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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)

concern

In this edit, I wrote:

          "present article

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, S_z) and Ψ(p, S_y), 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) 16:35, 6 February 2015 (UTC)

Well, you read it wrong. It is clear from the very first sentence in the lead that we are talking about one system, not an ensemble. Hence a pure state. From where do you get the idea that the article speaks about particle species and not states? Besides, that atom of yours is in a pure state, not a mixed state. YohanN7 (talk) 06:13, 7 February 2015 (UTC)

It is good to have a response from you.

It seems there are problems of communication here. Perhaps with patience and goodwill they can be solved.

I have no problem in accepting the obvious, that we are talking about one system that may consist of many particles, as stated in the sentence of the lead "There is one wave function containing all the information about the entire system, not a separate wave function for each particle in the system." I was not thinking in terms of ensembles. Perhaps I should add that your introduction here of the term 'ensemble' might need some attention if it becomes relevant. There may perhaps be potential for a communication difficulty in it.

You write "Hence a pure state." It my be here that there is a communication problem. Perhaps the Stern-Gerlach experiment may help to clarify. For the present I will limit my response to this, for the sake of comfort of discussion.

As I read the Stern-Gerlach experiment, they have an oven containing silver vapour. The vapour consists of single atoms. There is a small hole in the wall of the oven, through which atoms emerge in a beam, labelled here beam 0. The atoms in the beam travel independently of one another. Each atom is, for the present purposes, a single system. The beam passes through a Stern-Gerlach magnetic field. Some of the atoms go up, others down. These emerging sub-beams are here labelled beam 1u and beam 1d. Following The Feynman Lectures on Physics, each sub-beam is passed to another respective Stern-Gerlach magnet that has the same orientation as the primary one. From the 'up' secondary magnet, only one beam emerges, a second 'up' sub-beam, labelled beam 2uu. From the 'down' secondary magnet, only one beam emerges, a second 'down' sub-beam, labelled beam 2dd.

Then beam 0 is mixed with respect to the chosen Stern-Gerlach magnet orientation, because it will emerge from the primary Stern-Gerlach magnet in two nearly equal sub-beams, beam 1u and beam 1d.

Also, beam 1u is pure with respect to the chosen Stern-Gerlach magnet orientation, because it will emerge as a single beam 2uu from its second Stern-Gerlach magnet. Likewise, beam 1d is pure with respect to the chosen Stern-Gerlach magnet orientation, because it will emerge as a single beam 2dd from its second Stern-Gerlach magnet.

To judge from your above remarks, it may be better that I stop at this stage, to let you reply to what I have just written.Chjoaygame (talk) 07:26, 7 February 2015 (UTC)

definitions

I suggest you use the definitions in quantum state instead of experiments of thought. By "pure" is not meant "pure with respect to an observable" (whatever that means). There is not a trace of mixed states (as defined in the referenced article) in this article, and I consider the topic being off. YohanN7 (talk) 08:08, 7 February 2015 (UTC)

"Whatever that means." Feynman <Lectures, volume III, page 5–5>: "The answer is this: If the atoms are in a definite state with respect to S, they are not in the same state with respect to T—a +S state is not also a +T state. There is, however, a certain amplitude to find the atom in a +T state—or a 0T state or a −T state."[Feynman's format of —]Chjoaygame (talk) 03:04, 5 November 2015 (UTC)

A more general definition would be that measurement of (observable) S with certainty would yield a certain value s. Mathematically, the state is an eigenstate of (the Hermitian operator) S. (Using the same symbol for observable quantity and corresponding operator is standard.) I don't mind using the terminology "state pure with respect to observable" as long as we define it and don't confuse it with "pure state", which is defined elsewhere in Wikipedia. YohanN7 (talk) 11:10, 5 November 2015 (UTC)

Thank you for your reply. I think it would be valuable to actually put an explicit version of that 'elsewhere' definition right here, or very much better, in your own words, for clarity of comparison, if you have time. I am asking not for a mathematical abstract expression, such as that of David J. Griffiths, who says he intends not to tell what quantum mechanics means, but rather to tell "how to do it"; "shut up and calculate", to quote Mermin. I am asking for a physical expression, like Feynman's. Wikipedia is not a reliable source.Chjoaygame (talk) 14:18, 5 November 2015 (UTC)

