Talk:Wave function

From Wikipedia, the free encyclopedia
Jump to: navigation, search
          This article is of interest to the following WikiProjects:
WikiProject Physics (Rated C-class, Top-importance)
WikiProject icon This article is within the scope of WikiProject Physics, a collaborative effort to improve the coverage of Physics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
C-Class article C  This article has been rated as C-Class on the project's quality scale.
 Top  This article has been rated as Top-importance on the project's importance scale.
 
WikiProject Chemistry (Rated C-class, Mid-importance)
WikiProject icon This article is within the scope of WikiProject Chemistry, a collaborative effort to improve the coverage of chemistry on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
C-Class article C  This article has been rated as C-Class on the project's quality scale.
 Mid  This article has been rated as Mid-importance on the project's importance scale.
 

"wave function" vs "quantum state" ?, domain and codomain?[edit]

The lead sentence of the article say that a "wave function" represents "the" quantum state of a system. It would be better to say that a wave function represents a "type" or "kind" of "quantum state".

A "quantum state" is represented by a vector in a Hilbert space (according to the current Wikipedia article on "quantum state). Functions can certaily be vectors in a Hilbert space of functions, but my understanding is that not all "quantum states" are wave functions. The term wave function only applies when variables in the domain of the function include position, momentum, and time. Other sorts of functions and vectors can be "quantum states", but they are not called wave functions.

To a mathematician, if someone says "I have a function", the question "What are its domain and codomain" should have a completely straightforward answer. It would be nice to have a concise and precise statement of the domain and codomain of a wave function (or an admission that "wave function" refers to a jumble of different sorts of functions!). I would infer from the current article that the domain is a finite dimensional vector space over the real numbers and that the codomain is a finite dimensional vector space over the complex numbers.

Those accustomed to thinking of "states" in terms of classical physics, tend to think of a "state" as the condition of a system at a particular instant of time. it would be useful to point out that the "quantum state" represented by the wave function contains information about the current and future condition of the system as well as information about its past. The information about the future is valid only so long as the system is not disturbed. The information about the past is only valid back to the time when the wave function was created by some disturbance to a previous wave function.

Tashiro (talk) 16:42, 11 January 2015 (UTC)

It's not unusual to "isolate", say, the spin part of a wave function and call it the spin wave function. In this case the domain is a finite set and the range is ℂ. Formally one can proceed using tensor products. The same applies for other degrees of freedom. Stuff like this could go into a footnote (of the "visible" kind). This is what "n-tuples of complex numbers" hide for good and for bad. YohanN7 (talk) 17:36, 11 January 2015 (UTC)
Sorry to Tashiro for running off-topic, I split my previous post into the next section.
I thought "quantum state" refers to a collection of variables that characterize the (quantum) system, which involves coordinates or momenta and quantum numbers.
In any case "wave function" refers to the actual quantity solved from the Schrödinger equation. Since the wavefunction is a function of position or momentum, and the set of all wavefunctions for a particlar system can be conveniently enumerated by quantum numbers, the wavefunction can be thought of as "describing" the quantum state.
For this article, "the quantum state" instead of "a quantum state" is used since "the" refers to the system in question. For now I don't know how to change the wording... M∧Ŝc2ħεИτlk 23:13, 11 January 2015 (UTC)
Please be kind to mathematicians! If there is ambiguity or variety in the domain or codomain of "the" wave function then let this be pointed out in the article, even if it is not discussed in detail. (For example, if a wave function is a "function of position and momentum", does it become a function of time? Is it a function of time via the fact that both position and momentum are functions of time? Or is time an independent variable?) From a mathematical point of view, its baffling that there is an important function, but no statement (perhaps no agreement?) about the domain and codomain of the function. Tashiro (talk) 17:27, 12 January 2015 (UTC)
As you can see below, we intend to clarify the domain and codomain throughout, but in general it is a complex-valued function of all the position coordinates of all the particles (or momentum components), and time, and the spin projection quantum number for each particle along some direction. To answer your question about time, it is an independent variable in the wavefunction along with the position coordinates (or momentum components), and the observables (position or momentum) do not depend on time. This is the Schrödinger picture, the main theme of the article. You can move the time dependence around, see Dynamical pictures (quantum mechanics). Hope this helps, M∧Ŝc2ħεИτlk 17:55, 12 January 2015 (UTC)
Caveates: (To maybe remember when editing article)
  • There may be more than one discrete variable (isospin...).
  • Sometimes the discrete variables are ignored, and then the wave function is truly vector-valued as a function of configuration space (see below). This is mostly a notational matter, but potentially extremely confusing.
  • Equally potentially confusing is that x, y, z shouldn't be taken as points in space ⊂ spacetime, at least not when several particles are involved. The domain can generally be thought of (now ignoring discrete variables) as tensor products (Cartesian product works too AFAIK) of an appropriate number n of space, not spacetime, together with one copy of time. (This is configuration space.) Likewise for momentum, the latter wave function is just a 3n-dimensional Fourier transform.
  • For some applications the discrete variables are ignored as an approximation. (E.g. the usual Schrödinger equation for spin 12 particles.) Then the result is a scalar (complex valued).
  • Other domains are possible, the appropriate ones are related to the original ones by canonical transfromations.
YohanN7 (talk) 18:20, 12 January 2015 (UTC)
In other words: There is a Hilbert space of allowable states, and a wavefunction is a point in that space. But the space is different for different systems. For one spin-0 particle in 3D space, the space is (something like) L^2(\mathbb{R}^3) (see Lp space, but actually it may be a different function space than L2, I don't remember). For three spin-0 particles in 2D space, the space is (something like) L^2(\mathbb{R}^2) \otimes L^2(\mathbb{R}^2) \otimes L^2(\mathbb{R}^2) where \otimes is tensor product. For two spin-7/2 particle in 3D space, it's (L^2(\mathbb{R}^3) \otimes \mathbb{C}^{7}) \otimes (L^2(\mathbb{R}^3) \otimes \mathbb{C}^{7}). The term "wavefunction" is more-or-less a synonym of "pure quantum state", except that you don't normally call it a "wavefunction" if the Hilbert space is finite-dimensional. --Steve (talk) 20:17, 12 January 2015 (UTC)
I see that the term wave function is widely and perhaps variously used. I think some effort should go into surveying the possible sources and reporting the best.Chjoaygame (talk) 11:53, 13 January 2015 (UTC)
In the commonest or default usage, think a wave function is a solution of the Schrödinger equation for its defining Hamiltonian, with domain the Cartesian product of quantum configuration space and a time interval, and range the complex numbers. This ensures that it represents a pure state of a quantum system, and other goodies. (By the way, the Wikipedia articles that one would expect to provide definitions of the latter are respectively in need of repair and appalling. It is very far from obvious that this article should refer to them until they have been put into good shape.) Quantum configuration space has various forms and manifestations. I think it best to define the wave function simply in general terms as having it as domain. The specification of the quantum configuration space, in its own right, deserves a paragraph, section or article, and I think should, for clarity, be well separated from the definition of the wave function.
Perhaps some more general dynamical specification than 'a solution of the Schrödinger equation' may be mentioned. The Schrödinger equation is one way of stating quantum dynamics, but there are others. A wave function must be defined with respect to specified dynamics.
The wave function can be considered more abstractly, as a mathematical entity in its own right, not as a function as such. Then it can be viewed as a point in an abstract vector space (a function space as it happens, but that isn't the present focus of interest right here). That abstract vector space happens to have the structure of a Hilbert space, give or take some more details about Hilbert spaces. In the sense that it is a point in a vector space, one can call it a vector. There are other mathematical representations of quantum states, ways which go more directly to the view that they are points in an abstract vector space, without going into their structure as wave functions. I think it may be useful to say that this article will not call those other modes of representation by the name 'wave function', though often enough one encounters the latter usage in a loose way.
Configuration space for an n-particle system has 3n kinematic degrees of freedom or dimensions, which may be various, but are usually specified by 3n real numbers that have meanings pretty much the same as for classical mechanics, respectively three per particle, and it also has for each particle a spin degree of freedom, which is in general a spinor, but may be presented in a less general way. A spinor is a kind of object not encountered in classical mechanics, and may be represented in various ways. Configuration space is subject to transformations, which I suppose deserve an article of their own.Chjoaygame (talk) 11:53, 13 January 2015 (UTC)
Don't think we need to go very deep to sort out what domain and codomain is for a function. The literature also doesn't tell the obvious in each case. YohanN7 (talk) 12:45, 13 January 2015 (UTC)
Classical physics admits two basic types of object, particle and wave/field. A particle system has a trajectory that is a path in configuration space, without a spin. A wave/field has a physically valued displacement at every point in ordinary space, and that for every instant of time. It is customary to say that waves diffract and particles don't, but that distinction is made obsolete by the discovery of quantal transfer of energy/momentum, even in the old quantum theory without quantum mechanics. Once one has quantal momentum transfer, particles diffract. Many standard texts prefer to hide this elementary fact, for sociological reasons. For physics, what makes a wave is that it has a physical displacement at every point in 'space' at each instant of time. What makes a particle is that it is all at just one point, leaving the rest of space empty, at each instant of time. No interpretation is needed to distinguish classical wave from particle. Diffraction has nothing to do with it.
A quantum mechanical system of particles is not like a classical wave, because it is not specified by a physical displacement at every point in ordinary space. It is specified by an abstract scarcely physical displacement at every point in configuration space at every instant of time. There is at face value no hint of particulate character. Neither classical particle nor classical wave. Talk of wave-particle duality is sociological. The quantum concept of 'wave' is pure interpretation. The "wave" is imagined as a blurring of the pattern of particle detections. At present the lead says "just like" a classical wave. That is inaccurate and misleading, and is in the article for sociological not physical reasons. Quantum mechanical contact with physical reality is by calculations that predict particle counts in suitably placed detectors.Chjoaygame (talk) 15:21, 13 January 2015 (UTC)
Chjoaygame -- Despite your quotation marks, the words "just like" are not in the lead. It says "The wave function behaves qualitatively like other waves", which I think is fair and helpful and accurate, in the sense that "behavior" includes things like refraction, diffraction, interference, etc. It doesn't say "The wave function behaves unlike particles in the old quantum theory", and it also doesn't say "The wave function is fundamentally like other waves". --Steve (talk) 19:54, 13 January 2015 (UTC)
Sbyrnes -- thank you for this correction. Yes, you are right, I should have checked, it's the article Matter wave that has the objectionable phrase "just like", not this one as I mistakenly wrote just above. Still I think the present "qualitatively like" is misleading, and that the reasons refraction, diffraction, interference, etc. are sociological not physical, for the reason I gave.Chjoaygame (talk) 21:24, 13 January 2015 (UTC)

break for ease of editingChjoaygame (talk) 22:09, 22 January 2015 (UTC)[edit]

