Talk:Wave function/Archive 3

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Complex valued function?

"Typically, its values are complex numbers and, for a single particle, it is a function of space and time."

Is this true of any but a scalar particle? I know I'll probably spark a debate because this is not my area, but surely this is oversimplification to the point of confusion? For example, I'd expect any fermion to have a wavefunction that is spinor-valued, in which case complex numbers might come into it, depending on the representation chosen, but also might not. Shouldn't this description be changed? —Quondum 21:48, 15 March 2014 (UTC)

Yes, in general a spinor,or other things I don't know about, but we need to start from somewhere, and the first introduction to wavefunctions is always through complex numbers as far as I know. Presumably your'e thinking of multivector-valued wavefunctions in geometric algebra which have the "spinorial" behavior (according to Hestenes' use of the term "spinor")? For now, probably best left as complex numbers with a pointer saying other quantities are occur, maybe have a section on this too. M∧Ŝc²ħεИ_τlk 08:58, 16 March 2014 (UTC)

This, being an encyclopaedia and not a textbook, has rather strict guidelines about how it can present things. As such, the lead completely violates the guidelines. Even if one chose to take a pedagogical approach in the body of the text, I would have considered it necessary to make it clear that the approach is not the general one, and also to omit the invalid simplification from the lead. For example, one could say "Typically, for a single particle, it is a function of space and time."

I was thinking of spinors of any representation, be they Dirac's complexified vectors or either of the two most suitable GA representations, neither of which is an algebra over the complex numbers. From all perspectives (pedagogic and encyclopaedic), I think that it is important to maintain a clear distinction between what is being represented and the representation. I see the confusion in thinking that extends quite high (to some postgrads) when they use only one representation because they fail to disentangle these, for example with tensors and the Ricci calculus, where people are taught in effect that the collection of components is a tensor, and that the definition of a tensor is a set of components that transforms appropriately. Especially pedagogically, I feel that this is highly counterproductive because people are being taught the "arithmetic" of QM while glossing over its "mathematics", and encyclopaedically it is just plain wrong. —Quondum 16:05, 16 March 2014 (UTC)

Representation of what? Schrödingerian mechanics isn’t Lorentz/Poincaré invariant. It is Galilean-invariant. Usually, things that are Lorentz/Poincaré invariant are not called wave functions. These are fermionic fields and so. Incnis Mrsi (talk) 17:22, 16 March 2014 (UTC)

I can see this bogging down in undefined terminology. Would you care to put that statement ("things that are Lorentz/Poincaré invariant are not called wave functions") into the article? If true, it would certainly be helpful. Also, is the concept restricted to scalar particles? It would also help if this were made clear in the article. Coming from an outside perspective, I cannot even determine what the term wavefunction covers, and what it doesn't cover. With regard to representation, a particle is not a complex function of space and time, but it could be represented by a complex function. Or is the wavefunction by definition the complex value representation of a particle? The article seems to leave all these questions unanswered. —Quondum 06:03, 17 March 2014 (UTC)

Physics bogged down in terminology for a long time, but this is your “represented” verb that is undefined here. When I read about representations, I thought we talk about representation theory of symmetry groups. No, I won’t put the Lorentz/Poincaré invariance speculation into the article. The relativistic quantum mechanics uses hyperbolic (Lorentz-covariant) Hamiltonians with wave functions. Also, Wikipedia states:

Nevertheless, RQM is only an approximation to a fully self-consistent relativistic theory of known particle interactions because it does not describe cases where the number of particles changes; for example in matter creation and annihilation.

If you replace QM with wave function, then it will be consistent with my understanding. When you know how many (and which) particles do you have, then you have a wave function. When you don’t know it, then you have a field. Also, you can read here on a thought experiment that shows that “how many and which particles” concept is blurry, and it is related to your spinors. You can speak about spinor-valued wave functions only if you have some kind of Spin(3) symmetry. If you haven’t, then values are merely complex (finite-dimensional) vectors that represent some opaque states of the system. Yes, a particle, or a system of particles, is not obliged to have only one internal state. Incnis Mrsi (talk) 07:40, 17 March 2014 (UTC)

My use of the term representation is clearly confusing the matter, so let's drop the term. Whether we are dealing with (non-R)QM or RQM does not seem to be significant to the concept of a wavefunction. I agree that the concept of a wavefunction does not really apply to QFT. From what you say, you appear to consider solutions the single-particle Dirac equation as wavefunctions in the sense of the article, even if regarded as complex vector fields. This alone is enough to suggest that the description "its values are complex numbers" in the lead should be made more general. I'm suggesting that in the lead it should rather say something like "It is generally vector-valued and, for a single particle, it is a function of space and time." Or even simply "For a single particle, it is a function of space and time." The discussion of whether the vectors are considered to be over complex numbers or to be elements of a Clifford algebra need not be addressed in the lead. Scalar particles are simply a special case of vectors. —Quondum 18:40, 17 March 2014 (UTC)

Quondum, you refer to WP policies. One of the most important is to write from what most sources say: of course not like a textbook but as a plain English summary. Well, does anyone know how many introductory QM sources which do not use write wavefunctions as complex numbers or complex-valued vectors? Complex numbers are exclusively used in almost every single introductory QM book I've ever seen (anyone is welcome to contradict with examples). The only exception I can think of is when QM is written in the language of geometric algebra, which uses the field of real numbers. The restricted scope "introductory QM" is not meant to be pedagogic - this is one important level of sources we should use for this article.

As for:

"With regard to representation, a particle is not a complex function of space and time, but it could be represented by a complex function. Or is the wavefunction by definition the complex value representation of a particle? The article seems to leave all these questions unanswered"

the article has an ontology section linking to the main ontology article - saying the meaning of the wavefunction is unclear, and always has been from day 1. The sections on wavefunctions for one particle in 1d, and more particles in higher dimensions, all mention the Copenhagen interpretation, the most introductory interpretation of the wavefunction.

It's a bit unfair to say the lead totally violates the guidelines. It can be improved, but it gets the main points across. I'm not denying the article still needs work. But this discussion confuses me. M∧Ŝc²ħεИ_τlk 16:40, 22 March 2014 (UTC)

A sensible definition of what a wave function is might be that it solves a wave equation. This holds true in all cases I can think of. What a wave equation is is well defined as far as I know.

It is not true that wave functions aren't present in QFT. They are necessary to construct the Hilbert space. (Any complete set will do.) They just don't occupy the center of the stage; the dynamics (time evolution of the system) is moved to the operators on the Hilbert space. In addition, new operators are introduced to handle creation and destruction of particles. YohanN7 (talk) 23:40, 22 March 2014 (UTC)

QFT is beyond scope. I said nothing about the interpretation. I have evidently failed to communicate my point about the space from which the wave function may take its values (codomain), without which my comment about guidelines is also meaningless. I see little point in pursuing this. —Quondum 00:27, 23 March 2014 (UTC)

Quondum and YohanN7 - I was still thinking we could describe the role of the wavefunction in non-RQM, RQM, and QFT, all in a section for comparison. In all this time it just hasn't been done. Currently the article biases the Schrödinger picture. As I mentioned above, we could have a brief section on how operators act on the wavefunction in the Schrödinger picture (and related, briefly mention how statistical quantities like averages can be found), then go onto the Heisenberg picture, and possibly the interaction picture, again all in a section for comparison. No, we will not turn the article into a second Dynamical pictures (quantum mechanics), just an overview.