What do you want a physical expression for? "Pure state" or "Pure state with respect to observable"? If you are looking for "pure state" as opposed to "mixed state", then I'd suggest you go to the literature if you don't trust Wikipedia. I'm not suggesting Wikipedia as a reference for the article, but this is a talk page you know. I can only say that a superposition of eigenstates (more than one) of an observable is not a mixed state, but rather a pure state. But it is not a pure state with respect to the observable in question (as I am thinking about it here in this discussion). It may, however, be a pure state with respect to another observable. This is what Feynman alludes to. His S and T do not commute, hence if pure w r t S, then it is "mixed w r t T" (but not a mixed state). YohanN7 (talk) 12:19, 16 November 2015 (UTC)

Thank you for this reply. I value your care in this.

I was looking for a physical definition of a pure state. I have spent some effort on the literature, looking for a physical definition, but I find only mathematical ones, if you don't count 'pure with respect to a specified observable' as 'pure'. For definiteness, also I may refer to Wikipedia. There I am led to the article Quantum state, which is mathematical, not physical.

You advise me to "use the definitions in the article quantum state instead of experiments of thought". As I read this advice, it is to use mathematical definitions, not physical ones. My interest is in physics. Mathematics is useful for expression of physical ideas, but mathematics is not physics. I think physical ideas need expression in experimental terms as well as in mathematical terms. Without the experimental terms, they are undefined physically. I think experiments of thought are often near-enough to experimental expressions. With no experimental expression, we are not talking physics.

You write that "... a superposition of eigenstates (more than one) of an observable ... may, however, be a pure state with respect to another observable."

I agree. But I would go further and say that for a state to be pure, it must have an observable with respect to which it is pure. I am saying that a every pure state is pure with respect to some observable. Evidently you think I am mistaken in that?

There may be a problem here about the word 'ensemble'. It might have different meanings, depending on whether it was used in a physical or a probabilistic sense. In my view, a probability usually refers to an ensemble in a virtual sense used by probability theorists. Each single particle in an occasion of experiment has probabilities associated with it. Usually the probabilities are found in experiment by repeated, on occasion after occasion of experiment, practically identical preparations of single particles. By 'practically identical', I mean that definite identical practical steps are taken to prepare, on occasion after occasion of experiment, single particles. If there remain untaken further practical steps, by which the repeatedly prepared single particles could be further purified, then they are in a mixed state. If all possible further practical steps of purification have been taken, then the single particles are each in the same pure state. The repeatedly, on occasion after occasion of experiment, prepared single particles, constitute a probability theorist's virtual ensemble. A single occasion of preparation of a single particle does not provide experimental data for the determination of the relevant probabilities. In fact we cannot ensure that the practically identical particles are ultimately identical, whatever that might mean. I understand 'practically identical' in this sense to refer to single particles.

The single particles so prepared have not yet been observed. For observation, more is needed. In general, that 'more' is as follows. The single particles are passed, one by one, through some sorting process, and then detected by an array of particle detectors lying in the several output channels. The sorting process may be trivial, not actually doing anything. For a pure state, there is a definite sorting process, embodied in a definite physical device, symbolically designated by a definite linear operator, such that only one detector, of the several in the array, actually detects particles.

This is my idea of a physical definition of a pure state.

As I read you, the just-above story is, in your view, wrong. As I read you, you have not yet here proposed a physical definition of a pure state. Instead you advise me to be content with a mathematical one.Chjoaygame (talk) 22:00, 16 November 2015 (UTC)

Let us continue this discussion on either your or my talk page. You are welcome at my place.

I have not read the just-above story (I will, but not now). I can't see how you can argue with definitions. Definitions are there to establish terminology. You can't complain because you think the English word "pure" in "pure state" (through the definition) is misused in some linguistic or physical sense. You wrote,

I agree. But I would go further and say that for a state to be pure, it must have an observable with respect to which it is pure. I am saying that a every pure state is pure with respect to some observable. Evidently you think I am mistaken in that?.

You may be mistaken in that, depending on what you want. Mathematically you are right, because you can construct a Hermitian operator = observable with the state in question as an eigenvector. But its eigenvalues will in general have little to do with physically measurable quantities, which is what I guess you are after, in which case you are wrong.