Further confusion about the (or "a") domain and codomain of the wave function, is caused by the sentence "For a given system, the wave function is a complex-valued function of the systems degrees of freedom, continuous as well as discrete." This sentence has a link to the "Degrees of Freedom (mechanics)" Wikipedia article that says "degrees of freedom" is an integer that tells the number of state variables. By that definition "degrees of freedom" is not the set of state variables, it is the cardinality of that set. It would clearer to say that a wave function for a physical system is function of its state variables. However, that seems to be at odds with idea that the wave function represents the state. Do the state variables represent the state? If so, why is a function of the state variables needed to represent the state? Tashiro (talk) 18:33, 22 January 2015 (UTC)

Editor Tashiro, I think you are right to ask this question. Perhaps it is better that I do not expatiate on why I say that. But I may offer some pointers towards a partial answer.Chjoaygame (talk) 23:08, 22 January 2015 (UTC)
  • The wave function is defined with respect to a definite physical context. As a purely mathematical entity without physical context, it is either undefined or meaningless, as you please.Chjoaygame (talk) 23:08, 22 January 2015 (UTC)
  • Evidently here, the context is partly set by the words "For a given system". But those words are little more than a syntactic place filler. As far as I can see, we do not here have a useful definition for them. How is the system given? If the words are to contribute to setting the physical context, they must have physical content. Physical content necessarily involves empirical or experimental information. Physically, a system is given by describing, primarily in ordinary language, an experimental or observational activity. For example: "Place the telescope at the North Pole of the earth, and, at 00:00 UTC 31 Jan 2015, point it at the zenith. Take a photograph in the visible spectral range. Show us the photo, and tell us what you see in it." At least, something like that.Chjoaygame (talk) 23:08, 22 January 2015 (UTC)
  • Here, is the "system" specified by a one-off time interval of observation for a specified apparatus? Its result is a count from a detector. Perhaps the count is zero in that time interval. Or is the system specified somehow else? Etc..Chjoaygame (talk) 23:08, 22 January 2015 (UTC)
  • A wave function is a solution of the Schrödinger equation for a specified quantum Hamiltonian. The Schrödinger equation now needs to be defined. Let us write it as f(ψ) = 0. What kind of object is f ? What kind of object is ψ ? Chjoaygame (talk) 23:08, 22 January 2015 (UTC)
  • In order to define these terms, we need to recognize that quantum configuration space (the domain of ψ) depends on the system. This will determine the kind of operator that is needed to express f. Sometimes f can take scalar values, sometimes it has to take spinor values. Etc. That will determine the range of ψ.

I am asking questions here.Chjoaygame (talk) 23:08, 22 January 2015 (UTC)

Clarification on spin functions[edit]

Related to this the above thread - it would be nice to clarify about the decomposition of the wavefunction for a particle with spin into a product of a spin function and a space function, and explicate what the spin function is in this article. (Admittedly, I don't know. At uni we never ever once wrote a wavefunction "as a function of the spin quantum number", instead for spin states we just used braket notation and expressed them as column matrices, as is usual). Here is what seems to be correct...

This is really really really badly described in the literature, every book I have seen is so hand-wavy and unclear on this decomposition, or they just use braket notation.

Of course, feel free to complain on errors or bad presentation.

Aside: somewhere, maybe we could work in the phase factor for time-independent functions (in the Schrödinger picture)? I don't know... M∧Ŝc2ħεИτlk 19:03, 11 January 2015 (UTC)

Looks good, but I personally don't like the spin function being defined vector valued. This is not necessary, just organize its scalar values for every spin z projection in a column vector. But this is not important if what you wrote is from a reference. (Also, still don't like colons preceding equations (and still not talking about indentationFace-smile.svg)) YohanN7 (talk) 19:52, 11 January 2015 (UTC)
Looking closer, the spin dependence is hidden away in the ψ±1/2(x, t) (functions of coordinate space). Better to have
 \xi(s_z) = \left( \begin{matrix}\xi(1/2)\\ \xi(-1/2)\end{matrix}\right)
imo. Then you can have
 \xi_{1/2} = \left( \begin{matrix}1\\ 0\end{matrix}\right), \quad \xi_{-1/2} = \left( \begin{matrix}0\\ 1\end{matrix}\right)
as a complete set of basis functions. YohanN7 (talk) 20:07, 11 January 2015 (UTC)
Thanks for the reply. (Colons are just a habit - spelling, grammar, and punctuation can be fixed later).
The unfortunate thing is I have no reference to fall back on >_< . The books never say what the spin function is. They just write it as a function of the spin quantum number, but go on to just use the quantum number as an index to label components of a complex-valued vector.
To rephrase the confusion (likely not just for me): why write the wavefunction as a function of the spin quantum number, when all that's needed is to use the quantum number to label spin eigenstates? Well, the wavefunction is a function of all the system's degree's of freedom, but the spin dependence is not like the space or time coordinates.
I think you filled the gap very well by writing: Isn't it circular to write
 \xi(s_z) = \left( \begin{matrix}\xi(1/2)\\ \xi(-1/2)\end{matrix}\right)
since for a given sz, we have the corresponding component of the vector. ? When you substitute one value for sz, it looks meaningless like this
 \xi(1/2) = \left( \begin{matrix}\xi(1/2)\\ \xi(-1/2)\end{matrix}\right)\quad ?
One way or another it will be cleared up... Cheers anyway! ^_^ M∧Ŝc2ħεИτlk 23:13, 11 January 2015 (UTC)
You are right. Should have written
 \xi = \left( \begin{matrix}\xi(1/2)\\ \xi(-1/2)\end{matrix}\right),
but had my physicist hat on routinely confusing a function with a function given an argument. Math hat better for rigor here (though a real mathematician would still scream out loud with the latest version).
I understand the confusion. I have been there. The total wave function and in particular the spin function ξ really is a function of the spin quantum numbers. The spin quantum number also works as an index into a column vector. But then one should think about what an index is. From a bare-bones set theoretic approach it is a function from an indexing set, which in this case is just the set of spin quantum numbers. The difference between an index (or rather indexing function) is notational only. It is a function. There is usually one more difference in practice. The indexing set isn't really important. What matters is that the indexing set has the right cardinality. Only the range is important. Thus one can say that the spin wave function (time independent for simplicity here) is a function {−12, 12} → ℂ (which is an element of the function space {−12, 12}) or {down, up} → ℂ, both work. Once this is nailed down, one can just think of the spin quantum number as an index into a column vector and forget about it being a function. YohanN7 (talk) 00:02, 12 January 2015 (UTC)
The total wave function can then be thought of as Ψ: ℝ4 × {−12, 12} → ℂ, and I think it always factors (this requires a proof or a reference) as Ψ = ψ × ξ where ψ:ℝ4 → ℂ and ξ:{−12, 12} → ℂ, that is to say, the upper and lower components have the same spacetime dependence. YohanN7 (talk) 00:17, 12 January 2015 (UTC)
Whether or not the wavefunction factorizes, the mapping should probably be Ψ : ℝ4 × {−12, 12} → ℂ2, otherwise the wavefunction is not a 2d complex vector. The index set maps to the components of the vector, as you describe? If this is correct then for spin s the wavefunction is Ψ : ℝ4 × {−s, −s + 1, ..., s − 1, s} → ℂ2s + 1, again each spin quantum number maps to the components of the vector. M∧Ŝc2ħεИτlk 11:02, 12 January 2015 (UTC)
No, it definitely does not factor in general. (The first sentence in the box above, "A wavefunction for a particle with spin can be decomposed into a product of a space function and a spin function", is incorrect.) For example, a particle flying through a magnetic field can easily wind up in a superposition of (spin-up particle over here) + (spin-down particle over there). --Steve (talk) 00:22, 12 January 2015 (UTC)
You are right. I stand corrected. YohanN7 (talk) 00:31, 12 January 2015 (UTC)
Then what is the condition for the factorization? Whenever the position and spin are each affected by an external field like the magnetic field example, then it cannot be factorized, but what is the general condition?
Whether it factorizes or not, the set of allowed spin quantum numbers (for a given spin) as an index set seems helpful, not sure if we should add this to the article though... M∧Ŝc2ħεИτlk 00:55, 12 January 2015 (UTC)
At any rate, Ψ = ψ12 × ξ12 + ψ12 × ξ12, where the basis functions for the space of spin functions are used. YohanN7 (talk) 01:31, 12 January 2015 (UTC)
What is the × operation above? If you just mean complex-valued functions scalar-multiplying vectors like the above example (z-projection of spin):
\Psi=\psi_{1/2}(\mathbf{r},t)\begin{pmatrix}1\\
0
\end{pmatrix}+\psi_{-1/2}(\mathbf{r},t)\begin{pmatrix}0\\
1
\end{pmatrix}=\begin{pmatrix}\psi_{1/2}(\mathbf{r},t)\\
\psi_{-1/2}(\mathbf{r},t)
\end{pmatrix}
then yes, I agree.
When the factorization is possible is also badly described in the literature, but to a lesser extent. I think Landau & Lifshitz QM point out that in non-relativistic QM the wavefunction can always be factorized provided the particle is not in a field which influences both the position and spin of the particle - a magnetic field is an example, an electric field is a non-example, but don't have the book to hand now. It makes sense, since the non-relativistic SE for a potential that does not include a field coupled to the spin operators, the differential operators act on the space function and leaves the spin function separate. IMO this should be in the article if we agree. In RQM it is probably never possible. M∧Ŝc2ħεИτlk 10:30, 12 January 2015 (UTC)
No, I mean complex-valued functions multiplying each other in the range. The domain of each term is 4 × {−12, 12} and the range is for both factors in each term. Note that ξ±12 are functions. The result is (in terms of standard notation) exactly what you wrote above, and illustrates well how indexing functions can be confused by notation that hides that they are functions.
Disentangling what factorizes and what does not requires (besides examination of the Pauli equation for spin 12) references, and L&L is as good as any. I have it too, but can't get to it right now. There is also a passage in Shankar's book (my copy in the same place as my L&L) where he examines the dynamics of a spin 12 particle (at rest?) in a magnetic field exclusively using the a time-dependent spin wave function. YohanN7 (talk) 11:31, 12 January 2015 (UTC)
I don't have Shankar's book, but if he describes the magnetic field example well, then by all means add it.
Thanks, this is clearer, but it looks like you're using the Cartesian product in an expression for complex numbers. To summarize:
Ψ = ψ12 × ξ12 + ψ12 × ξ12
Ψ : ℝ4 × {−12, 12} → ℂ
and to be extremely sure... ψ±1/2:ℝ4 → ℂ and ξ±1/2:{−12, 12} → ℂ.
(Off-topic again but related, IMO all the "definitions" should have the mapping notation to explicate the domain and range so there is no ambiguity. I did include this years ago, but it was removed, and thought back then it was fine removed since it may have over-complicating things. But for the sake of a few unfamiliar symbols, it would be better to be rigorous). M∧Ŝc2ħεИτlk 12:24, 12 January 2015 (UTC)
100% right. The Cartesian product is unfortunate - juxtaposition is even more unfortunate. The tensor product symbol is misleading too. I'll try to find a standard notation for this sort of multiplication of functions, different domains, common range. YohanN7 (talk) 12:50, 12 January 2015 (UTC)
Sorry, but what throws me off are the terms ψ12 × ξ12 and ψ12 × ξ12.
Based on your description, each of the ψ and ξ are functions with their domains defined as you say and and codomains the complex numbers, fine.
But what are the terms with the × products? These are surely the components of the vector, whose domains are 4 × {−12, 12} as you say, but their codomains are 2 and I should have written Ψ : ℝ4 × {−12, 12} → ℂ2.
Am I getting or getting there, or too annoying? Face-smile.svg M∧Ŝc2ħεИτlk 12:48, 12 January 2015 (UTC)
Actually, you just went back to square one. The spin function is complex valued, not vector valued. It assumes a complex number for each value of the spin z-component. The vector thingie is just notation. YohanN7 (talk) 12:55, 12 January 2015 (UTC)
To see this beyond any doubt, plug in definite values for x, y, z, t, sz in the argument to Ψ. You get a complex number (or you really get it wrong). Then plug in x, y, z, t, −sz. You get a different complex number. If you want to, you can organize these two numbers in a 2 × 1 matrix (rectangular scheme), commonly confused with a vector. I don't know if these descriptions help, but I don't know how to communicate it otherwise. YohanN7 (talk) 13:00, 12 January 2015 (UTC)
This might be useful: The Ψ viewed as a function of spacetime only is indeed vector valued. YohanN7 (talk) 13:10, 12 January 2015 (UTC)
OK, I misinterpreted again. Now that you confirmed that Ψ is a complex number, it is clearer, but just because you use an (undefined) operation that makes sense to you, doesn't mean everyone else will know what it means.
Nevertheless, these descriptions help, so thanks again. Now we just need to update the article, which I'll try later today. M∧Ŝc2ħεИτlk 14:50, 12 January 2015 (UTC)
We just need to define the operation. If φ:A → ℂ, χ:B → ℂ then define φχ:A × B → ℂ; φχ(a, b) ≡ φ(a)χ(b). The only problem is references. As you have noted, the references suck badly (and besides, the introductory QM books I have aren't in my present location). But maybe L&L is partly available online? It should make clear at least what the spin wave function is. YohanN7 (talk) 15:31, 12 January 2015 (UTC)
L&L QM is available on internet archive if one dares to look. The copy on Google books will not have all the pages. M∧Ŝc2ħεИτlk 17:55, 12 January 2015 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── I honestly think a section in the article covering this clearly would be very helpful. Let us wait to see if further comments pop up here, an independent sanity check is always good before inserting OR into articles. I don't plan to add anything myself (too much to do in other articles), but I'll be happy to copy-edit. Also, since I think this is all clear (nowadays), I might not convey it clearly enough, even if I try. YohanN7 (talk) 15:44, 12 January 2015 (UTC)