Quondum, no, you've raised a good point. There is a bias with complex numbers, sure, and having a variety of mathematical languages would remove the bias. Since you raised the point, others will probably ask the same question. I'm simply not sure how to do this yet.

Thanks. (P.S. YohanN7 sorry about not checking in Griffiths yet, I couldn't find it last week). M∧Ŝc²ħεИ_τlk 07:54, 23 March 2014 (UTC)

Wave functions take their values in Cⁿ for n ≥ 1, and this covers the rare cases (e.g Majorana fermions, and certain potentials in one dimension (forgot details, see Landau and Lifshits)) when they can be taken as real-valued. Why talk about guidelines? I can't honestly see the point in having a huge debate about this. The GA representations are clearly fringe (unless something revolutionary has happened the last few years), so they can be left out. YohanN7 (talk) 09:40, 23 March 2014 (UTC)

This isn't just about guidelines, it's about content.

If it helps, I saved up enough to order Wienberg's vol 1 Quantum theory of fields, Landau and Lifshitz's vol 2 Classical theory of fields, and Doran and Lasenby's Geometric algebra for physicists (which includes QM formulated using geometric algebra). All fairly expensive, but there is no way round that, and they're top quality books so it's just about tolerable. Next week it will be the easter break too. Within the next week the books hopefully arrive and I get more time, it will be easier to make edits. M∧Ŝc²ħεИ_τlk 10:03, 23 March 2014 (UTC)

Excellent choices. The L&L book also covers general relativity (guess you knew this, much much better than MTW). YohanN7 (talk) 10:20, 23 March 2014 (UTC)

You might want to have a look at Field Quantization by Walter Greiner et al. Weinbergs attitude is that he doesn't quantize any pre-existing theory. But is quite useful to see that view too. It contains hundreds of detailed calculations, and it has a highly unusual description of classical field theory (Poisson-bracket formulation). Only $20 at amazon. YohanN7 (talk) 12:46, 23 March 2014 (UTC)

Greiner's books are OK (the classical mechanics, QM, RQM, QED, QCD, and Field quantization (if I recall correctly) are all at the library). But for now Weinberg's first volume is enough. Thanks for the suggestion. Yes, while MTW is a well-known and widely used classic, some bits are inadequate (e.g. the angular momentum sections are a bit vague, the relativistic heat conduction is not thorough, and there doesn't seem to be any description of the relativistic D, H, M, P fields in matter, all these are remedied either by Pauli or Tolman). LL's classical theory of fields seems to be loaded: SR, EM, GR using Lagrangian mechanics and field theory. M∧Ŝc²ħεИ_τlk 14:11, 23 March 2014 (UTC)

Mentioning that Cⁿ may be the space is reasonable, mainly just not implying a limitation to C. No need (and it may be unwise especially in the lede) to mention GA or any specific space (vectors, matrices etc.), but preferably don't imply their exclusion by the wording. And forget my mention of guidelines: I was hoping to aid understanding of rather than to push my point. And my point does not imply any effort in adding detail about spaces, only wording that does not create misconceptions. We know that isomorphisms exist; we don't need them to be enumerated. —Quondum 16:33, 23 March 2014 (UTC)

I'll have a very quick go at tweaking the lead along these lines. M∧Ŝc²ħεИ_τlk 16:56, 23 March 2014 (UTC)

Not so quick, please. First, wave mechanics is a dab page. Second, although it is true that a particle with spin has a spinor-valued wave function, I’d not assert that a spinless particle is always scalar. Which spin has a (non-rotating) ammonia molecule? With ¹⁴
₇N
¹
₁H
₃ it certainly isn’t zero (even number of electrons and odd number of nucleons), but I do not see a reason for it not to be spinless with an appropriate isotope composition. So, is it a scalar particle, is it really? Incnis Mrsi (talk) 18:24, 23 March 2014 (UTC)

Nice example. As far as I can make out it is a superposition of two states ("left" and "right" handed).

So what is the best wording for the lead then? M∧Ŝc²ħεИ_τlk 22:29, 23 March 2014 (UTC)

I hope the lead is better now. Anyone should boldly edit themselves. M∧Ŝc²ħεИ_τlk 23:15, 23 March 2014 (UTC)

Thank you: your edit is a massive improvement, and makes the lede both understandable and essentially complete in its description. —Quondum 18:10, 23 March 2014 (UTC)

Relation between wave functions

I tried to explain a little. There are probably bugs in the new equations, possibly a missing minus sign (the plane wave), and possible a missing factor of one over two pi (delta function normalization). These things depend on conventions, including for the Fourier transform. Can anyone check what Griffith (the ref in the section) says? YohanN7 (talk) 22:47, 15 March 2014 (UTC)

Nice work. Except for one quibble: we now have the inner product of momentum states before the concepts of the inner product is introduced even in the position representation, so these should be moved down into the inner product section. I don't have Griffith's QM book now but may be able to check sometime in the week. M∧Ŝc²ħεИ_τlk 08:58, 16 March 2014 (UTC)

I have noticed the mentioned quibble. Will make an attempt later today to fix it. YohanN7 (talk) 10:33, 16 March 2014 (UTC)

Oops, I missed the time dependence. But, should we instead state that it is the solution of the time-independent Schrödinger equation? Time dependence is mostly disturbing in this section. YohanN7 (talk) 03:22, 24 March 2014 (UTC)

I moved the momentum inner product content down, and indicated the time-independent Schrödinger equation. M∧Ŝc²ħεИ_τlk 06:58, 24 March 2014 (UTC)

Hydrogen wave animation

May I destroy [1]? This flickering picture that does not, actually, “animate” anything useful, only distracts attention and wastes the processor time. Incnis Mrsi (talk) 13:36, 29 March 2014 (UTC)

It does not add anything; it is merely an attempt at a different presentation of what is already there. From the description "hydrogen wave animation" I had expected the evolution of a wavefunction in time, but it is nothing of the sort. I agree with Incnis that it should be removed. —Quondum 16:10, 29 March 2014 (UTC)

Limited by the speed of light?

I'm not a quantum mechanic (can't handle those teeny, tiny wrenches) so forgive me if I'm not putting this question correctly, but I wonder if someone could say (and maybe put in the article) something about how causality and the speed of light relate to the time-dependent wavefunction. I assume the wavefunction is causal? If the boundary conditions or the potential are altered in one place, do the resulting changes in the wavefunction propagate to other places at the speed of light? And what about the delayed quantum eraser experiment? Thanks. --Chetvorno^TALK 06:38, 27 October 2013 (UTC)

An interesting question.