Here is one attempt at definition of "pure state": Every vector in Hilbert space represent a pure state. (Mixed states are described not by state vectors, but by density matrices.) YohanN7 (talk) 10:11, 17 November 2015 (UTC)

Thank you for this. Following your suggestion, I will reply on my talk page.Chjoaygame (talk) 10:44, 17 November 2015 (UTC)

With respect

With respect, Wikipedia is not a reliable source. I am not willing to accept it as such for the present discussion. I have offered what I think is a suitable paradigm for a definition of a pure state. It seems you think it not a good definition. I think the next step in our discussion would best be that you would say in your own words how you would define a pure state. Also I should perhaps back up my definition. I will do so in due course.

I am a little puzzled that you seem to object to my using the Stern-Gerlach experiment as I have done here. I am working from Chapter 6 of volume III of Feynman, Leighton, Sands, as I noted in my comment. A rather similar approach is taken by B.-G. Englert's posthumous edition of Schwinger's Quantum Mechanics: Symbolism of Atomic Measurements, Springer (2001).Chjoaygame (talk) 09:54, 7 February 2015 (UTC)

From the article that you don't trust:

Mathematically, a pure quantum state is represented by a state vector in a Hilbert space over complex numbers, ... .Griffiths, David J. (2004), Introduction to Quantum Mechanics (2nd ed.), Prentice Hall, ISBN 0-13-111892-7 If this Hilbert space is represented as a function space, then its elements are called wave functions.

There's your pure state with reference complete with page number and all. If you worry about what this has to do with physics, then you can turn to the first postulate of QM that says precisely that (pure) physical states are represented by vectors in Hilbert space.

For mixed states, I suggest you look up paragraph 14 in L&L. (They put quotation marks around "mixed". (B t w, Landau invented the density matrix afaik)). You can also read p 68 and on in Weinberg (he doesn't baptize mixed states, but his description is as any good description).

Your "definitions" are homemade and completely unintelligible to me. I'll not discuss them. YohanN7 (talk) 10:42, 7 February 2015 (UTC)

Yes, you give mathematical definitions, which are about mathematical representations. But my concern here is as to their physical meaning, which neither the Wikipedia article you cite, nor your definition, make any attempt to address. According to Rosenfeld, a reliable source on Bohr's thinking and Copenhagenism in general, without physical context, which you avoid, the mathematical formulas are physically meaningless. Since you avoid considering the physical meaning, and I am largely concerned with it in combination with the formulas, it is hardly surprising that that you find my thoughts not worth discussing. I am sorry about this.Chjoaygame (talk) 13:38, 7 February 2015 (UTC)

"species"

Chjoaygame, for one thing, "species" in this physical/chemical context usually means "which subatomic particle/composite subatomic particle/atom/molecule/compound", e.g. electrons, hydrogen atoms, helium nuclei, silver atoms in a beam, etc. . What does this term have anything to do with "wave function", considering all the particles in the system regardless of what they are, are described by the wave function? And no, this article should not be about ensembles and mixed states. M∧Ŝc²ħεИ_τlk 14:52, 7 February 2015 (UTC) I added the underlined segments, typed the above very fast. M∧Ŝc²ħεИ_τlk 16:45, 7 February 2015 (UTC)

The system is composed initially of some species or generalized species in some state, and the particular wave function of interest is specific for that. Yes, I agree this article should not be about ensembles and mixed states.Chjoaygame (talk) 16:02, 7 February 2015 (UTC)

How is the wave function "specific" for this or that species? It isn't. What does matter are quantities which affect statistics, most obviously the spin of the particles, and other quantum numbers, but also the number of particles, and the number of spatial/momenta dimensions (e.g. 1d, 2d, and 3d and higher dimensions, all have various effects on statistics), and there could be others. M∧Ŝc²ħεИ_τlk 16:45, 7 February 2015 (UTC)

The wave functions for the hydrogen atom in its various states are different from those for the hydrogen molecule in its various states. That's what I mean by specific.Chjoaygame (talk) 18:26, 7 February 2015 (UTC)

What about hydrogen-like atoms or ions? The family of wave functions is not so different to that of the hydrogen atom is it? Similarly for analogues of the hydrogen molecule, what about systems which have particles of different masses and charges but the same spin and statistics and are subject to the same potential as for the hydrogen molecule? M∧Ŝc²ħεИ_τlk 18:43, 7 February 2015 (UTC)