Clarification in general[edit]

I think that a wave function can be viewed in two lights: as a ordinary function on an experimentally specified configuration space, or as an abstract point in a Hilbert space or a complex projective space or whatever.
On one hand, considering it for a particular experimental set-up, the wave function takes values in the complex plane. It is just a function appropriate to that set-up with the appropriate particular configuration space, with no thoughts about its abstract life in some abstract space. When the experimental set-up is changed, the configuration space may quite likely change according to a suitable transformation, and perhaps the Hamiltonian. A new wave function is needed for the new set-up. But the range (co-dimension) is still the complex plane.
On the other hand, when a wave function is considered as an abstract point in an abstract space, it can refer to a wide diversity of experimental set-ups with a corresponding diversity of appropriate particular configuration spaces, and is a much more abstract object than the wave function for a particular experimental set-up. In this abstract view, a transformation of the configuration space will induce a transformation on the abstract point. It is then neither its value that is being transformed, nor its structure as an ordinary function with a complex numbered co-domain; it is its home and citizenship as an abstract mathematical entity that is changed. The abstract space might be a complex projective space, it might be a vector space, even a Hilbert space, whatever. If it happens to be a vector space, for example a Hilbert space, then the transformed wave function will transform as a tensor or whatever under suitable conditions.
As I see it, these are two distinct stories. Failure to observe this kind of distinction is a source of vast reams of peer-reviewed academic literature that I consider would not provide reliable sourcing, to say the least. I think even generously funded research projects and careers are built on it. Maybe. I think there is so much of it that its sheer weight and prolixity give it legs.
As for OR, which is just above referred to thus: "an independent sanity check is always good before inserting OR into articles." Well, OR is forbidden, even for editors who are infallible and omniscient. It is part of the duties of an editor to produce good reliable sources. Above I read "This is really really really badly described in the literature, every book I have seen is so hand-wavy and unclear on this decomposition, or they just use braket notation." If so, it may not be easy to find good sources, but that is still part of the job.Chjoaygame (talk) 09:08, 14 January 2015 (UTC)
We realize references are important, but wouldn't you rather have a clear presentation than what the sources say (on the particular case of spin functions, and decomposition of a wave function into spin and space functions)? The above section is not about inserting OR, but clarify what the sources are saying (or should say), which is why it appears as OR.
About the "two lights", as in wave functions as elements of function spaces, or vectors (kets) in vector spaces, I have nothing to say, except these are presented in the article reasonably well as is. M∧Ŝc2ħεИτlk 11:02, 14 January 2015 (UTC)
What can I say? The rules of the game, Wikipedia editing, are binding for ordinary editors but not for those with PhDs in quantum field theory nor for those who are omniscient and infallible? I have a feeling that a longer reply here from me would not be useful.Chjoaygame (talk) 11:51, 14 January 2015 (UTC)
On further thought, perhaps I can make a useful reply here. It seems that my meaning was not conveyed by my just above comment that starts "I think that a wave function can be viewed in two lights".
In my comment I was distintuishing two viewpoints on the term wave function: (1) as a ordinary function on an experimentally specified configuration space; (2) as an abstract point in a Hilbert space or a complex projective space or whatever.
It seems that you have read me as intending to distinguish between wave functions: (a) as elements of function spaces; (b) as vectors (kets) in vector spaces. This was not the distinction I was intending to indicate. From my point of view the vector spaces (b) are pretty much the same thing as function spaces (a), endowed with vector properties. A function in a function space usually has special criteria for citizenship.
The objects to which I was pointing in (1) were not to be considered as elements of functions spaces at all. They were to be considered just as functions, without placing them as citizens in a function space. They come in diverse forms and do not respectively have common features that would put them easily as citizens in some specified function space. Their forms are as diverse as are possible experimental set-ups which they describe.
It seems that experiments on non-relativistic spinless particles can be described by wave functions just considered as functions in a general sense, as in (1), from a configuration space that is pretty much the same as a classical configuration space, into the complex plane. They are the sort of thing dealt with in Schrödinger's 1926 papers.
It seems that experiments that involve spin cannot be so simply described. One needs to go to more abstract things, such as kets, and such as points in a function space, or in a vector space.
Kets are not Schrödinger's 1926 wave functions. As I understand it, they were not clearly defined till Dirac's 1939 paper. The configuration space which is the domain of the ingredient functions is now inescapably non-classical because it has spin degrees of freedom. In 1924 Kronig showed that an electron with spin 1/2 would explain anomalies in the Zeeman effect, but Pauli was so scathing about this idea, that Kronig did not publish it. Nevertheless, the spin appeared in quantum mechanics in a purely formal way in Pauli's 1927 paper. Early quantum mechanics did have an algebraic version, called matrix mechanics, and Dirac's 1926 paper expressed an algebraic approach. Dirac in 1958 (4th edition, p. 80) explicitly distinguishes the terms 'wave function' and 'ket': "A further contraction may be made in the notation, namely to leave the symbol \rangle for the standard ket understood. A ket is then written simply as ψ(ξ), a function of the observables ξ. A function of the ξ 's used in this way to denote a ket is called a wave function.[Dirac's footnote: The reason for this name is that in the early days of quantum mechanics all the examples of these functions were of the form of waves. The name is not a descriptive one from the point of view of the modern general theory.] The system of notation provided by wave functions is the one usually used by most authors for calculations in quantum mechanics." (By the way, I actually met Dirac in person, long ago.) Weinberg's 2013 Lectures on Quantum Mechanics uses Dirac's distinction: "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. This is essentially the approach of Diracs's ″transformation theory″."<Weinberg, S. (2013). Lectures on Quantum Mechanics, Cambridge University Press, Cambridge UK, ISBN 978-1-107-02872-2, page xvi.> Landau and Lifshitz wait till page 188 to introduce spin into their quantum mechanics.
As for literature, on my shelves is a text, Zare, R.N. (1988), Angular Momentum: Understanding Spatial Aspects in Chemistry and Physics, Wiley, New York, ISBN0-471-85892-7. Zare recommends earlier works: Rose, M.E. (1957), Elementary Theory of Angular Momentum, Wiley, New York; Edmonds, A.R. (1957), Angular Momentum in Quantum Mechanics Princeton University Press, Princeton NJ; Brink, D.M., Satchler, (1962), Angular Momentum, Clarendon Press, Oxford UK.
I read in a 2001 text by Elliot Leader as follows
"For a free particle the spin degree of freedom is totally decoupled from the usual kinematic degrees of freedom, and this fact is implemented by writing the state vector in the form of a product, one factor referring to the usual degrees of freedom and the other to the spin degree of freedom. Thus for a particle of momentum \mathbf p,
|\mathbf p ; sm \rangle= |\mathbf p\rangle \otimes |sm \rangle
or, equivalently, for the wave function,
\psi_{\mathbf {p} ; sm} (\mathbf x)=\phi_{\mathbf {p}} (\mathbf x)\eta_{(m)}
where \eta_{(m)} is a (2s+1) -component spinor and \phi_{\mathbf {p}} (\mathbf x) is a standard Schrödinger wave function."<Leader, E. (2001), Spin in Particle Physics, Cambridge University Press, Cambridge UK, ISBN 0-521-35281-9, p. 2.> Needless to say, the text continues at length from here.
If the reader of Wikipedia wants to know about the more abstract approaches, in terms of kets, or in terms of Hilbert spaces, there are Wikipedia articles entitled Bra–ket notation and Spin (physics). The present article is entitled Wave function.
Between wave functions and kets, we are looking at a significant step of level of abstraction beyond the 1926 Schrödinger account. Many controversies about quantum mechanics fail to recognize the step, and I think they are in consequence a waste of time. I think the newcomer Wikipedia reader deserves an explicit heads-up about this further step of abstraction. I think it would confuse the average reader to conflate wave functions with kets or Hilbert space vectors.Chjoaygame (talk) 09:32, 15 January 2015 (UTC)
It is not clear if you want to exclude Hilbert spaces and the use of bra-ket notation (your point (2) above), but in any case they must stay because they are needed in the formulation of wave functions, even if there are entire articles on them. While I appreciate the references (which you are welcome to add), it is also not entirely clear if you want the article to be Schrödinger's original formulation, your point (1) above. It would help if your posts were shorter, that's a wall of text.
A ket (including tensor products of them) may actually be referred to as a "wave function" as well. It took a long time to carefully relate the bra-ket notation to the functional analysis approach in this article, so it should be kept in, the main article on the notation has more details and generality. M∧Ŝc2ħεИτlk 13:27, 15 January 2015 (UTC)
It's is probably not a good idea to look very deeply into each and every QM book (needless to say, papers from the 1920's) for mathematical descriptions the abstract Hilbert space of QM and related Hilbert spaces. They will sometimes be whimsical - and never precise. As far as I am concerned, Dirac's bra-ket notation is just that. Notation. The abstract Hilbert space is certainly needed. How else could we say that the same state has many wave function representations. YohanN7 (talk) 13:39, 15 January 2015 (UTC)
Perhaps "A ket (including tensor products of them) may actually be referred to as a "wave function" as well." Perhaps "This is really really really badly described in the literature, every book I have seen is so hand-wavy and unclear on this decomposition, or they just use braket notation." Perhaps there are "whimsical and imprecise QM books". That some writers may be imprecise or conflative is not a reason for us to imitate them. Our job is to find and report sources that are precise and reliable. We should keep looking till we find such.