Quantum entanglement itself does not violate special relativity because information is not transmitted "in" the wavefunction. For two observers to agree on the outcome of an experiment, one observer has to send a signal to the other, which is limited by c. For time-dependent potentials, presumably there would be time-dependent phase factors in the wavefunction related to the potential, like the Aharonov–Bohm effect.

In any case it would be a good addition to the article, thanks for raising this, will look into it. M∧Ŝc²ħεИ_τlk 07:03, 27 October 2013 (UTC)

In non-relativistic QM causality isn't a problem, while the speed of light certainly is a problem. In relativistic QFT, this can dealt with using the cluster decomposition principle, which is weaker (I believe) than a strict causality requirement, but has the advantage that one need not care about whether fields are measurable in any sense. This principle (roughly) implies that quantum fields commute at space-like distances, in turn yielding relativistic field equations, both for states and field operators. YohanN7 (talk) 17:12, 3 April 2014 (UTC)

domain of the wave function

I have put into the lead some brief remarks on the domain of the wave function. To save editors the trouble of checking my sources, I copy quotes from them here.

"For Schrödinger's waves move not in ordinary space but in configuration space, that has as many dimensions as the degrees of freedom of the system (3N for N particles)."

"The waves do not have immediate reality, of the kind we earlier ascribed to the waves of Maxwell's theory. One must interpret them as probability waves and therefore expect a sudden change at every observation."

"Now, Schrödinger's work first of all contains some misunderstanding of the usual interpretation. He overlooks the fact that only the waves in configuration space (or the "transformational matrices") are probability waves in the usual interpretation, while the three-dimensional matter waves or radiation waves are not."

"Bohr unequivocally rejected Schrödinger's interpretation of wave mechanics. Indeed, he immediately accepted Born's probabilistic interpretation, which was published just before Schrödinger's visit to Copenhagen."

"Schrödinger's idea of identifying the Ψ wave of a system in configuration space at first shocked me very greatly, because, configuration space being a pure fiction, the conception deprives the Ψ wave of all physical reality. For me the wave of Wave Mechanics should have evolved in three-dimensional physical space. the numerous and brilliant successes that resulted from adopting Schrödinger's point of view obliged me to recognize its value;"

"This dq is often called an element of volume in the configuration space of the system; for one particle, dq coincides with an element of volume dV in ordinary space."

"The three parameters needed to identify the location of the particle are denoted by ${\vec {q}}$, the vector whose components are equal to its three Cartesian coordinates in Euclidean space. However, the vector ${\vec {q}}$, as a dynamical variable, does not denote a point in physical space but a point in configuration space. For two particles, a system of 6 degrees of freedom, the 6 needed dynamical variables are the Cartesian coordinates ${\vec {q}}_{1}$ and ${\vec {q}}_{2}$ which identify the locations of particles #1 and #2 in physical space which make up a vector in the six-dimensional configuration space.".

References

Chjoaygame (talk) 14:51, 31 July 2014 (UTC)

Basic interpretation

I think there is a need for a basic interpretation of the wave function before introducing the Copenhagen interpretation, and even measurements. I am thinking about things that are often misinterpreted. For instance (in a toy model), if a wave function is a superposition of two delta-like functions located at A and B (in position space), it does not mean any of the following:

• The particle is either at A OR B.
• The particle is both at A AND B.

Both statements are provably not true (actually, some "plausible" statements of this nature are not true/false but plain nonsense), using Hilbert space theory alone (they lead to contradictions of mathematical and even logical nature), but both of these interpretations are easy to be picked up by a novice, especially when fed the Copenhagen interpretation. (Note, no measurements are involved in the statements above) An example of a correct statement would be

• The particle is NOT anywhere else (C) than A or B.

A great source for stuff like this is Robert Griffith's Consistent Quantum Theory. This book is about an alternative (other than Copenhagen) interpretation of quantum theory, but the stuff I mentioned above is basic and independent of any interpretation. Well, my question is, should we have a "basic" section here or in a separate article, or does one already exist that handles matters like these? YohanN7 (talk) 20:12, 28 August 2014 (UTC)

Oh yes. I'd say, "the particle is at A" and "the particle is at B" are two "quantum possibilities"; and a quantum possibility is more real than a classical possibility, but less real than a classical reality. This is extremely hard to understand, since it is a new ontological category.Boris Tsirelson (talk) 06:22, 29 August 2014 (UTC)

What should be the scope of such a section? Should it appear in this article?

In my view, the presentation should be simplistic (use a "toy model"), define a little terminology, and make comparisions with classical mechanics (phase space formulation) for the defined terms, that would include "physical property of a system" as "subset of phase space with associated indicator function" contra "subspace of Hilbert space with associated projection operator". That would be a start. The inspiration comes from Griffith's book. I don't know to what extent he is biased, but it would surprise me if there are different views around, because this is from the very earliest chapters. YohanN7 (talk) 11:43, 31 August 2014 (UTC)

Spin

This article too presents the misconception that spin is a consequence of the Dirac wave functions. (Other common and incorrect formulations include that spin is a consequence of the Dirac equation.) The truth is that spin is a consequence of the symmetry group of special relativity, the Poincaré group, more precisely its subgroup, the Lorentz group — even more specifically it originates in its subgroup SO(3), the rotation group, which is doubly connected.

More accurate would be to say that the Dirac equation is a consequence of (the existence of) charged spin 1⁄2 quantum particles coupled to the EM field via minimal coupling with parity inversion symmetry (without the latter the Weyl equations arise). Dirac spinors have spin 1⁄2, they don't predict it. Paraphrased, "The DE doesn't predict spin 1⁄2, but spin 1⁄2 predicts the (free) DE". YohanN7 (talk) 19:09, 28 August 2014 (UTC)

I want to emphasize that Dirac himself surely would have objected to the current article's formulation. He was well aware of the order of events. He actually published the Bargmann-Wigner equations a decade (or more, 1933?) before Bargmann and Wigner. Moreover, the investigations into the infinite-dimensional representations of the Lorentz group was instigated by him. That is, it is hardly to dishonor Dirac to take away the "prediction-of-spin-status" from the equation or its solutions.

The great thing that the Dirac equation actually did predict was the existence of positrons, or, in general, antiparticles for every particle obeying the DE. (Dirac initially speculated that they may be protons, this at a time before the discovery of the neutron. Someone else (don't remember who) showed that the positron must have the same mass as the electron, hence couldn't be a proton.) YohanN7 (talk) 11:43, 29 August 2014 (UTC)

For what its worth - I wrote (pretty much all of) the history section in relativistic wave equations. Perhaps we could take out the (trivial) reference to the Dirac wavefunction in this article (which could be potentially extended on RQM and other bits of abstract algebra), and concentrate on the history over there first, all in one place, before spreading misconceptions elsewhere? M∧Ŝc²ħεИ_τlk 00:48, 1 September 2014 (UTC)

Inner product (again)

The definition of the "inner product" suffers from an annoying and confusing error omission that is present in most basic QM texts(, and some math texts too). The "inner product" is not an inner product, it is a semi-inner product. The resulting "norm" is a pseudo-norm semi-norm, and its "metric" is a pseudo-metric (not semi-metric).