Back to your original point, it seems you are the one who has conflated "state" with "species" while reading the article. The article doesn't say, in what you term "ordinary language", that a state refers to "species of quantum system", or "species of quantum entity", or "kind of particle", "or some such". It also doesn't say "species of quantum system can be prepared in several different states". It is also completely irrelevant, circular, and meaningless to say "a pure quantum state is described by a particular eigenfunction of the operator with respect to which it is pure". M∧Ŝc²ħεИ_τlk 18:53, 7 February 2015 (UTC)

Key to my concern

The key to my concern here was and still is that, in the traditional or older conception, a wave function represents a pure state, while there is no state pure with respect both to to position and to momentum. Position and momentum purity are physically, or in actuality, incompatible, and cannot co-exist in nature; their mathematical expressions as operators do not commute. Thus a 'state', to which "there corresponds two wave functions, Ψ(x, S_z) and Ψ(p, S_y), both describing the same state", is an abstract kind of object, at a level of logical abstraction further from actuality than the older or traditional wave function that represents a particular physical state. We seem perhaps to be somewhere near what Alfred North Whitehead, co-author of Principia Mathematica, called 'the fallacy of misplaced concreteness'.Chjoaygame (talk) 16:35, 6 February 2015 (UTC)

States are no more abstract than wave functions as explained in the article. They are only clearer as what what is meant. There is a one-to-one correspondence between states and wave functions in different representations (as also explained in the article). Wave functions have not changed one bit since the old days. They (each) provide a complete description of a system in a given state.YohanN7 (talk) 06:19, 7 February 2015 (UTC)

Further edits

Since my post copied above, there has been extensive editing which might be construed as related to my post copied above. Read as they stand, these edits seem to relate to it, but they seem to me not to resolve the problem altogether.

For comparison, the article currently reads:

• Wave functions corresponding to a state are accordingly not unique. This has been exemplified already with momentum and position space wave functions describing the same abstract state. This non-uniqueness reflects the non-uniqueness in the choice of a maximal set of commuting observables.
• The abstract states are "abstract" only in that an arbitrary choice necessary for a particular explicit description of it is not given. This is the same as saying that no choice of maximal set of commuting observables has been 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 representation yielding two different, but entirely equivalent, explicit descriptions of the same state for a system with no discrete degrees of freedom.
• Corresponding to the two examples in the first item, to a particular state there corresponds two wave functions, Ψ(x, S_z) and Ψ(p, S_y), both describing the same state. For each choice of maximal commuting sets of observables for the abstract state space, there is a corresponding representation that is associated to a function space of wave functions.Chjoaygame (talk) 16:35, 6 February 2015 (UTC)

It is very nice of you to quote the article in addition to old posts by yourself. But no, your old post was just too confused to have influenced the editing. YohanN7 (talk) 06:22, 7 February 2015 (UTC)

Consequences

I think this indicates a radical problem in the article, as follows. I will start by referring to

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

Weinberg on Dirac's notation

Apart from the short section on pages 57–58 about Dirac's bra-ket notation, throughout the book, Weinberg uses the notation ( . , . ), not Dirac's bra-ket notation.Chjoaygame (talk) 16:35, 6 February 2015 (UTC)

So? Each to his own. YohanN7 (talk) 06:23, 7 February 2015 (UTC)

A distinction drawn by Weinberg

Weinberg writes on page xvi:

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

Weinberg is at fault in logic here. He is conflating [physical] states with their [mathematical] representatives. Properly speaking, there is no inner scalar product of states, because they are physical, not mathematical, objects. There are inner scalar products between vectors, and dual vectors, which are mathematical objects. That it is proper for Wikipedia to fault Weinberg in logic is apparent to a careful Wikipedia editor reading Weinberg's Section 3.7 on "Interpretation of Quantum Mechanics", in which on page 95 he writes "We can live with the idea that the state of a physical system is described by a vector in Hilbert space, rather than by numerical values of the positions and momenta of all the particles in the system, but it is hard to live with no description of physical states at all, only an algorithm for calculating probabilities." Here Weinberg does not conflate. The aforementioned conflation is thus seen as a mistake by Weinberg. Part of a Wikipedia editor's mandate is to ensure reliability of sources. This may require survey of many candidate sources.