I think it would confuse the average reader to conflate the wave functions of Schrödinger with the kets of Dirac. That distinction is not merely notational. True, there is a notational difference. Nevertheless, the distinction is conceptual, as stated precisely above by Weinberg, and by Dirac, and by Leader, who I think are reliable. The distinction should be made explicitly.Chjoaygame (talk) 19:16, 15 January 2015 (UTC)
Have you read wave function#Wave functions as elements of an abstract vector space? What specific improvements would you make there? M∧Ŝc2ħεИτlk 19:33, 15 January 2015 (UTC)
  • The section to which you refer begins "The set of all possible wave functions (at any given time) forms an abstract mathematical vector space." I would qualify the phrase 'all possible'. All possible with respect to what range of possibilities? Unqualified, the phrase is vague to a point of near meaninglessness from a physical perspective. This article is about physics. A mathematician may dismiss concern about the physical meaning, but the term 'wave function' is primarily of interest for its physical meaning. The article Quantum state does a poor job of this and I think it unsafe for this article to rely on it.
Pretty much from the beginning of the article, the term wave function is treated as primarily referring to elements of a vector space, but only far down in the article is this concept clarified by the section to which you refer. Till then the reader is left to be mystified about it. So I would bring that section much earlier in the article.
I would, for the sake of the reader, in the newly early-placed section, make more explicit the distinction (1) versus (2) that I mention above. Vast clouds of "interpretive" drivel get wings through this distinction being ignored or slighted. For example, as I read him, Editor YohanN7 views it it as merely notational, that is to say, trivial, in effect unimportant. The distinction is in some ways like that between individuals, species, and genera. The Wikipedia reader should have somewhere to find this distinction made clearly enough for him to have a tool to begin to see through the clouds of drivel. I think this is a good place to supply that need. Moreover, with the distinction made clear, some sections of the article could be simplified.
I would also make the distinction clear in the lead.Chjoaygame (talk) 22:03, 15 January 2015 (UTC)
By conflation you probably mean
|\Psi\rangle = \langle x|\Psi\rangle = \Psi(x)?
The expressions
|\Psi\rangle, \langle x|\Psi\rangle = \Psi(x), \langle p|\Psi\rangle = \Psi(p),
all refer to the same state but belong in one sense to different Hilbert spaces. I think this 100% agrees with what both Weinberg and Dirac says. Perhaps I take this too lightheartedly and the article needs to be sharpened? On the other hand, this article has problems of its own and need no further burdens. Why not beef up Quantum state instead? YohanN7 (talk) 22:37, 15 January 2015 (UTC)
By conflation I mean that the phrases 'wave function' and 'element of a function space' are used more or less interchangeably. That would confuse a new reader. With respect, a mathematical formula is a mathematical formula, and a phrase is a phrase.
There is a school of thought that, for physics, when the prepared pure state of interest is changed, then the state is changed. A transformation is regarded as having a physical meaning. Niels Bohr used teraliters of ink saying so. In some places in Wikipedia, it is enough to undo an edit if it even might suggest a departure from the Copenhagen interpretation. Here, as far as I can see, the Copenhagen interpretation is a laughing matter. It is often said that Niels Bohr was one of the perpetrators (whoops, I mean fathers, creators, architects) of the Copenhagen interpretation. It may be considered as taking a point of view to dismiss him as a silly old fool. Of course he is right to have so dismissed that airhead Albert Einstein (would I dare hint otherwise?). Just for the record, I think that the phrase 'Copenhagen interpretation' is a source of confusion, as did Heisenberg.
As I read you, you have two hats, a mathematician's and a physicist's. With respect, there is also such a thing as a Wikipedia editor's hat.
I think the present article is confusingly constructed as I have just above indicated. I would think a mathematician would be concerned if an equals sign joined elements of different spaces. I am saying that it also has significance for physics.Chjoaygame (talk) 02:07, 16 January 2015 (UTC)
The article may be confusing, but nothing is as confusing as youFace-smile.svg. I don't understand one bit of what you are trying to say, except maybe that you are perhaps upset with something I wrote. I certainly don't see why you would be. YohanN7 (talk) 02:23, 16 January 2015 (UTC)
Perhaps we could take a break at this point.Chjoaygame (talk) 02:47, 16 January 2015 (UTC)
  • Looking back over the history of this article, I find this edit. It seems to mark a stage in the development of the article. Before it, the views of the wave function were pretty much as I indicated above with my distinction (1) vs. (2). The general approach admitted a rather concrete view of the wave function as a multivariable function, and a more abstract view in terms of function spaces. After it, the general approach became more abstract, and the more concrete view became clothed in abstract terms.
The electron probability density for the first few hydrogen atom electron orbitals shown as cross-sections. These orbitals form an orthonormal basis for the wave function of the electron. Different orbitals are depicted with different scale.
In the earlier stage, the reader had a good heads-up of the steps in degree of abstraction. In the later stage, the step had become draped in abstract garments, and the body underneath was less visible.
Looking back at the above talk page material, I see something similar. I labelled a distinction (a) vs. (b), "as in wave functions as elements of function spaces, or vectors (kets) in vector spaces". I see this distinction as coming from a more abstract approach than the one of the earlier stage of the article. The earlier approach admitted a more concrete view of the wave function, which I have above called 'the Schrödinger wave function'. One pictures such a concrete view as in a figure that is currently in the article. The functions illustrated in that figure can be viewed as elements of a function space, but that figure does not illustrate them as points in that space. More concretely, it illustrates them as densities in a physical space. I suggest moving that figure up into the section headed Wave functions and function spaces. Moreover, I don't think it helps at this stage to talk about function spaces at all. The notion of function space begins to be pedagogically relevant with the introduction of the inner product, which is further down in the article. The comment about Sobolev spaces is from a more abstract viewpoint.Chjoaygame (talk) 03:05, 17 January 2015 (UTC)
That whole section is a total disaster. The only thing that could make sense is the "requirements" subsection. But this is problematic too, see post by Tsirel a bit up. YohanN7 (talk) 07:57, 17 January 2015 (UTC)
I beefed it up a bit. The the "requirements" subsection must be pruned substantially. It essentially "requires" what turns out to be a non-Hilbert space. YohanN7 (talk) 10:08, 17 January 2015 (UTC)
Nice edits, although the "requirements" section is actually what many sources say, feel free to rewrite in any case.
In the lead somewhere it should clarify domain, codomain, and function spaces side by side. Maybe something like this (details are later in the article):
"For a given system, the wavefunction is a complex-valued function of the degrees of freedom. Since wave functions can be added together and multiplied by complex numbers to obtain more wave functions, and an inner product is useful and important to define, the set of wave functions for a system forms a function space, and the actual space depends on the system's degrees of freedom.". M∧Ŝc2ħεИτlk 10:30, 17 January 2015 (UTC)
I made an attempt (second paragraph) based on your suggestion. YohanN7 (talk) 16:05, 17 January 2015 (UTC)
Great, thanks, but there is no need to say it is a summary since the lead is the summary of the article. Preserving your new second paragraph, I'll cut out some repetition and condense the wording if its ok. M∧Ŝc2ħεИτlk 16:23, 17 January 2015 (UTC)
You have a point (that I was aware of beforehand) about mentioning summary. I put the wording there to soften the blow for the apprentice. In that paragraph, there is an inpenetrable (is that word English, Firefox says it isn't) wall for a junior undergraduate from the chemistry department. Can we say something like "In condensed form, bla bla ..." or something equivalent? The point being that the reader shouldn't lose all hope already in the second paragraph. YohanN7 (talk) 18:15, 17 January 2015 (UTC)
Impenetrable.Chjoaygame (talk) 20:05, 17 January 2015 (UTC)
In the second paragraph (maybe the whole lead), the most likely sentence to throw the reader off would probably be
"The topology of the space is that generated by the metric."
Do we need to mention this point in the lead? It is in the main text of the article. The rest of the paragraph is very well-written IMO and should probably stay as is.
Again - no need to mention "in condensed form" because the lead is a condensed form (summary) of the article, any reader should expect that. M∧Ŝc2ħεИτlk 10:17, 18 January 2015 (UTC)
I thought too that the sentence was the weak spot. I don't want to dump "topology" altogether because it is needed (informally) for the title function space. I tried a rewrite. YohanN7 (talk) 00:07, 19 January 2015 (UTC)

Archiving[edit]

I might have screwed up when putting a MiszaBot template here. It did archive, but the gods themselves only know to where. YohanN7 (talk) 00:50, 12 January 2015 (UTC)

It filled up an old archive, Archive 2, then created Archive 3. Stuff still here. YohanN7 (talk) 11:02, 12 January 2015 (UTC)

Phase space[edit]

From article:

In the common formulations of quantum mechanics, the wave function is never a function of both the position and momentum of a particle at any instant, because of the Heisenberg uncertainty principle; if the position of the particle is known exactly, the momentum is not known at all, and vice versa. For a particle in 1d, we can never write a wave function as Ψ(x, p, t). Taken together, x and p are called phase space variables. However, it is possible to construct a phase space formulation of quantum mechanics, using different mathematics and physical interpretations, in a way that does not violate the uncertainty principle.