There are standard ways (all equivalent I believe) of dealing with this. One way is to declare points at a zero distance apart to be equivalent. Then by passing to the quotient w r t this equivalence relation, one obtains a true inner product space. It's elements are equivalence classes of functions from the original "inner product space".

Before I (possibly) do anything, I'd like your opinion on whether this conceptually important, but practically unimportant, issue is worth mention in the article. IMO, QM books being generally crappy mathematically should not be a reason for this article to make the same mistakes.

There are probably three categories of students (of QM books or this article). The first category never notices the problem. The third category does notice the problem and also its solution. The second category will be left puzzled. I belonged once to the second category. The instructor did some hand-waving when I asked him (he really didn't know I think). YohanN7 (talk) 12:20, 3 April 2014 (UTC)

I'm not too sure how you're applying the metric. We're dealing with a Hilbert space, so the vectors are not 4-vectors to which a metric tensor would apply for this inner product. On a scalar (multi-particle) field, this seems to be unambiguously an inner product (it is always positive for a nonzero field used for both arguments, and the equivalence class issue should not occur), and it is not clear to me that for a spinor field it will be any different. But I'm not familiar with the area at all, just looking at it from the outside, so really I'm just asking. —Quondum 20:00, 3 April 2014 (UTC)

We are not dealing (yet) with a Hilbert space (or metrics for that matter). Hilbert spaces have inner products. What is defined here on a function space is a semi-inner product, since

$\left\langle \Psi ,\Psi \right\rangle =0$

does not imply

$\Psi =0.$

In other words, you have missed the axiom that distinguishes an inner product from a semi-inner product. It has nothing to do with spin or 4-vectors. (This puts you in category one of above :D)YohanN7 (talk) 20:19, 3 April 2014 (UTC)

What reference are you using for the definition of a semi-inner product outside of WP? Do you know any refs for semi-inner products in the context of QM? (Admittedly I'm in your categories 1 and 2 above). M∧Ŝc²ħεИ_τlk 20:24, 3 April 2014 (UTC)

See A course in Functional analysis by John B Conway or General topology by Willard. The latter explains how to systematically obtain a metric space from a pseudo-metric space. There is probably a note too in our article on L^p-spaces. (L² isn't a Hilbert space before the mentioned identification and taking quotients.) YohanN7 (talk) 20:34, 3 April 2014 (UTC)

And oh, the integration must be taken as Lebesgue-integration. Riemann-integration will fail in some technical aspects to yield what we need, but this is probably definitely outside of the scope. YohanN7 (talk) 20:34, 3 April 2014 (UTC)

OK, thanks, I'll look into these. This article really has always been in turmoil... M∧Ŝc²ħεИ_τlk 20:52, 3 April 2014 (UTC)

The omission is no disaster, it is systematically made in the literature. The difference between the math literature and the QM literature is that the math literature takes it as an obvious standard thing to do (the metric identification procedure, that is, and it usually gets a parenthetical remark, math students are supposed to have seen it done). The QM literature should not be able to do the same thing, but it does, leaving some of us, including me, confused for a while. YohanN7 (talk) 21:17, 3 April 2014 (UTC)

Remark on Lebesgue vs Riemann integration: My ref is Byron and Fuller, Mathematics of Quantum and Classical Physics, a book I don't have at where I am now. I think Lebesgue integration is required for completeness of the inner product space (so that it is a Hilbert space), but the details have faded from my memory. At least, it is crucial for some proofs. Might perhaps be found in Conway, but that book by default uses Lebesgue-integration, so that it passes without mention. YohanN7 (talk) 21:36, 3 April 2014 (UTC)

For the purposes of this article, the following should be enough:

• A remark, and link to Lp space#Lp spaces
• A clarification that this is of no practical concern, i.e. which function one takes to represent an equivalence class doesn't matter.

Only the latter point has a problem. Such a statement needs to be sourced. YohanN7 (talk) 21:51, 3 April 2014 (UTC)

Then again, is it really possible for two continuous functions to differ from each other on a set of Lebesgue measure zero? A continuous function is completely determined by its values on a dense set (e.g the rationals). This requires some thought. I'd say no, so if attention is restricted to continuous square-integrable functions at the outset, then we do have a Hilbert space. Edit: Probably no, see below. YohanN7 (talk) 23:45, 3 April 2014 (UTC)

No, this procedure will probably not work either. I think I can imagine a Cauchy-sequence of continuous square-integrable functions with the limit being a discontinuous function. (Modify an example given here: Inner product space#Examples.) We get an inner product space, but not a Hilbert space. Adding further smoothness requirements probably wont help. YohanN7 (talk) 02:21, 4 April 2014 (UTC)

I think the conclusion may be this: With this choice of inner product, the space of functions should be the space of square-integrable functions period. No continuity assumptions allowed. Only then will we get a Hilbert space (upon taking an appropriate quotient). There are other possible Hilbert spaces to make use of, e.g. the Sobolev spaces. These are mentioned in the article. But, for these, the inner product given here is not the right one. YohanN7 (talk) 13:54, 4 April 2014 (UTC)

Careful authors would say that elements of L^2 are equivalence classes of functions, although this is seldom emphasized when it's expedient not to do so. I recently read a paper where the author placed "equivalence classes of" in parentheses at one point. That seems to be about as far as I'd be willing to push things. Of course, for continuous functions such as those that appear in quantum mechanics, it makes no difference whether you identify functions if they are equal outside a set of measure zero. I think a rather enlightening perspective is that in quantum mechanics it isn't really the Hilbert space itself that one sees, but rather its dual. But the continuous dual space of the space of all square integrable Lebesgue measurable functions (which is a nonhausdorff topological vector space with semi norm give by the usual hilbertian inner product) *is* a Hilbert space. No magic identifications are needed. This also explains why distributions have such a big role to play in the theory. Sławomir Biały (talk) 14:44, 4 April 2014 (UTC)

"Methods of Modern Mathematical Physics: Functional analysis", Volume 1, by Michael Reed, Barry Simon: page 17: ...an element of $L_{1}$ is an equivalence class ... when $f\in L_{1}$, the symbol $f(x)$ for a particular x does not make sense ...

Then they explain that for a continuous function $f(x)$ is unproblematic, since two continuous functions cannot be equivalent (unless equal everywhere).

But then on page 40, "Example 4", $L_{2}$ consists of functions!

Boris Tsirelson (talk) 15:15, 4 April 2014 (UTC)

Thank you for your replies. I have follow up questions, the most itching one being this: Suppose we confine ourselves to square-integrable continuous functions. I can see that the articles inner product is a true inner product. We get an inner product space. But I don't think we have a Hilbert space. since it appears to me to exist Cauchy sequences (in this space, w r t the metric induced) that converges to a discontinuous function. True or false?

