9847-6, p. 6:

With more particular reference to physics the principle of causation may thus be stated: No matter at what time an experiment be performed, its result is the same if the original conditions are identical. ... On the other hand, it may be supposed that it is a category of a priori rank, a fact which lies at the very basis of experience as the condition for its possibility.

Reciprocity

Looking more at the appearance of Hermitian operators in this article:

In the lead:

Reciprocity is closely related to the concept of Hermitian operators from linear algebra, applied to electromagnetism.

In the article, there is no particular section that focuses on the difference between Hermitian and complex-symmetric matrices. Nor is a particular reason given for mentioning Hermitian matrices in this article. Perhaps there is a reason?

In the section headed , there article reads:

For any Hermitian operator ${\hat {O}}$ under an inner product $(f,g)\!$, we have $(f,{\hat {O}}g)=({\hat {O}}f,g)$ by definition, and the Rayleigh-Carson reciprocity theorem is merely the vectorial version of this statement for this particular operator $\mathbf {J} ={\hat {O}}\mathbf {E}$: that is, $(\mathbf {E} _{1},{\hat {O}}\mathbf {E} _{2})=({\hat {O}}\mathbf {E} _{1},\mathbf {E} _{2})$. The Hermitian property of the operator here can be derived by integration by parts. For a finite integration volume, the surface terms from this integration by parts yield the more-general surface-integral theorem above. In particular, the key fact is that, for vector fields $\mathbf {F}$ and $\mathbf {G}$, integration by parts (or the divergence theorem) over a volume V enclosed by a surface S gives the identity:

$\int _{V}\mathbf {F} \cdot (\nabla \times \mathbf {G} )\,dV\equiv \int _{V}(\nabla \times \mathbf {F} )\cdot \mathbf {G} \,dV-\oint _{S}(\mathbf {F} \times \mathbf {G} )\cdot \mathbf {dA} .$

This identity is then applied twice to $(\mathbf {E} _{1},{\hat {O}}\mathbf {E} _{2})$ to yield $({\hat {O}}\mathbf {E} _{1},\mathbf {E} _{2})$ plus the surface term, giving the Lorentz reciprocity relation.

Could this be shortened this to:

For a finite integration volume, the surface terms from an integration by parts yield the surface-integral theorem above. In particular, the key fact is that, for vector fields $\mathbf {F}$ and $\mathbf {G}$, integration by parts (or the divergence theorem) over a volume V enclosed by a surface S gives the identity:

$\int _{V}\mathbf {F} \cdot (\nabla \times \mathbf {G} )\,dV\equiv \int _{V}(\nabla \times \mathbf {F} )\cdot \mathbf {G} \,dV-\oint _{S}(\mathbf {F} \times \mathbf {G} )\cdot \mathbf {dA} .$

It is notationally convenient here to define the coupling $(\mathbf {F} ,\mathbf {G} )=\int \mathbf {F} \cdot \mathbf {G} \,dV$ for vector fields $\mathbf {F}$ and $\mathbf {G}$.

This identity is then applied twice to $(\mathbf {E} _{1},{\hat {O}}\mathbf {E} _{2})$ to yield $({\hat {O}}\mathbf {E} _{1},\mathbf {E} _{2})$ plus the surface term, giving the Lorentz reciprocity relation.

without detracting from the article?

The reason for introducing the idea of Hermitian operators is to do with the cases of magnetic fields. Should this be done just at the point where it becomes relevant and necessary, or should it permeate the mathematical formulation throughout the article?

substance in itself

Aristotle said "It is because of their underlying all things and all other things being predicates of them or in them that primary substances are especially called substances. ... Primary substances are especially called substances in the chief sense because of their unerlying all things."

Perhaps "there exists no possible world containing nothing but the surface of an apple." You are arguing that the surface of an apple is not a substance. I think that the surface of an apple can change? If someone thought that

I accept that a philosopher might postulate a hypothetical world containing nothing but an apple. But to view such a world as 'possible' is a feat perhaps possible only for a philosopher. Under duress, I would allow that a such world is perhaps 'conceivable', or 'imaginable'. But for me, such a world is actually impossible. It would be far from the actual ordinary world, to which I think 'substance' refers. I don't see 'substance' as a synonym for 'entity'. I would admit that

Whitehead (1929), page 81: "But this complete experience is nothing other than what the actual entity is in itself, for itself."

Page 93: "This is the direct denial of the Cartesian doctrine, “. . . an existent thing which requires nothing but itself in order to exist.”"

Locke, page 280: "Our ideas of particular sorts of substances. Whatever therefore be the secret abstract nature of substance in general, all the ideas we have of particular distinct sorts of substances are nothing but several combinations of simple ideas, coexisting in such, though unknown, cause of their union, as makes the whole subsist of itself."

Page 578: "For we are wont to consider the substances we meet with, each of them, as an entire thing by itself, having all its qualities in itself, and independent of other things;"

kim

Spinoza, page 88

By substance, I understand that which is in itself and is conceived through itself; in other words, that, the conception of which does not need the conception of another thing from which it must be formed. (Ethics and Selected Letters, p. 31)

Aquinas, page 117

The general mode of being expressed by “one” pertains to every being in itself (in se); “one” adds to being a negation, for it signifies that being is undivided.

Kant, page 356

Kant maintains that the objects of our knowledge are only appearances, and that things in themselves are unknowable.

Leibniz, page 366

And since it has relations (at least trivial ones) with every other substance in, and every fact about, the whole world of which it is a part, it follows that each substance contains within itself a complete expression of its universe, and thus corresponds perfectly with every other substance.

E.J. Lowe (2003), The Oxford Handbook of Metaphysics, ed. M.J. Loux, D.W. Zimmerman, Oxford UK, ISBN 978-0-19-825024-1, pp. 78–79

It is on the individuation of substances that I shall concentrate in what follows, partly because this has received the most attention in the literature on individuation and partly because of its intrinsic interest. ... For although it is, of course, causally impossible for an organism to survive for any appreciable length of time in isolation from other, wholly distinct substances—because of the demands of its metabolic processes—it does not appear to be metaphysically impossible for an organism to be a solitary occupant of space throughout its existence, however brief, and this suffices for it to qualify as an individual substance according to the independence criterion that I have just proposed.

'Metaphysical possibility' is a technical concept, with technical relationship to ordinary language 'possibility', and to 'ontological independence'.

Peter van Inwagen, ibid. p. 134

′Aussersein′ may be translated as 'independence [sc. of objects] of being'.

It is not always an easy thing to translate philosophy between ordinary languages such as English and German.

real thing

"reale Dinge (entia realia)"

"Boundaries and Things"

sock

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