I think this may be misleading to some degree. See Quantum harmonic oscillator#Ladder operator method. YohanN7 (talk) 12:45, 14 January 2015 (UTC)

And in that example link where is the wavefunction a function of position and momentum? M∧Ŝc2ħεИτlk 13:42, 14 January 2015 (UTC)
Did you truly expect to find any? I wouldn't have put it as mildly as I did if there were any. YohanN7 (talk) 13:45, 15 January 2015 (UTC)
Then what is misleading? Yes, you can use both the position and momentum operators together to solve ladder-like problems like the quantum SHO (an incredibly boring system but nevertheless important and useful), but this has nothing to do with writing a wave function as a function of both position and momentum. M∧Ŝc2ħεИτlk 14:16, 15 January 2015 (UTC)
I think it actually does have something to do with it. It is an example of a canonical transformation. But this is of minor importance, I just thought the formulation was a bit to strong on the emphasis of never mixing x and p. The new rule should be to not mix (the classical quantities) a and a*. At least I think so, I haven't done the problem in years. The operators â and â certainly each commute with themselves. Then again, the virtue of Dirac's method is that the Schrödinger equation (corresponding to the a or a*, there should be one since there is one for both x and p) doesn't have to be solved. I am admittedly uncertain here, it was a long time ago I looked into this. YohanN7 (talk) 16:25, 15 January 2015 (UTC)
"Mixing" x and p is allowed for the operators (obviously e.g. commutation relations, orbital angular momentum). The wave function as a function of both position and momentum is a separate thing. What I mean is the wave function is not a function of the full phase space, but just position (and time, spin etc.) or momentum (and time, spin etc.). M∧Ŝc2ħεИτlk 18:08, 15 January 2015 (UTC)
I know what you mean. The article might put it too strongly. I believe there is a wave function depending on either a or a*, for a = cx + ikp (where c and k are constants) describing the dynamics equally well. Do you say this is wrong? YohanN7 (talk) 18:27, 15 January 2015 (UTC)
I don't know. Maybe it is possible somehow using canonical transformations? I have done them in the Hamiltonian formulation of classical mechanics, but never in quantum mechanics. It just doesn't make sense for the wave function to depend on x and p because of the uncertainty principle, and what becomes of the relation between position and momentum space wavefunctions (Fourier transform)? M∧Ŝc2ħεИτlk 18:51, 15 January 2015 (UTC)
Yes, canonical transformations is what I'm talking about, see post four levels up. It is a wave function of x and p in a sense. It is a wave function of the single variable cx + ikp, much like an analytical function is a function of x and y, constrained to z = x + iy, not a function of x and y varying independently, but still a function of x and y. Again, I don't say the formulation in the article is wrong, it just might be misleading, ruling out what we are discussing here. I have also not seen the motivation (in the article) that it is the HUP that would rule out Ψ(x, p, t). If I'm right here, you can have Ψ(x, p, t), it must just be constrained to be of the form Ψ(cx + ikp) which does depend on x and p. YohanN7 (talk) 19:21, 15 January 2015 (UTC)
For now, best to delete the phase space section. It can be re-introduced later. There is no motivation for the exclusion in this article since the section is just a short digression, but how could you have a function of two observables which do not commute (in this case x and p)? If you know all the position coordinates how do you know the momenta? What meaning does Ψ(cx + ikp) have then? I don't know. If you have a source it may be interesting to add this to the article. M∧Ŝc2ħεИτlk 19:39, 15 January 2015 (UTC)
The supposed wave function is not a function of variables whose canonically associated operators do not commute. From above, the operators â and â certainly each commute with themselves (because every operator commutes with itself). No wave function Ψ(a, a*) exists (probably because â and â do not commute). Few things in QM have a sensible interpretation, this is quite general.
This is not intended for the article. Do you see now why the section of (the previous version of) the article could be misunderstood, provided I am right? YohanN7 (talk) 20:37, 15 January 2015 (UTC)
Maybe it is misleading, but I can't see the point in debating this further. M∧Ŝc2ħεИτlk 21:16, 15 January 2015 (UTC)
Then don't reply is a manner sneezing me off as your final comment. YohanN7 (talk) 21:48, 15 January 2015 (UTC)

Still not right[edit]

This formula,

\begin{align} \Psi(\mathbf{r},t,s_z) & = \psi_{-s}(\mathbf{r},t)\xi_{-s}(s_z) + \psi_{-s+1}(\mathbf{r},t)\xi_{-s+1}(s_z) + \cdots \\
& + \psi_{s-1}(\mathbf{r},t)\xi_{s-1}(s_z) + \psi_{s}(\mathbf{r},t)\xi_{s}(s_z) \,, 
\end{align}

is really saying nothing. The (2s + 1) spin functions referred to in the above formula are a complete set of basis spin functions,

\xi_{s_z}(s'_z) = \delta_{s_z,s'_z}.

The formula then essentially says

\Psi(\mathbf{r},t,s_z) = \psi_{s_z}(\mathbf{r},t),

which is entirely correct, but not a superposition of different spin states as the preceding text may indicate. YohanN7 (talk) 13:27, 14 January 2015 (UTC)

Based on this and this I was tempted to suggest the Kronecker delta expression above but held off since then you'd say that was wrong or too restrictive since the spin functions are complex valued. M∧Ŝc2ħεИτlk 13:42, 14 January 2015 (UTC)
Once again, here is everything in one place, for one particle with spin s:
\Psi : \mathbb{R}^4\times\{-s,-s+1,\ldots,s-1,s\} \rightarrow \mathbb{C}
\psi_{s_z} : \mathbb{R}^4 \rightarrow \mathbb{C}
\xi_{s_z} :  \{-s,-s+1,\ldots,s-1,s\} \rightarrow \mathbb{C}
\Psi(\mathbf{r},t,s_z) = \psi_{-s}(\mathbf{r},t)\xi_{-s}(s_z) + \psi_{-s+1}(\mathbf{r},t)\xi_{-s+1}(s_z) + \cdots \psi_{s}(\mathbf{r},t)\xi_{s}(s_z)
 \xi_{s_z}(s'_z) = \delta_{s_z,s'_z}
Are we agreed on this much for the case of the z-projection? If this is correct, then everything presented together in the first place would have saved reams of posts.
What would ξ be for the spin quantum number in any direction (for concreteness in the direction of a unit vector n(θ, φ) using standard spherical coordinate angles)? For that general case it would probably be a complex-valued linear combination of Kronecker deltas, but need to come back to this later.
M∧Ŝc2ħεИτlk 14:03, 14 January 2015 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── What makes you think I'd automatically say you're wrong? Not a habit of mine. And I wouldn't object very loudly to the statement ℝ ⊂ ℂ.
The ground level is this:

\Psi : \mathbb{R}^4\times\{-s,-s+1,\ldots,s-1,s\} \rightarrow \mathbb{C}
\Psi(\mathbf{r},t,s_z) \equiv \psi_{s_z}(\mathbf{r},t) \equiv \xi_{\mathbf{r}, t}(s_z)
\psi_{s_z} : \mathbb{R}^4 \rightarrow \mathbb{C}
\xi_{\mathbf{r}, t} :  \{-s,-s+1,\ldots,s-1,s\} \rightarrow \mathbb{C}

[On occasion this factors,

\Psi(\mathbf{r},t,s_z) = \psi_(\mathbf{r},t)\xi(s_z) = \phi_(\mathbf{r})\zeta(s_z, t),

and spin dynamics can be studied in isolation.]

Then one might want to define

\xi_{s_z} :  \{-s,-s+1,\ldots,s-1,s\} \rightarrow \mathbb{C}; \quad \xi_{s_z}(s'_z) = \delta_{s_z,s'_z},

and proceed to introduce the vector notation. YohanN7 (talk) 14:55, 14 January 2015 (UTC)

New draft[edit]

Any spin[edit]

Assuming you still mean:
\begin{align} \Psi(\mathbf{r},t,s_z) \equiv \psi_{s_z}(\mathbf{r},t) \equiv \xi_{\mathbf{r}, t}(s_z) & = \psi_{-s}(\mathbf{r},t)\xi_{-s}(s_z) + \psi_{-s+1}(\mathbf{r},t)\xi_{-s+1}(s_z) + \cdots \\
& + \psi_{s-1}(\mathbf{r},t)\xi_{s-1}(s_z) + \psi_{s}(\mathbf{r},t)\xi_{s}(s_z) \,, 
\end{align}
then the article will be updated later. Before that, I'd like to provide a transition from the "scalar form" to the vector form using an analogy the reader may understand and based on what you just said. The links above (especially the first) actually seem to describe this better than even the best books like LL. M∧Ŝc2ħεИτlk 20:01, 14 January 2015 (UTC)
Link looks good. Suggestion: Associate to each function ξsz a vector χsz (tne image of ξsz ⊂ ℂ2s + 1) to make the transition to the vector notation. (I still don't see the benefit of the long expansion of Ψ before introducing the ξsz, since only one term survives.) YohanN7 (talk) 20:36, 14 January 2015 (UTC)
Why would we introduce (a particular choice of) basis spin functions without referring to the expression that includes them? Why include the long expression at all if only one term survives? Because it parallels the linear combination of the column vector after it. The expression should really be done for an arbitrary direction and the choice to the z-component be made. Correct me if wrong, but the spin functions are not the same for all directions and neither are the eigenvectors of the spin operator in other directions. Feel free to make edits. M∧Ŝc2ħεИτlk 21:04, 14 January 2015 (UTC)
Ok! YohanN7 (talk) 21:10, 14 January 2015 (UTC)
If the "Ok!" refers to you editing (and no doubt sharpening/correcting), then I'll temporarily stay out of the way so we don't edit conflict, but I will come back to this soon. M∧Ŝc2ħεИτlk 21:22, 14 January 2015 (UTC)
No, not editing, ok means I understand your explanation. (I better not edit because I'm too tired for that. Besides, it looks good on the surface of things at least.) YohanN7 (talk) 21:37, 14 January 2015 (UTC)
Ok! (as in I'll carry on editing). M∧Ŝc2ħεИτlk 21:51, 14 January 2015 (UTC)
M∧Ŝc2ħεИτlk 20:01, 14 January 2015 (UTC)
Begins to look pretty good. YohanN7 (talk) 21:06, 14 January 2015 (UTC)

