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Clarification on spin functions

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...

Does this look good to insert where spin functions are introduced? Not more - see new box(es) below

A wavefunction for a particle with spin can be decomposed into a product of a space function and a spin function: $\Psi (\mathbf {r} ,t,s_{z})=\psi (\mathbf {r} ,t)\xi (s_{z},t)$ The above decomposition is useful, because sometimes it is convenient to consider spin without the space dependence.[QM E. Merzbacher 3rd edn] For general time-dependent problems, time is a parameter in each function. For time-independent problems the time-dependence can be excluded from ψ and ξ since the Schrödinger equation determines it to be a phase factor e−iEt/ħ where E is the energy eigenvalue corresponding to the wavefunction ψ, so in this case: $\Psi (\mathbf {r} ,t,s_{z})=e^{-iEt/\hbar }\psi (\mathbf {r} )\xi (s_{z})\,.$ First, consider the case of sz = 1/2, since this is a case of practical interest for many real particles (all leptons and quarks are the known elementary which constitute matter, are spin 1/2). As always, the space function ψ(r, t) is a complex-valued function of continuous variables (position vector of the particle) and time is a parameter. The spin function is different, it takes in sz and returns a complex-valued vector. The spin function must be vector-valued since the components of the vector correspond to different spin states. Since sz is a discrete variable, each vector must be given separately: $\xi (1/2,t)={\begin{pmatrix}\alpha _{1}\left(t\right)\\\alpha _{2}\left(t\right)\end{pmatrix}}\,,\quad \xi (-1/2,t)={\begin{pmatrix}\beta _{1}\left(t\right)\\\beta _{2}\left(t\right)\end{pmatrix}}$ and both the vectors must be linearly independent. (The vectors do not have to be written as column matrices, but it is convenient and conventional to do so, since spin operators are conventionally displayed as matrices). It is natural to take these spin functions to be the corresponding eigenvectors of the spin operator. For the z-projection of spin 1/2, we have $\xi (1/2)={\begin{pmatrix}1\\0\end{pmatrix}}\,,\quad \xi (-1/2)={\begin{pmatrix}0\\1\end{pmatrix}}$ since these are the eigenvectors of the z-component spin operator with eigenvalues +ħ/2 and −ħ/2, respectively, with the interpretation of ξ(1/2) as the "spin-up" function, and ξ(−1/2) as the "spin-down" function. Returning to the full wavefunction, the quantum number selects the components: $\Psi (\mathbf {r} ,t,1/2)=\psi (\mathbf {r} ,t)\xi (1/2)={\begin{pmatrix}\psi (\mathbf {r} ,t)\\0\end{pmatrix}}\,,\quad \Psi (\mathbf {r} ,t,-1/2)=\psi (\mathbf {r} ,t)\xi (-1/2)={\begin{pmatrix}0\\\psi (\mathbf {r} ,t)\end{pmatrix}}$ Now, the particle could be in either state, so the complete wavefunction for the particle is not only ψ(r, t)ξ(sz) for any sz, but must be a superposition of both spin states, a complex vector: $\Psi =\psi _{1/2}(\mathbf {r} ,t)\xi (1/2)+\psi _{-1/2}(\mathbf {r} ,t)\xi (-1/2)=\psi _{1/2}(\mathbf {r} ,t){\begin{pmatrix}1\\0\end{pmatrix}}+\psi _{-1/2}(\mathbf {r} ,t){\begin{pmatrix}0\\1\end{pmatrix}}={\begin{pmatrix}\psi _{1/2}(\mathbf {r} ,t)\\\psi _{-1/2}(\mathbf {r} ,t)\end{pmatrix}}$ where the subscripts label the space functions corresponding to the spin states. The above wavefunction is not a vector in the elementary sense, but a spinor. Spinors transform differently to vectors. The above can be extended to particles of any spin. Again for the z-component of spin, there are 2s + 1 vectors: $\xi (s)={\begin{pmatrix}1\\0\\\vdots \\0\end{pmatrix}}\,,\quad \xi (s-1)={\begin{pmatrix}0\\1\\\vdots \\0\end{pmatrix}}\,,\quad \cdots \quad \xi (-s)={\begin{pmatrix}0\\0\\\vdots \\1\end{pmatrix}}\,,$ since these are the eigenvectors of the z-component spin operator with eigenvalues ħs, ħ(s − 1), ..., −ħ(s − 1), −ħs, respectively. Since the particle could be in any spin state, the entire wavefunction is a superposition of all spin states, again a complex vector (really, a spinor): $\Psi =\sum _{s_{z}}\psi _{s_{z}}(\mathbf {r} ,t)\xi (s_{z})={\begin{pmatrix}\psi _{s}(\mathbf {r} ,t)\\\psi _{s-1}(\mathbf {r} ,t)\\\vdots \\\psi _{-s}(\mathbf {r} ,t)\end{pmatrix}}$ In the case of spin projections along other axes, say the x or y directions, or any direction, the corresponding eigenvectors for the component of the spin operator along that direction would be used instead. The above formalism is general for non relativistic quantum mechanics. In relativistic quantum mechanics and relativistic quantum field theory, the wavefunctions are different, constructed out of Dirac spinors.