Surely, true; and moreover, every element of L² is the limit of some sequence of continuous functions (that is, corresponding elements of L²). Indeed, given $f$ and $\varepsilon$, we define $f_{\varepsilon }$ by $f_{\varepsilon }(x)={\frac {1}{2\varepsilon }}\int _{x-\varepsilon }^{x+\varepsilon }f(t)\,dt$ and get $f_{\varepsilon }\to f$ (in L²) as $\varepsilon \to 0$; and each $f_{\varepsilon }$ is continuous. Boris Tsirelson (talk) 18:27, 5 April 2014 (UTC)

Or alternatively, use Fourier series; its finite sums are continuous, and converge to a given element of L² (on a finite interval; and this is enough, in the limit of long interval). Boris Tsirelson (talk) 19:06, 6 April 2014 (UTC)

@Slawomir: Let L² denote the set of all square integrable Lebesgue measurable functions (no identifications) and let an asterisk denote the set of continuous linear functionals on it. You are saying that L^2* is a Hilbert space? Hmm, this reminds me of something I have read, but not understood, in Conway. YohanN7 (talk) 15:47, 4 April 2014 (UTC)

Yes, that's precisely what I'm saying. Sławomir Biały (talk) 17:25, 4 April 2014 (UTC)

Thank you both. It is clear now. It remains only for me to come up with a sentence or two for the article. YohanN7 (talk) 19:41, 7 April 2014 (UTC)

So I finally took care of this. The beefing out can be found in a popup. YohanN7 (talk) 11:47, 7 November 2014 (UTC)

recent edits by Maschen

Editor Maschen has been making several edits which I do not wish myself to try to fiddle with. But I would point out some things that I think Maschen should fix.

It is not good to use footnotes in Wikipedia articles. Either something deserves to be in the main text, or it doesn't have a place at all. The scope for ever-expanding confusion with footnotes is obvious. So I have moved Maschen's footnote into the main text.

On the same topic, I have also moved the recent footnote by Editor YohanN7 into the main text.

A sentence that begins "Actually, ..." is very often a stylistic failure or error, for reasons that I will not here try to expound. In a nutshell, I think Maschen should remove the word 'actually' from the relevant text sentence, and make what other adjustment he thinks best. A problem is that Maschen's edit is in the lead, and it may be making the lead grow like Topsy. Perhaps the footnote stratagem was intended to by-pass this problem. I think it won't do. Somehow Maschen should deal with this.Chjoaygame (talk) 04:20, 6 December 2014 (UTC)

Not all text is directed to all readers. In fact, most readers will only be confused with having technicalities in the main flow of text. Yet, the content of the footnote does belong in the article since without it, the claims of the article are provably inconsistent. A footnote is the ideal middle path. The interested readers (less than 1% will notice the problem with the definition w/o the made qualifications) can pursue the matter by opening the notebox (which, b t w can be distinguished from an ordinary citation through its name = nbx).

A citation needed tag is never alone sufficient for deletion of content. Especially, since in this case, a citation was in place. YohanN7 (talk) 05:59, 6 December 2014 (UTC)

On the other hand, I am not sure that Maschen's nb at the top should stay at all. I do not know what it means.

This cost me about an hour because the reverts managed to re-revert each other having me to manually set things straight. Please discuss on the talk page before you make any more edits. YohanN7 (talk) 06:37, 6 December 2014 (UTC)

@Chjoaygame: I can see that you are at it again. The disputed chunk of text was, by design, created as a footnote, and a footnote it should remain. You wrote in your last revert that it was an issue for Maschen. He has had his say already. It is a footnote. Please discuss here before you make any more edits. YohanN7 (talk) 14:46, 6 December 2014 (UTC)

Sorry for the delay. Here is the edit in question for reference.

Basically, the objection is to the nb in the lead, on bosons described by spinors. That's fine, but I thought the flow would be fine to say "fermionic wavefunctions are spinorial while bosonic wavefunctions are tensorial", with a digression that spinors can describe particles of any spin (after all tensors can be replaced by spinors can't they?).

But maybe it is also excessive wording. I'm neutral on how we do this. Perhaps the best solution is to delete the nb and replace the statement by

"in general, the wavefunction for a particle with spin is a spinor"

(no reference to abstractions and unfamiliar terminology like "spinor rank" etc. keep it simple, this is elaborated later on in this article and other articles anyway). I'll make that change, any alternative wording is fine, people should feel free to edit.

For completeness, my only other edit to the lead was the deletion of the underlined part:

"The wave of the wave function, however, is not a wave in physical space; it is a wave in an abstract mathematical "space", which can be represented as 'configuration space', or can be represented as 'momentum space', and in this respect it differs fundamentally from water waves or waves on a string."

since in the next paragraph, the position space and momentum space wavefunctions are described anyway.

Hope that clarifies things, I'm fine if there's any other objections. Best, M∧Ŝc²ħεИ_τlk 23:04, 6 December 2014 (UTC)

Just to offer an alternative, if we wanted to reinstate mention of fermions and bosons:

"In general, all particles can be divided into two distinct groups according to their spin; bosons have integer spin (0, 1, 2, 3, ...) while fermions have half-integer spin (1/2, 3/2, 5/2, ...). Spinorial wavefunctions can describe particles with any spin, although tensorial wavefunctions can describe bosons only."

M∧Ŝc²ħεИ_τlk 23:25, 6 December 2014 (UTC)

On the technical side of affairs: Tensors (in that they transform under integer spin reps) are not spinors and spinors are not a generalization of tensors in any way. It is another matter that a, say, vector field has spin 1. It doesn't make it a spinor. They always concern half-integer spin. Then, of course, you can take the tensor product of two spin representations and extract irreducible modules with various integer spin (tensors). This is also another story. YohanN7 (talk) 05:29, 7 December 2014 (UTC)

You'll notice in the series of recent edits I never said spinors "are" tensors but can replace tensors (or wording to that effect). M∧Ŝc²ħεИ_τlk 09:24, 7 December 2014 (UTC)

How? YohanN7 (talk) 11:11, 7 December 2014 (UTC)

... Because in physics there are sources (e.g. Penrose and Carmeli's Theory of Spinors) which suggest spinors can be used where tensors cannot. If it isn't the case in general then it isn't the case. I'm not to going to debate over this. M∧Ŝc²ħεИ_τlk 13:10, 7 December 2014 (UTC)

I'm not saying Penrose or Carmeli are wrong, but I do say that you have with 100% certainty misunderstood what they are saying based on what you have written here and edited into the article. Wave functions governing integer spin particles simply aren't spinors or "spinorial in nature" - unless you use some weird non-standard GA terminology. YohanN7 (talk) 13:52, 7 December 2014 (UTC)