Spin-1/2 followed by any spin[edit]

Just in case someone raises it, we probably should work everything in the box above for the spin-1/2 case then mention how to extend it for any spin, since this is the simplest case, but results for any spin seem much more interesting (to me at least). I know I have constantly flitted between spin-1/2 and any spin, but it was keeping in lines with previous discussions. M∧Ŝc2ħεИτlk 22:45, 14 January 2015 (UTC)

The spin-1/2 case followed by higher spin is the one that should be in the article in principle, but as one can see, the above box is too long and has errors, especially for the projection in any direction, so panic not - it will not be inserted in the article any time soon till I trim and correct it. M∧Ŝc2ħεИτlk 10:30, 17 January 2015 (UTC)

tensor product[edit]

I am puzzled by the deletion of the definition of tensor product. It seems to me that the tensor product is of deep conceptual importance for quantum mechanics. For example, I think Elliot Leader's above-quoted statement, using the tensor product, is helpful in clarifying what you have been discussing about how to represent spin states. I accept that in a sense he is there not using it to combine states of distinct systems. I think it has also far wider use.Chjoaygame (talk) 02:21, 16 January 2015 (UTC)

Originally I intended to take the reader up to speed with the operation in the context of this article, before it is used later for the many particle states and position-spin states afterwards, but in the end it looked just like the tensor product section in the bra-ket article.
But perhaps it can be rewritten better, so let's reinstate it. In case there is strong consensus to delete it can be deleted again. M∧Ŝc2ħεИτlk 10:56, 16 January 2015 (UTC)
The tensor product section should definitely precede the multi-particle sections, and possibly the spin section as well. YohanN7 (talk) 14:11, 16 January 2015 (UTC)

new caption for diagram[edit]

I have changed the caption for the diagram of the harmonic motions. The changes show how the diagram illustrates this particular article, as distinct from the article for which the caption was originally written.Chjoaygame (talk) 07:52, 17 January 2015 (UTC)

If you're referring to this edit the caption is far too long for a caption... It seemed fine before, but I will not personally revert it. M∧Ŝc2ħεИτlk 10:35, 17 January 2015 (UTC)

Basis[edit]

Removed this for now:

Since linear combinations of wave functions obtain more wave functions, the set of all wave functions W = {Ψ(x, t)} is an infinite dimensional vector space over the field of complex numbers. To form a vector space basis B, we need a maximal set of wave functions ψ1, ψ2, ... in W which are linearly independent: each one of them is not a linear combination of the others, for example ψ1z2ψ2 + z3ψ3 + ... and ψ2z1ψ1 + z3ψ3 + ..., etc., for any complex numbers zn and every function in W is a linear combination of functions in B. This linear independence allows a linear combination of ψ1, ψ2, ... to uniquely construct an arbitrary wave function in W:
 \Psi(x,t) = \sum_n a_n \psi_n(x,t)
In this way, Ψ(x, t) can be viewed as an infinite dimensional vector, where the complex-valued coefficients an are the components of the vector. The choice of which wave functions to use as a basis is not unique, but if a change of basis is made, the components an need to change to compensate.

It is too shaky. (What kind of basis, how does it relate to our Hilbert space? How does it relate to free particle solutions?) YohanN7 (talk) 13:09, 17 January 2015 (UTC)

More problems to overcome associated to "basis". We need to find separate words for "basis" as in vector space (or Hilbert space) and "basis" as in the x and p and other combination of "observables". This is, atm conflated (partly by me) in the article. YohanN7 (talk) 18:04, 17 January 2015 (UTC)

Belated reply - it may be shaky and does not explicate anything about free particle solutions, but the idea was to pitch to the reader, in the simplest possible way, the idea that wave functions form vector spaces (linear combinations of wave functions are wave functions). I'm neutral on the deletion, since now it is in the lead. M∧Ŝc2ħεИτlk 10:04, 18 January 2015 (UTC)
It needs a rewrite, not permanent deletion. YohanN7 (talk) 10:13, 18 January 2015 (UTC)
In the literature the x and p (and others) "bases" are usually called "representations", not sure if that is misleading though. M∧Ŝc2ħεИτlk 10:43, 18 January 2015 (UTC)
That solves the problem. Thanks for pointing out. Sometimes it is just too obvious to see for oneself. YohanN7 (talk) 23:41, 18 January 2015 (UTC)

Historical information[edit]

I have given historical information here. At that time, 1905, Planck did not admit the notion of a photon. The eponym "Planck–Einstein relation" occurs in a very few otherwise reliable sources, but currency of it is largely an invention of Wikipedia editors which we do not need to copy here. It is hardly necessary to give de Broglie's name to Einstein's recognition of the momentum of the photon.Chjoaygame (talk) 19:49, 17 January 2015 (UTC)

Are you sure about the last point? (The "Planck–Einstein relation" is of course nonsense). De Broglie is attributed the glory of having suggested the de Broglie wavelength for massive particles in practically every reference there is. YohanN7 (talk) 20:07, 17 January 2015 (UTC)
Hm. The name "Planck–Einstein relation" might on second thought be appropriate to mention. (Hey, YOU named the article that wayFace-smile.svg) Planck didn't suggest photon quanta (while he did find his own constant), so it should be attributed to Einstein. The de Broglie wavelength is the definite name in the massive case. I think attribution should go to him as well. YohanN7 (talk) 20:19, 17 January 2015 (UTC)
Yes, it's horrible. I did it with reluctance and distress, to try to escape something even worse. People did a Google search and found some popularizing sources that called it the 'Planck relation', and were insistent for it. Very few reliable sources use that one, and I hoped to avoid Wikipedia giving it currency by circular quotation. Planck had a quantum for emission that belonged to a purely heuristic virtual simple harmonic oscillator. More he thought ε = nhν, not just ε = . He did not think of photons till years later, after Einstein's ideas. Amongst reliable sources, I found scarce consistency, but two used the eponym 'Planck–Einstein', so I used it reluctantly as the least worst reliably sourced escape option.
As for de Broglie, since you think it matters, what about putting it in a sentence of its own?
Say we add 'From Einstein's idea for the photon, in 1923 Louis de Broglie generalized to a 'wavelength' for massive particles'?Chjoaygame (talk) 21:27, 17 January 2015 (UTC)
I have edited to that effect (or something like it). YohanN7 (talk) 21:33, 17 January 2015 (UTC)
Ok. Date fix.Chjoaygame (talk) 21:37, 17 January 2015 (UTC)
On closer reading, I am suggesting some changes.Chjoaygame (talk) 21:56, 17 January 2015 (UTC)

lead paragraph change proposal[edit]

With due respect to the opinion of Editor Sbyrnes, I am unhappy with the simile that the wave function is like water and string waves. Dirac's footnote more or less explicitly says it isn't. I would like to change that lead paragraph to

The wave functions are just the solutions of the Schrödinger equation, determined by the quantum mechanical Hamiltonian that defines the system. The Schrödinger equation is mathematically a type of wave equation. This explains the name "wave function", and gives rise to wave–particle duality. The wave of the wave function is not a wave in physical space; it is a wave in an abstract mathematical "space", and in this respect it differs fundamentally from water waves or waves on a string.[1][2][3][4][5][6][7]
  1. ^ Born, M. (1927). Physical aspects of quantum mechanics, Nature, 119, 354–357.
  2. ^ Heisenberg, W. (1958). Physics and Philosophy: the Revolution in Modern Science, Harper & Row, New York, p. 143.
  3. ^ Heisenberg, W. (1927/1985/2009). Heisenberg is quoted by Camilleri, K. (2009), (from Bohr, N. (1985), Collected Works, edited by J. Kalckar, volume 6, 'Foundations of Quantum Mechanics I 1926–1932, North-Holland, Amsterdam, p. 140), Heisenberg and the Interpretation of Quantum Mechanics: the Physicist as Philosopher, Cambridge University Press, Cambridge UK, ISBN 978-0-521-88484-6, p. 71.
  4. ^ Murdoch, D. (1987). Niels Bohr's Philosophy of Physics, Cambridge University Press, Cambridge UK, ISBN 0-521-33320-2, p. 43.
  5. ^ de Broglie, L., (1960). Non-linear Wave Mechanics: a Causal Interpretation, Elsevier, Amsterdam, p. 48.
  6. ^ Landau, L.D., Lifshitz, E.M. (1958/1965). Quantum Mechanics: Non-relativistic Theory, translated from Russian by J.B. Sykes and J.S. Bell, second edition 1965, Pergamon Press, Oxford UK, p. 6.
  7. ^ Newton, R.G. (2002). Quantum Physics: a Text for Graduate Students, Springer, New York, ISBN 0-387-95473-2, pp. 19–21.

Moreover, since the wave function is defined by its being a solution of the Schrödinger equation, I would like to put that paragraph second in the lead, to complete the definition of the wave function before discussing it.Chjoaygame (talk) 20:34, 17 January 2015 (UTC)

change to lead sentence proposed[edit]

I would like to alter the following sentence of the lead: "It is a central entity in quantum mechanics and is important in all modern theories, like quantum field theory incorporating quantum mechanics, while its interpretation may differ."

Instead, I would prefer:

It is a central entity in quantum mechanics, important in all modern quantum theories, including the quantum theory of fields.