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

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

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

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

ψ(x) = (Φ_x,Ψ)."

Weinberg writes on page 57:

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

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

See also Cohen-Tannoudji et al. on page 147, as below. And Zettili 2009.Chjoaygame (talk) 16:35, 6 February 2015 (UTC)

As noted in the next section, and as pointed out by another editor, I mistakenly wrote above "inner product". The proper term in this context, according to reliable sources, is 'scalar product'.Chjoaygame (talk) 05:57, 28 December 2015 (UTC)

Problem in the article

Though it is interesting and informative, the present article has an element of ambivalence. The present lead writes "For a given system, once a representation corresponding to a maximal set of commuting observables and a suitable coordinate system is chosen, the wave function is ..." This is not far from the wording of Bransden & Jochain <1983, page 69> "Once a given basis — also called a representation — has been chosen, Ψ is completely specified by its components c_j ." This wording is strongly suggestive that an abstract state vector wave function (Chjoaygame (talk) 09:04, 28 December 2015 (UTC)) refers to a definite pure state, as opposed to an abstract class of states, which seems to be the concept of the rest of the article.

Besides this there is a larger problem. Evidently, along with Cohen-Tannoudji et al. and with Zettili, Weinberg regards Schrödinger's wave mechanics as less general than Weinberg's own approach in terms of Hilbert space. And he regards the wave function as belonging to Schrödinger's wave mechanics. Weinberg is one of the main sources cited by the article. But Weinberg's distinction (between Hilbert space state vectors and wave functions,Chjoaygame (talk) 09:04, 28 December 2015 (UTC)) is largely rejected by the present version of the article, which mostly conflates wave functions with abstract vectors. The citations are therefore inaccurate or misleading. The viewpoint of the present version of the article, conflating wave functions with Hilbert space vectors, is thus original research, or perhaps merely erroneous. Such is not permitted by Wikipedia policy.Chjoaygame (talk) 16:35, 6 February 2015 (UTC)

Which citation is inaccurate or misleading? Wave function are Hilbert space vectors as explained in both the lead and in the section Wave functions and function spaces. This you can find in every reliable source (at the basic level). It is taught in the kindergarten of quantum mechanics. The Schrödinger equation is linear. Therefore its solutions can be added and multiplied with scalars. There is an inner product on this function space (given in every reliable source). YohanN7 (talk) 06:46, 7 February 2015 (UTC)

Possible remedies

To this unfortunate error, two remedies suggest themselves. One is to give the article some new title that accurately reflects its deeply abstract content, such as perhaps Hilbert space vectors. Then a fresh article should be written that is truly titled Wave function. The other remedy that suggests itself is a thorough rewrite of the present article, to make it conform its advertised title, Wave function, as distinct from 'Abstract Hilbert space vectors'.

It seems that the present error arose partly because of a desire to build the article on the most abstract lines. The error arose partly also because the article was written largely with apparent disregard for the usual Wikipedia policy that in-line sources should be well supplied. Care would be needed in the rewrite to give accurate in-line reliable sources.

There is imaginable a possibility that a temptation might be felt to cast the new Wave function article again according to the abstract conceptual structure of the present article, arguing that the concept of the wave function is partly outdated. That this might happen is suggested by the fact that the present article already has a subsection that proposes that the topic Wave function is not a leading idea of the quantum theory of fields, which prefers to work with field operators. Obviously, experts in quantum field theory would be in a good position to create a new article entitled Quantum field operators or some such. That would leave the title Wave function comfortably free to concentrate on that topic, unconstrained to try to cover or be based on the distinctly more abstract concept of Hilbert space vectors.

A question still perhaps remains unresolved. Is the [generalised] 'state' of a system to be defined in an abstract modern way as the set of all possible [particular] states of the species of particle, or is the state to be defined in the older traditional directly particular way, by the actual initial condition of the particles as they emerge from the chosen preparatory device into the experimental registratory field, pure (and described by a wave function) or mixed (and described by a density operator) as the case may be? I would favour the choice that the article entitled Wave function should take as its primary focus the traditional wave function that refers to an actually possible particular state, with perhaps a short section explaining how the traditional concept has been more or less superseded by a more sophisticated abstract concept of a class of possible states.Chjoaygame (talk) 16:35, 6 February 2015 (UTC)

The article puts wave functions into context, namely quantum mechanics of yesterday and today. You seem to disapprove of that. Regarding inline citations, well, there could be more. But, and this is important, the subject of the article is basic and in the reference list there are more than six classic textbooks (I have read only six of them in the reference list) on quantum mechanics where the reader can look up for himself. If you want inline citations added, well, add them.