A spin j representation of the rotation group, whether j is integral or not, can be written as a symmetrized product of 2j spin 1⁄2-representations. See Van der Waerden notation or (better) Landau & Lifshitz Quantum Mechanics and Quantum Electrodynamics. This is an example of taking tensor products of spin representations and extracting irreducible modules with various, sometimes integer, spin. For instance, relativistic fields belonging to the (A, B)-representation of the Lorentz group are written with 2A two-valued (1⁄2, 0) indices (undotted) and 2B two-valued (0, 1⁄2) indices (dotted). This does not make the resulting space a space of spinors in the sense of normal usage of the word, and especially not in the sense of the spinor article. I don't know if this is precisely what the Carmeli and Malin are doing (have only parsed it quickly, seems to be a nice book), but they are surely doing something to the same effect. Well, either that or they say that the symmetry groups of physics are SU(2) and SL(2, C) instead of SO(3) and SO(3, 1), which is permissible (see Weinberg p 90.) provided you modify quantum mechanics accordingly. This still doesn't lead to spinors in the sense of Wikipedia. It is the other way around. The SU(2) and SL(2, C) are simply connected and have as such no goofy projective (spin) representations at all. They are called spin representations, and their elements spinors, because the groups in the guise of being universal covers of SO(3) and SO(3, 1) are called Spin(3) and Spin(3, 1). While your edits didn't say that spinors are tensors, they did say that tensors are spinors (In general, the wavefunction for a particle with spin is a spinor.), which is misleading, so you @Maschen: can hardly blame me and be in a bad mood because I reversed those changes. YohanN7 (talk) 11:53, 9 December 2014 (UTC)

What now?

I was not bitter/angry/whatever about "your reversion" and never will, I said "I'm neutral on how we do this". If anything the original wording (by me) was "The wave function is spinorial for fermions, namely particles with half-integer spin (1/2, 3/2, 5/2, ...), or tensorial for bosons, particles with integer spin (0, 1, 2, 3, ...).". But never mind. M∧Ŝc²ħεИ_τlk 19:39, 9 December 2014 (UTC)

Well, you said you didn't want to discuss it if I thought it was incorrect, which is kind of strange because In general, the wavefunction for a particle with spin is a spinor is a pretty wild claim (and unheard of) to put into the article I have nothing against your original formulation of above, that could have been kept unless our friend the nb-hater had come in between.

By the way, the "dotted-undotted" formalism is widely used and badly covered in Wikipedia. Van der Waerden notation could be extended substantially with stuff from Carmeli & Malin, and L & L. YohanN7 (talk) 20:32, 9 December 2014 (UTC)

Born's conditions

This website refers to the 'requirements' as "Born's Conditions". I'm not sure if that is a common term. RJFJR (talk) 17:33, 17 December 2014 (UTC)

Wasn't it Born that came up with the probability density interpretation? In that case, the square-integrability could be attributed to him, but referring to the whole package to Born's conditions would require a reference. YohanN7 (talk) 05:36, 18 December 2014 (UTC)

I am not a physicist, and cannot vote here; but anyway, I think it is a fuss, to formulate, once and for all, conditions on the wave function.

If we do not idealize the conditions, then everything is smooth (infinitely differentiable, at least). Though, we always idealize, at least a little, since otherwise we have no quantum mechanics (but rather quantum field theory, or even worse) and therefore no wave function.

The truth is, the Hilbert space of state vectors, and operators ("observables") on it, rather than the wave function and its values at points.

The Hilbert space is complete; thus all functions of L₂ must be here.

Dealing with bounded operators we can use all vectors.

But many observables are unbounded operators; their domains are dense in the Hilbert space, but not the whole space. This is the origin of restrictions. And it is hardly a good idea, to fix once and for all the "right" collection of "physically natural" observables and formulate the conditions accordingly. Relevant observables depend on context, and the same holds for conditions.

Boris Tsirelson (talk) 07:26, 18 December 2014 (UTC)

You are of course right. The problem is that the description (in the article) is all over the undergraduate literature. (I had similar concerns with the "inner product" as defined in the article. Those concerns ended up with an "nb" in the text.) Perhaps we should, in this case, focus more on what is actually correct than what is most often presented in the literature. I think so because this is a central article. The "truth" can of course be referenced too, but the difficulty of the article increases with its rigor. Perhaps there is a middle path. One could relegate the technical details to an "nb", and edit the text to talk about "physically realizable" wave functions. Physicists often talk about two Hilbert spaces, Hilbert space and physical Hilbert space. The latter isn't always a Hilbert space AFAIK, e.g. superpositions of states of half-integer and integer spin are assumed not to exist (hence not even a vector space). Just a thought. YohanN7 (talk) 11:46, 18 December 2014 (UTC)

About "physical Hilbert space", it seems, you mean superselection. Yes, indeed, we should not add vectors from different superselection sectors. We also do not need such sums, since anyway, observables never intermix these sectors. All my arguments should be applied only within a sector. But still, a sector is a Hilbert space. It is complete. It must contain non-smooth functions, etc. Boris Tsirelson (talk) 12:21, 18 December 2014 (UTC)

Yes, but my point is that we could perhaps retain the "requirements" as describing physically realizable states that have a reasonable interpretation, while leaving Hilbert space, with all its states, intact (complete). It does not represent my opinion (I don't have one), nor does it represent logical arguments, it is just a though, since the formulations exist in the literature. YohanN7 (talk) 13:03, 18 December 2014 (UTC)

Moreover, there is a kind of complementarity here. The less idealized observables, the more state vectors. Position measuring devices cannot have infinite resolution. Moreover, if not so much idealized, they are described by smooth functions (on the space) rather than points or even (smoothly bounded) domains. And if so, then the corresponding restrictions on the wave function disappear! No more values at points, and no more continuity. Just an equivalence class... The same holds for momentum observables, and whatever. (Is this my OR or POV?) Boris Tsirelson (talk) 08:00, 18 December 2014 (UTC)

Probably, the most "universally relevant" observables are Hamiltonians. Free-particle Hamiltonians. Interaction Hamiltonians (two-particle interactions, and many-particle interactions if needed). Ultimately, every measuring device must be a combination of particles, interacting with the object via these Hamiltonians. Though, in practice, classical external fields are a very convenient idealization. Boris Tsirelson (talk) 08:28, 18 December 2014 (UTC)

Now it may seem that a wave function must belong to the domain of the Hamiltonian(s). But wait; the Hamiltonian is nothing but the generator of dynamics. The self-adjoint generator of the one-parameter unitary group of evolution operators. Being continuous, evolution operators are well-defined on the whole Hilbert space. If a vector belongs to the domain of the Hamiltonian, then its time evolution is a classical solution of the Schrodinger equation. Otherwise it is not; so what? it is a generalized solution (the limit of a sequence of classical solutions, convergent in the metric of the Hilbert space).