I think we can survive on this shorter version.Chjoaygame (talk) 20:54, 17 January 2015 (UTC)

Seems fine with me. M∧Ŝc2ħεИτlk 10:06, 18 January 2015 (UTC)

Dirty work[edit]

Having full references inline sucks. It is highly uneconomical and strongly discourages (at least me) people to put in new citations. We probably need few new sources, but more inline citations. I'll do the dirty work of moving all references to a reference section and introduce Harvard citations (I think they are called so). This makes it so much simpler to cite and avoids senseless duplication. No time table. You are allowed, even encouraged, to help. YohanN7 (talk) 21:43, 17 January 2015 (UTC)

Not sure what triggered this? Not even clear what you mean by your above remarks? I just put the references here above where I did to avoid format complications on the talk page. Surely you have more urgent things to do?Chjoaygame (talk) 21:51, 17 January 2015 (UTC)
On looking at what you done so far, on bended knee I beg you to stop, and undo it.
Though Editor Maschen feels strongly for them, I think footnotes are very undesirable in Wikipedia. I am not a Wikilawyer, but I have a feeling I am not alone in this. Footnotes open the way for all kinds of slipshod editing and admission of material of dubious relevance and other abuses. I think the present (before your latest) arrangement is good. The template method has advantages and disadvantages. Horses for courses.Chjoaygame (talk) 22:08, 17 January 2015 (UTC)
I'm neutral if anything on footnotes, and don't feel strongly on them. A few don't cause any harm, but then the article should the details. My rule of thumb would be: only digressions which disrupt the main flow of text should be in the footnotes. Otherwise it should be in the main text. M∧Ŝc2ħεИτlk 10:11, 18 January 2015 (UTC)
What on earth are you talking about? The visual appearance is identical as far as relevant information goes. That is what and in where. The reader eager to see details about the publication can click on the appropriate link and get taken to the reference section. It is far superior. (Who the hell wants to see isbn numbers and doi's in a popup?) It is just easier to maintain and use for further inline literature references. YohanN7 (talk) 22:33, 17 January 2015 (UTC)
Ok, ok. Now I see what you intend. Panic no longer.Chjoaygame (talk) 22:41, 17 January 2015 (UTC)
At first your intention was not clear to me. Now looking at what you are doing. Yes, I very much agree with your plan for separate sections for the specific citations (page x, pages y–z, etc.), and for the full bibliographic details for the source books or articles for the citations. I have done it on several articles. It makes it much easier to give many different specific citations from one book or article.
I have to admit I am only lukewarm for the citation template method, as distinct from handwork. The templates can be too rigid for some complicated citations or unusual sources.
I do not know of standards in this. Briefly glancing around, I see diverse ways of doing things.
But what very much concerns me is the risk of opening of the floodgates for 'notes'. That is why I posted the heading 'citations' instead of 'notes'. I am strongly of the view that if something is worth a place, it is worth a place in the body of the text. If it doesn't fit right there, it usually means that the article structure needs fixing. A footnote is in my view a bad way to deal with such problems. I seem to recall that I learnt this from some good Wikipedia source, but I don't recall the detail, and I am not a Wikilawyer. That is why I was upset at the header 'notes'. I still think it is an invitation to abuse.Chjoaygame (talk) 23:20, 17 January 2015 (UTC)
We already have a dedicated section for what you dislike, namely the remarks section. These are footnotes in the traditional sense. Footnotes are standard in good articles when appropriate, as here. The biggest structural problem with the article at present is a one mile long figure caption in the lead. This is not standard. YohanN7 (talk) 23:36, 17 January 2015 (UTC)
Yes, this article does have footnotes, which I think is a bad idea. But it seemed at first to me that your present plan (that I now see as good) to separate citations and bibliography looked as if you were introducing a second footnote scheme. As for what is standard, I don't know much about that. I am inclined to do what seems best for the particular article, and let the standard follow that good lead. I am sorry you find my new caption offensive.Chjoaygame (talk) 02:34, 18 January 2015 (UTC)

state or species ?[edit]

How should this article define the term 'quantum state', and consequently the 'wave function? Below are some findings from possibly reliable sources.Chjoaygame (talk) 23:31, 21 January 2015 (UTC)

Born[edit]

"A knowledge of ψ enables us to follow the course of a physical process in so far as it is quantum-mechanically determinate; not in a causal sense, but in a statistical one. Every process consists of elementary processes which we are accustomed to call transitions or jumps; the jump itself seems to defy all attempts to visualize it, and only its result can be ascertained. This result is, that after the jump, the system is in a different quantum state. The function ψ determines the transitions in the following way: every state of the system corresponds to a particular characteristic solution, an Eigenfunktion, of the differential equation; for example the normal state the function ψ1, the next state ψ2, etc."<Born, M. (1927). Physical aspects of quantum mechanics, Nature 119: 354–357.>

It seems that Born thought of ascertained results of determinate physical processes in terms of probabilistic successions of jumps between quantum states as physical objects that correspond with mathematical entities called eigenfunctions.Chjoaygame (talk) 23:31, 21 January 2015 (UTC)

Bohr[edit]

"As a more appropriate way of expression, one may advocate limitation of the use of the word phenomenon to refer to observations obtained under specified circumstances, including an account of the whole experiment."<Bohr, N. (1948). On the notions of complementarity and causality, Dialectica 2: 312–319.>

Bohr's thought continued to develop long after the early days. He eventually settled on the idea of a 'phenomenon'. He refers to the just-quoted paper in his celebrated attack on Einstein in the 1949 Schilpp book. Here below, Rosenfeld and Wheeler note this culminating concept. In ordinary language, I would say that by 'phenomenon', Bohr means 'process observed and described'. He is not referring to what I would think of as Einstein's idea of a natural process that happens whether or not someone later observes it. Obviously, an account or description of an unobserved process is to a large extent a theoretical speculation. Quantum mechanics is a method of description of experiments. Bohr thinks it ineluctably involves preparation and detection as ingredients of phenomena. The preparation is specified by a generation of an initial 'state' and the detection determines the specification of a final 'state'. Sometimes they are the same. The quantum 'states' are specified in terms of appropriate 'configuration' spaces. Unlike classical mechanics using states in phase space, quantum mechanics using 'configuration' space 'states' cannot in general support deterministic predictions, although the Schrödinger equation itself is deterministic as noted by von Neumann, and a 'phenomenon' is a determinate actual physical entity.Chjoaygame (talk) 23:31, 21 January 2015 (UTC)

Heisenberg[edit]

"But that a revision of kinematical and mechanical concepts is necessary seems to follow directly from the basic equations of quantum mechanics. .... But what is wrong in the sharp formulation of the law of causality, "When we know the present [state] precisely, we can predict the future," is not the conclusion but the assumption. Even in principle we cannot know the present [state] in all detail."<Heisenberg, W. (1927). The physical content of Quantum kinematics and mechanics, Zeit. Phys.43: 172–198, translated in Wheeler, Zurek (1983) pp. 62–84.>

I have inserted the items [state] to bring out the relevance of Heisenberg's remarks here to the notion of quantum state. Also the kinematics are the description of the 'state'. As Dirac points out below, what we can know is determined by our mode of construction of the artificial state (e.g. our necessary choice of momentum space or configuration space, or whatever) that we observe. That is the ineluctable limitation on knowledge of state to which Heisenberg is referring, imposed by the quantum mechanical kinematics. A quantum phenomenon becomes determinate only when it has been detected, as pointed out below by Rosenfeld. Its initial condition as specified by quantum kinematics does not determine it. This contrasts with the classical kinematics which allow a state description that supports exact determination of the future.Chjoaygame (talk) 23:31, 21 January 2015 (UTC)

Rosenfeld[edit]

"It is only too true that, isolated from their physical context, the mathematical equations are meaningless: but if the theory is any good, the physical meaning which can be attached to them is unique. .... The wholeness of quantal processes necessitates a revision of the concept of phenomenon. Since the concepts which in classical theory describe the state of a physical system are actually subject to mutual limitations, they can no longer be regarded as denoting attributes of the system. Their true logical function is rather to express relations between the system and certain apparatus of entirely classical (i.e. directly controllable) character which serve to fix the conditions of observation and register the results. A phenomenon is therefore a process (endowed with the characteristic quantal wholeness) involving a definite type of interaction between the system and the apparatus."<Rosenfeld, L. (1957). Misunderstandings about the foundations of quantum theory, pp. 41–45 in Observation and Interpretation, edited by S. Körner, Butterworths, London.>

To make a definite actual physical entity, a phenomenon, quantum physics requires that both initial and final conditions be determinate. Quantum kinematics defines a quantum 'state' that supplies only the initial, not the final, condition. That enforces its probabilistic character. (Perhaps I may remark that Einstein was not sure that Nature works by preparing pure states and detecting final states as required by quantum mechanics. Indeed, it is obvious that Nature supplies only mixed states.)Chjoaygame (talk) 23:31, 21 January 2015 (UTC)

Kramers[edit]

"A physical situation which is characterised by a solution of the Schrödinger equation of the form ψ = φ exp (−iEt/ħ) with normalizable ψ and which thus in accordance with the quantum postulate E = corresponds to a well defined energy of the system under consideration is called a stationary state of the system."<Kramers, H., (1937/1956). Quantum Mechanics, translated by D. ter Haar, North-Holland, Amsterdam, pp. 58–59.>

In the olden days they tried to define their terms. Kramers distinguished the physical situation from its mathematical characterisation.Chjoaygame (talk) 23:31, 21 January 2015 (UTC)

Messiah[edit]

"When the system is in a state represented by a wave of type II.34, 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)." <Messiah, A. (1961). Quantum Mechanics, volume 1, translated by G.M. Temmer from the French Mécanique Quantique, North-Holland, Amsterdam, page 72.>

As I read this, Messiah has in mind two entities, a physical object in a quantum state, and a mathematical object that lives in a function space. He thinks the mathematical object "represents" the physical object.Chjoaygame (talk) 23:31, 21 January 2015 (UTC)

Weinberg[edit]

"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. This is essentially the approach of Diracs's ″transformation theory″."<Weinberg, S. (2013). Lectures on Quantum Mechanics, Cambridge University Press, Cambridge UK, ISBN 978-1-107-02872-2, page xvi.>

Evidently, Weinberg agrees with the view of Messiah that there are two kinds of object, physical and mathematical. He calls the physical ones "states" and the mathematical ones "vectors" or "wave functions". The mathematical ones "represent" the physical ones. It seems he has important points of agreement with Dirac.Chjoaygame (talk) 23:31, 21 January 2015 (UTC)

Dirac[edit]

"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."<Dirac, P.A.M. (1940). The Principles of Quantum Mechanics, fourth edition, Oxford University Press, Oxford UK, pages 11–12.>

No mention here of mathematical objects. Dirac is referring to physical objects. He distinguishes between an instantaneous state and a state with an indeterminate duration in time. The state so defined is physically indeterminate because it it not actually observed by detection. That is the meaning of 'undisturbed'. An indeterminate state does not define a physical phenomenon, such as is intended by Wheeler in his well-known aphorism

"Had quantum mechanics stopped here, its deepest lesson would have escaped attention: ″No elementary quantum phenomenon is a phenomenon until it is a registered (observed) phenomenon.″"<Wheeler, Zurek (1983), page xvi.>

Here Wheeler is referring to statements such as the following by Bohr:

"... every atomic phenomenon is closed in the sense that its observation is based on registrations obtained by means of suitable amplification devices with irreversible functioning such as, for example, permanent marks on the photographic plate caused by the penetration of electrons into the emulsion ..."<Bohr (1958) quoted by Wheeler, Zurek (1983), page viii.>

Evidently, for Wheeler and Bohr, a quantum mechanical phenomenon is an actual physical entity, a fully determinate process, with a finite time duration, with no remaining unrealized potential possibility. Such is not a quantum state as defined by Dirac.