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∧Ŝc²ħεИ_τlk 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 indentation)) 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 s_z, we have the corresponding component of the vector. ? When you substitute one value for s_z, 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∧Ŝc²ħεИ_τlk 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 {−1⁄2, 1⁄2} → ℂ (which is an element of the function space ℂ^{{−1⁄2, 1⁄2}}) 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 × {−1⁄2, 1⁄2} → ℂ, and I think it always factors (this requires a proof or a reference) as Ψ = ψ × ξ where ψ:ℝ⁴ → ℂ and ξ:{−1⁄2, 1⁄2} → 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 × {−1⁄2, 1⁄2} → 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} → ℂ^{2s + 1}, again each spin quantum number maps to the components of the vector. M∧Ŝc²ħεИ_τlk 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∧Ŝc²ħεИ_τlk 00:55, 12 January 2015 (UTC)

At any rate, Ψ = ψ_1⁄2 × ξ_1⁄2 + ψ_−1⁄2 × ξ_−1⁄2, 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∧Ŝc²ħεИ_τlk 10:30, 12 January 2015 (UTC)

No, I mean complex-valued functions multiplying each other in the range. The domain of each term is ℝ⁴ × {−1⁄2, 1⁄2} and the range is ℂ for both factors in each term. Note that ξ_±1⁄2 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 1⁄2) 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 1⁄2 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 = ψ_1⁄2 × ξ_1⁄2 + ψ_−1⁄2 × ξ_−1⁄2

Ψ : ℝ⁴ × {−1⁄2, 1⁄2} → ℂ

and to be extremely sure... ψ_±1/2:ℝ⁴ → ℂ and ξ_±1/2:{−1⁄2, 1⁄2} → 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∧Ŝc²ħεИ_τlk 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 ψ_1⁄2 × ξ_1⁄2 and ψ_−1⁄2 × ξ_−1⁄2.

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 ℝ⁴ × {−1⁄2, 1⁄2} as you say, but their codomains are ℂ² and I should have written Ψ : ℝ⁴ × {−1⁄2, 1⁄2} → ℂ².

Am I getting or getting there, or too annoying? M∧Ŝc²ħεИ_τlk 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, s_z in the argument to Ψ. You get a complex number (or you really get it wrong). Then plug in x, y, z, t, −s_z. 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∧Ŝc²ħεИ_τlk 14:50, 12 January 2015 (UTC)

We just need to define the operation. If φ:A → ℂ, χ:B → ℂ then define φ ∗ χ:A × B → ℂ; φ ∗ χ(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∧Ŝc²ħεИ_τlk 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

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∧Ŝc²ħεИ_τlk 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∧Ŝc²ħεИ_τlk 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∧Ŝc²ħεИ_τlk 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 you. 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∧Ŝc²ħεИ_τlk 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∧Ŝc²ħεИ_τlk 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∧Ŝc²ħεИ_τlk 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)