In this sense, ultimately, a wave function is an arbitrary element of the relevant Hilbert space. Wow! Boris Tsirelson (talk) 09:51, 18 December 2014 (UTC)

True, the expected value of coordinate, momentum, energy, or any other unbounded variable is ill-defined for arbitrary state vector. The reason is rather mathematical, not physical: generally, the expectation is ill-defined for an arbitrary probability distribution. True, we like to calculate various expected values; and in order to do this, we need some conditions. However, we never observe an expected value in an experiment (just because it is too sensitive to the tail of the distribution). Remarkably, in quantum theory, everything empirically available depends continuously on the state vector. Not only evolution operators (being unitary) are continuous; but also the probability for a measurement outcome to belong to a given interval is continuous (being always of the form $\langle A\psi ,\psi \rangle$ with operator A satisfying $0\leq A\leq 1$, therefore bounded). Boris Tsirelson (talk) 11:04, 18 December 2014 (UTC)

undue weight given to Nature paper by Pusey Barrett Rudolph 2012 in 'Ontology' section

I think the present account of the 2012 Nature paper by Pusey Barrett Rudolph gives it undue weight. In a brief section such as the current one headed Ontology, it is inappropriately recherché. The paper is, charitably read on its merits, of marginal value. And not suitable for an account, even a one-sentence one, appearing here. Editor Sbyrnes very kindly gave an interpretation of it. While admirable, that interpretation seems more original research or editorial commentary than immediate reporting of the source. I think the felt call for such interpretive explication would be better responded to by simply removing the present account of the paper.Chjoaygame (talk) 16:08, 6 December 2014 (UTC)

I assume you are talking about this: On the reality of the quantum state

While the paper is authentic, I can't say whether it has any value or is worth mentioning here. The sentence in question has been around in the article since (too!) shortly after the publication. You are probably right. YohanN7 (talk) 16:53, 6 December 2014 (UTC)

I think that the paper is quite explicit that it is only talking about pure states and I think it is straightforward paraphrasing rather than commentary to say that the paper proves "whenever two observers both think that a system is in a pure quantum state, they will always agree on exactly what state it is in". The other part "(but this may not be true if one or both of them thinks the system is in a mixed state)" is definitely editorial the way it's written now. But it could be fixed by moving the citation:

...whenever two observers both think that a system is in a pure quantum state, they will always agree on exactly what state it is in [26] (but this may not be true if one or both of them thinks the system is in a mixed state [??])

where [??] is a different reference, probably to any decent description of EPR and/or mixed states in a QM textbook.

I think this is valuable information for readers, and uncontroversial, and that it can be properly cited even if it isn't properly cited right now. So I vote against the option of deleting it altogether.

It would also help if, instead of just citing Pusey et al. for the first clause, we also cite a textbook or two. I do think that this fact about pure states has been well-known and uncontroversial since the early 20th century, even though Pusey et al. try to make it sound like new progress. --Steve (talk) 14:50, 7 December 2014 (UTC)

A reference appeared in the form of a pdf from the same user that put in the original disputed piece of text (very shortly after the referenced paper was published). On closer inspection, this was the only non-peer reviewed paper appearing in the article (until the pdf). I therefore removed the text. We should require a doi at the very least for this article. Otherwise it will give a rather silly impression when the next reference is Einstein. YohanN7 (talk) 23:27, 1 January 2015 (UTC)

It may be of interest to glance at this. Also at this.

For me here, the question is not about the validity or admirability or notability or precision of citation of the Pusey Barrett Rudolph paper. It's about its place in the present context, or elsewhere.

The words 'ontology' and 'ontic' are relevant here, but they are not the commonest words in the quantum physicists' vocabulary. In my search, they are not to be found in Steven Weinberg's Lectures on Quantum Mechanics. In the Wikipedia article Interpretations of quantum mechanics, the word 'ontology' appears once and the word 'ontic' seven times. In this particular article, for this particular section headed Ontology, what level of detail is appropriate?Chjoaygame (talk) 01:29, 2 January 2015 (UTC)

You are right, there may be two issues here. I reverted (+ deleted some) solely on the grounds that the papers lacked weight based on the observations that

• Every other reference in this article is peer-reviewed or a published well-known book
• The reference (Pusey Barrett Rudolph paper) in question came into the article very quickly after publication.
• The same person now tries to amend even pdf files together with speculative results (that may or may not be found in the pdf)

Papers from arxiv may or may not generally be appropriate for an article. It depends on the article and the level of potential controversy. In this case, the paper fails in both respects imo (not peer-reviewed, at least no DOI, and dealing with a historically controversial and still debated subject). We really should set a bar here. YohanN7 (talk) 01:50, 2 January 2015 (UTC)

As far as I can see, the paper was published in Nature, as linked above, but the citation in the article does not give that link; instead it links to arXiv.

The question that concerns me is not about the virtues or citation details of the paper. It is about its place in this section of this article, as opposed to perhaps some more appropriate place?Chjoaygame (talk) 01:57, 2 January 2015 (UTC)

(ec) Copy from above::It may be of interest to glance at this. Also at this.

Okay, the second link gives a doi; doi:10.1038/nphys2309. The question remains, is it notable? PhD students usually do not have their own Wikipedia pages for that matter, unless they are truly outstanding or simply get lucky and slip through the net. For me this is still about notability. YohanN7 (talk) 02:03, 2 January 2015 (UTC)

For me, I at present wish to avoid commenting on the article's notability for Wikipedia in general at large. For the present context in the present section of the present article, I think it is not notable.Chjoaygame (talk) 02:12, 2 January 2015 (UTC)

I understand. Notability criteria are such that even those without knowledge can assess notability. Original research is of course not allowed. Then there are, I think they call it that, primary sources (we quote original research published elsewhere in peer-reviewed journals). These should to the extent possible be avoided. Whenever the primary sources itself gets quoted in the literature (preferably textbooks), then you might get something notable. What we have had in the article was a primary source whose notability was unknown. Maybe I'm too quick to judge, but it seems peculiar how quickly after (before?) publication it appeared here. YohanN7 (talk) 02:46, 2 January 2015 (UTC)

Immediately after publication comes press releases and newspaper stories. Somebody (probably not a subject expert) read one of those stories and then added a sentence describing this paper to this article. This kind of thing happens very frequently on wikipedia, and is unfortunate but not unusual or "peculiar". When I saw that new addition, I thought about deleting the sentence, but instead decided to edit it to describe the paper more accurately.