A physical entity that is indeterminately defined can have future adventures only probabilistically. Being indeterminate, it cannot have a determined future. This contrasts with a determinate classical physical object, which can have a determined future. That is one difference between Dirac's quantum 'state' and a classical ordinary language physical state.

Nevertheless, Dirac's state is defined as restrictively as is theoretically possible for a quantum system. This makes it a pure state. The pure state is not that of a raw natural object, such a an atom of silver vapour escaping through a small hole in an oven wall. No, it is an artificially prepared state. Even though it is not yet observed, it is still causally conditioned by the observer, not in a native state. For example, it might have been prepared in a definite state of uniform motion in a nearly straight line if it is observed in a place in space where there is nearly no gravity. Then it is in a momentum eigenstate. It has no definite position. Its momentum can be measured by its angle of deflection by a diffraction grating and detection by a suitably placed device.Chjoaygame (talk) 23:31, 21 January 2015 (UTC)

present article[edit]

The present article says

  • Wave functions corresponding to a state are not unique. This has been exemplified already with momentum and position space wave functions describing the same abstract state.
  • The abstract states are "abstract" only in that an arbitrary choice necessary for a particular explicit description of it is not given. This is analogous to a vector space without a specified basis.
  • The wave functions of position and momenta, respectively, can be seen as a choice of basis yielding two different, but entirely equivalent, explicit descriptions of the same state.
  • Corresponding to the two examples in the first item, to a particular state there corresponds two wave functions, Ψ(x, Sz) and Ψ(p, Sy), both describing the same state.

As I read it, this part of the present article flatly contradicts the consensus of Born, Bohr, Heisenberg, Rosenfeld, Kramers, Messiah, Weinberg, and Dirac.

As I read it, the present article uses the term "state", where ordinary language would speak of 'species of quantum system', or 'species of quantum entity', or 'kind of particle', or some such, and would say that a species of quantum system can be prepared in several different states, respectively pure with respect to several observables = quantum analysers = operators, each pure state with its own respective wave function. And, as I read it, the present article rejects the idea that a pure quantum state is described by a particular eigenfunction of the operator with respect to which it is pure. Rather, for the present article, the "state" extends over all possible operators for the species or kind of particle. The notion of a mixed state seems to have faded out.

An example of a species of system would be typified by an unfiltered atom emerging from a hole in the wall of an oven containing metal vapour. It is in a mixed state. Filtering it with some device such as a Stern-Gerlach magnet will split the beam into several sub-beams in states respectively pure for that filter.

The confusion may perhaps arise by using the phrase "abstract state" to mean 'class of pure states in which a species of system can be prepared'. Alternatively, perhaps the just above eight cited authors are obsolete, and quantum mechanics has changed since their days?

Dirac defines a state by the most restrictive possible set of conditions, making it a pure state, while the present article's "abstract state" seems to extend over the least restrictive possible range of conditions.Chjoaygame (talk) 23:31, 21 January 2015 (UTC)

For the record, the present article states
  • Basic states are characterized by a set of quantum numbers. This set is a set of eigenvalues of a maximal set of commuting observables. I'd say it is pretty much spot on. YohanN7 (talk) 15:05, 23 January 2015 (UTC)
Thank you for this comment. I will think it over.Chjoaygame (talk) 00:32, 24 January 2015 (UTC)

Max Born on Dirac[edit]

According to Max Born

"It may be mentioned in conclusion that the fundamental idea behind Heisenberg's work was worked out by Dirac (1925) in a very original way, and that in 1964 he put views to the effect that although Heisenberg's and Schrödinger's approaches are perfectly equivalent in ordinary (non-relativistic) quantum mechanics, this is not the case in quantum field theory. Here Heisenberg's method turns out to be more fundamental."<Born. M. (1969). Atomic Physics, eighth edition, translated by J. Dougall, R.J. Blin-Stoyle, J.M. Radcliffe, Blackie & Son, London, p. 130.>

Likely Born is referring to <Dirac, P.A.M. (1964). Lectures on Quantum Mechanics, Belfer Graduate School of Science, Yeshiva University, New York.>

Perhaps this may help in planning the conceptual structure of this or other articles.Chjoaygame (talk) 11:46, 25 January 2015 (UTC)

Is this about the Schrödinger picture and the Heisenberg pictureYohanN7 (talk) 12:58, 25 January 2015 (UTC)
I ought to have given some context. It is about Schrödinger's wave mechanics and Heisenberg's matrix mechanics.Chjoaygame (talk) 13:07, 25 January 2015 (UTC)
Yes. It would help tremendously if you supplied some context to your mountain of quotes. Taken in isolation, they are useless, especially the older ones from times when terminology (and knowledge and interpretation) was different from today.
Maybe something for the history section.
The Schrödinger picture and the Heisenberg picture must definitely go into the article. In one case the wave function is time dependent, in the other it is not. In QFT the Heisenberg equation in the Heisenberg picture is taken as the master equation. The corresponding Schrödinger picture equation is a derived quantity, often, but not always, the same equation as obtained using Schrödinger picture RQM. (This can be taken as further evidence of RQM being an both incorrect and incomplete reconciliation of QM and SR.) YohanN7 (talk) 13:29, 25 January 2015 (UTC)
The Schrödinger, Heisenberg, and Dirac (interaction) pictures are mentioned in the article wave function#time dependence.
Sorry once again for negligence, there is a huge amount of talk to read through, and still more to rewrite, I'll catch up and try some over the next few days... M∧Ŝc2ħεИτlk 13:59, 25 January 2015 (UTC)

Conceptual structure[edit]

This clears up slight mystery for me. It has seemed in the past to me that Heisenberg's matrix mechanics<Razavy, M. (2011). Heisenberg's Matrix Mechanics, World Scientific, Singapore, ISBN 978-981-4304-10-8.> just disappeared into the ether of history. And it has seemed a little puzzling that the Heisenberg picture somehow appeared out of nowhere. Now I see what happened. The Heisenberg matrix formalism went into the Dirac theory, where it appeared as the Heisenberg picture. It is common enough to read that Heisenberg's matrix mechanics and Schrödinger's wave mechanics were equivalent.<plenty of references for this if appropriate>. But they are not quite so, as pointed out by N.R. Hanson.<Hanson, N.R. (1961). Are wave mechanics and matrix mechanics equivalent theories?, Czech. J. Phys. 11: 693–708.> and later by Dirac, and then by Born, as noted above.

The "quantum mechanics all the way" noted by User:YohanN7 is Heisenberg's scheme, first in the guise of his matrix mechanics, then in the guise of the Heisenberg picture in the Dirac 〈bra|ket〉 formulation. It seems that Schrödinger's wave mechanics did not make it all the way.Chjoaygame (talk) 19:36, 25 January 2015 (UTC)Chjoaygame (talk) 20:14, 25 January 2015 (UTC)

The difference between the Heisenberg and Schrödinger pictures is, in the context, just a mathematical triviality (with practical computational consequences of course). If there is any real difference between matrix mechanics and Schrödinger's wave mechanics, I couldn't tell since I don't know matrix mechanics. I have acquired Dirac's 1925 paper though. Even his early papers feel more modern in touch and style than most papers from much later times (meaning readable). YohanN7 (talk) 20:39, 25 January 2015 (UTC)
The referenced paper abstract questions the validity of the proof that MM and WM are equivalent. It says too that Born proved them equivalent as physical theories. YohanN7 (talk) 20:48, 25 January 2015 (UTC)
Interesting.
Matrix mechanics is quantum mechanics pretty much as Heisenberg invented it in 1925. But Heisenberg was no pure mathematician and had never heard of matrices. One could say he invented them all over again by himself for the purpose. Born and Jordan were in close touch with him and were familiar with the matrix as a mathematical object, and recognized it in his work. Then they wrote joint papers. Some practical calculations were done with matrix mechanics in its original form, but very soon Schrödinger's wave mechanics appeared and was much more congenial to work with. It was generally accepted that Schrödinger had satisfactorily /demonstrated-proven-established/ that his wave mechanics and Heisenberg's matrix mechanics were equivalent. In a sense then the Heisenberg matrix version disappeared from sight. But it was still there as the Heisenberg picture in the Dirac 〈bra|ket〉 formulation. (By 'Dirac 〈bra|ket〉 formulation' I do not mean 'Dirac picture'.)
In the Dirac 〈bra|ket〉 formulation, the equivalence of the Schrödinger and Heisenberg pictures is as you say pretty much a mathematical triviality. But by 1964, Dirac had thought things over some more, and wrote what Born interpreted as a statement that the Heisenberg version is "more fundamental" (in my quick read of it I didn't find those exact words in Dirac 1964). The mathematically trivial proof that the Schrödinger and Heisenberg pictures are equivalent works for the non-relativistic case, but there is as you rightly say no valid relativistic version of Schrödinger's wave mechanics, so no question of relativistic equivalence arises. It seems Hanson saw signs of this but it wasn't made clear till Dirac 1964. It was the 1964 Dirac lectures that made Born become aware of it and see that his former view of equivalence worked only for the non-relativistic case. Enough for now.Chjoaygame (talk) 00:11, 26 January 2015 (UTC)
This is, you say, in Dirac's QM book? Do you know in which chapter? (It's against my principles, but I know a (very probably illegal) copy of it floating around on the net, and I'll sneak a peek nonetheless.) YohanN7 (talk) 00:30, 26 January 2015 (UTC)
After 'edit conflict' message.
Looking quickly at the fourth edition 1958 of Dirac's Principles of Quantum Mechanics, I see him on pages 111 and 112 apparently inventing the terms 'Schrödinger picture' and 'Heisenberg picture'. The 1964 Lectures on Quantum Mechanics is what Born was referring to.Chjoaygame (talk) 00:51, 26 January 2015 (UTC)
It seems you are saying above that there is a valid Schrödinger picture also for the valid relativistic case, which is the quantum theory of fields. The Schrödinger picture in the quantum theory of fields cannot be validly derived from a Schrödinger-like "relativistic quantum theory", because there is none. It must be derived from the Heisenberg picture, which is the primary and only reliable way to construct new formulas in the quantum theory of fields. The Schrödinger picture will be different for different inertial reference frames.Chjoaygame (talk) 15:01, 26 January 2015 (UTC)Chjoaygame (talk) 15:08, 26 January 2015 (UTC)