I really don't like the PBR paper. I think that at its core, it describes and justifies a true fact -- if a system is in a pure quantum state then there can be no subjectivity about which pure state it is in (apart from a phase factor) -- but they describe the fact in a misleading way, and paint the fact as controversial when nobody (AFAICT) has ever denied it. And I don't believe for a second that PBR are the first people to state or justify this fact. On the contrary, this fact is implicit in every description of quantum mechanics I've ever seen, and I bet that it's explicitly stated and justified in some textbook or other ... but I couldn't find any examples in my 5 minute search. :-P

Anyway if I ever find a better older reference then I will put the sentence back, until then it's OK with me that it's deleted. --Steve (talk) 19:34, 2 January 2015 (UTC)

I am glad to see the above comment by respected Editor Sbyrnes321. I think he is rightly concerned to put into the article a very important and valuable thing, indeed I think one of the keys to quantum physics. But exactly what and how has been the problem. With respect, I would like to try to help by making a suggestion. I think the key idea that Sbyrnes321 is concerned to put in is that wave functions can be pure or mixed, and that decomposition of a given mixed wave function into components is not unique, while a pure case is already in a sense fully decomposed so that it cannot be further decomposed. I think that putting the Pusey Barrett Rudolph 2012 paper into a section entitled Ontology is a poor means to that good end. Perhaps a good means to that end would be a section entitled Pure and mixed wave functions.Chjoaygame (talk) 22:55, 2 January 2015 (UTC)

I have had a shot at it. I will not feel offended by further edits.Chjoaygame (talk) 03:32, 3 January 2015 (UTC)

Pure and mixed states as refered to in Puseys paper are defined mathematically, see Quantum entanglement#Pure states. Your section demonstrate more the superposition principle. This can be done (and is done) in fewer words. YohanN7 (talk) 04:57, 3 January 2015 (UTC)

I know a mixed state, but what the hell is a "mixed wave function"?? Boris Tsirelson (talk) 06:45, 3 January 2015 (UTC)

Even after not referring to "mixed wave functions", the described experiment does not enlighten pure and mixed states as described in quantum state. The experiment yields the same result when the intensity is turned down so that no entanglement is even possible, i.e. involving pure states only.YohanN7 (talk) 08:11, 3 January 2015 (UTC)

Thank you for these valuable comments. I have responded accordingly.Chjoaygame (talk) 08:31, 3 January 2015 (UTC)

I find your edits very confusing. I have read dozens of QM textbooks and still cannot figure out what you mean by "quantum analyzers". I doubt that a typical reader of this article (who is new to QM) will be able to make any sense of it either. I have never seen this sort of definition for "wavefunction". I think it was better before. --Steve (talk) 16:58, 3 January 2015 (UTC)

thank you for this comment

Thank you for this comment. It is indeed a problem for a newcomer to make sense of articles on such matters. In particular you say that my term quantum analyzer is inscrutable, even to an expert. My term is intended as an ordinary language version of the QM term of art "measurement". The latter term in ordinary language as I read it refers to a combination of a quantum analyzer and particle detector. The Qm term of art "measurement" is in my opinion very definitely a term of art, not part of the ordinary language, even of the ordinary language of science, indeed even of the ordinary language of physics. The particle detector part is needed to produce numbers that can be tested against the probabilistic predictions of the mathematical theory. I guess that is not a worry to you.

But indeed the term quantum analyzer may be criticized. An ordinary language word is needed that labels the class of physical arrangements or devices that split the beam into the respective sub-beams that are fed to the particle detectors. The classic example is Newton's prism by which he split a daylight beam into coloured beams. The next classic example is perhaps the Stern-Gerlach magnet that splits the beam into two sub-beams. Another example is a calcite crystal that splits a beam into two polarized sub-beams. Another example is a half-silvered mirror. In general, a quantum analyzer analyzes beams of quantal entities or particles into sub-beams of quantum eigenstates. It is of interest that you have read dozens of QM textbooks and still cannot figure out what I mean here.

It is not too easy to define a wave function in physical terms. Since this is an article on quantum physics, I think it appropriate to try to do it. According to Dirac (4th edition), there should be a one-to-one correspondence between a mathematical formalism and a collection of experimental items. Dirac starts his definition of a wave function by saying that it is superposable (Section 5, pp. 14-18). This is only the first of a list of defining criteria. The next item on his list of definitive criteria (Secction 10, pp. 36-38) is the mathematical operators and their corresponding "measurements". Dirac writes "Any result of a measurement of a real dynamical variable is one of its eigenvalues." Further, "The question now presents itself - Can every observable [mathematical operator] be measured? The answer is theoretically yes. In practice it may be very awkward, or perhaps even beyond the ingenuity of the experimenter, to devise an apparatus which could measure some particular observable, but theory always allows one to imagine that the measurement can be made." Further, "It is often very difficult to decide mathematically whether a particular real dynamical variable satisfies the condition for being an observable or not, ... However ... good reason on experimental grounds ... even though mathematical proof may be missing." Both superposition and existence of eigenstates are needed for the definition of the wave function, as in the next paragraph.

Perhaps an unusual aspect of my definition is that it tries to fully obey the injunction of Niels Bohr, which he used to defeat the EPR paper. The very definition of the physical objects necessarily includes the method of measurement. The physical eigenstates, that are discriminated by the quantum analyzer, correspond one-to-one with the mathematical basis. The mathematical version of this is that the wave function must be defined with respect to some basis of the relevant Hilbert space. I have preferred the physical approach in an article on physics, especially in the light of Dirac's warnings about the mathematical concepts. Most texts on this subject start with a mathematical chapter or two, mainly on Hilbert spaces, before embarking on a definition. I think a physical approach is more appropriate for Wikipedia, which cannot easily make the reader do his Hilbert space homework. I think the texts assume that the reader obviously knows that the vector space basis corresponds with the physical eigenstates. Apart from a physical basis, the QM state, according to Bohr, is undefined. Like "measurement", a QM 'state' is a term of art, not part of the ordinary language. A classical state belongs to the particle apart from all else. That is why the classical state concept is misleading for QM. It is a regrettable relic of habitual classical thinking to presume that a QM 'state' belongs to the particle apart from the permitted set of operators and their associated eigenstate basis defined by the quantum analyzer. I have tried to tell that to the reader in physical terms.Chjoaygame (talk) 18:21, 3 January 2015 (UTC)

Perhaps I should add that my thinking was significantly guided by page 29 of Julian Schwinger's (2001) Quantum Mechanics: Symbolism of Atomic Measurements, edited by B.-G. Englert, Springer, ISBN 3-540-41408-8. He writes

"In checking out my impression of undergraduate courses, I happened to glance through a particular elementary textbook and found this statement:

The laws of quantum mechanics cannot be derived, any more than can Newton's laws or Maxwell's equations. Ideally, however, one might hope that these laws could be deduced, more or less directly, as the simplest logical consequence of some well-selected set of experiments. Unfortunately, the quantum mechanical description of nature is too abstract to make this possible.

"Despite the last pessimistic assertion, I propose to present just such an ideal induction (the more accurate term) of the general laws of quantum mechanics from a well-selected set of experiments — indeed from a single type of experiments."

Schwinger goes on to work directly with Stern-Gerlach magnets without mention of Hilbert space and so forth. I followed that lead because this is an article about physics.Chjoaygame (talk) 19:36, 3 January 2015 (UTC)

Englert, in his own Lectures on Quantum Mechanics in three short volumes, volume 1, Basic Matters, ISBN 981-256-790-9, uses this approach of Schwinger to introduce the ideas. I think it is pretty close to how Feynman introduces the ideas too.Chjoaygame (talk) 20:30, 3 January 2015 (UTC)