What I object to is that the abstract Hilbert space "context" so dominates the article that it is not easy to find in the article an explicit and direct description of a wave function as conceived by Schrödinger and indicated by Weinberg.Chjoaygame (talk) 09:12, 28 December 2015 (UTC)

Again, your last paragraph, the very last sentence in fact, again (see also further up, same blunder) illustrates your misunderstanding of the subject. This is not meant as an insult, it is actually well meant. Please, read up on the subject. Read any book, not just the history sections. It is really not very advanced what is treated in the article, and you have understood wrong that a state is referring to something else than (what you call) "wave function that refers to an actually possible particular state". Right now you are just wasting your own time, and hinder people from touching the article. If I would have known that this was coming, then I wouldn't have touched the article with a barge pole to begin with.

You say that my words are a "blunder". I think you miss the point. You write that the Hilbert space state vector is "abstract". The meaning of the word 'abstract' is that something has been omitted. But you do not concede that something has been omitted, and you don't seem to recognize the nature of the omission. The state vector alone, without a specified basis, does not specify a physical state. Yes, many abstract calculations can proceed without such specification, and that is why the abstract or symbolic (Dirac's word) formulation is useful, and is the reason why the 'transformation theory' works. But Dirac also points out that the fuller, less abstract, form is often handier for specific problems. The two forms are different, not to be conflated as you do to them.Chjoaygame (talk) 09:26, 28 December 2015 (UTC)

Chjoaygame, a vector is a vector whether a basis is given or not. Different bases – same vector, same physical state. No basis – still same vector, and, again, still same physical state. This is what you have to conclude when you speak of the state vector. If you have a problem with a formulation or another in the article (of course there may be errors in the formulation here and there), please edit instead of trying to change established terminology in mathematics and physics world wide. YohanN7 (talk) 13:45, 28 December 2015 (UTC)

You continue to argue as if the mathematical formula were not an abstraction. That is a fallacy of misplaced concreteness. The vector encodes some but not all information about the physical state, but is not itself a physical state. The basis encodes further information about the physical state. It would be unhelpful if I were to make fiddling edits to the article when there remains disagreement about substantial principles. Perhaps some progress may be possible on my talk page, on our unfinished investigation, which seems to have stalled.Chjoaygame (talk) 14:17, 28 December 2015 (UTC)

The state vector encodes all information about the physical state. It is not a physical state. It corresponds to one. QM 101. YohanN7 (talk) 14:29, 28 December 2015 (UTC)

If you still don't like the reasoning, think of a particular physical state first. This corresponds to a state vector in Hilbert space (by the zeroth postulate of QM). No basis needed to have a vector. The abstractness lies in that in order to describe the state mathematically, we have to resort to a basis. YohanN7 (talk) 14:25, 28 December 2015 (UTC)

I think it a more likely way forward that we continue our stalled investigation on my talk page, as you proposed when your computer problem interrupted it.Chjoaygame (talk) 15:28, 28 December 2015 (UTC)

The article currently contains the following: "Wave functions corresponding to a state are accordingly not unique." Your above remarks also use the ill-defined term "correspond". The article does not deal satisfactorily with the non-uniqueness. This is because the thinking of the article is primarily from a purely mathematical mindset, not a physical point of view.

This problem is not confined to this article. For example, the current version of the article Quantum state has as its leading definition: "A pure quantum state is a vector". As I read it, this contradicts your above distinction: "It is not a physical state. It corresponds to one. QM 101." As it happens, I agree with your distinction just made, not with the cited current version of the Quantum state article. The latter was posted with the edit summary: "in QM pure states are vectors, period.".

I am not so happy with your remark "QM 101". Since I think we are debating principles, I would prefer some kind of reference such as "There is the symbolic method, which deals directly in an abstract way with the quantities of fundamental importance (the invariants, &c., of the transformations) and there is the method of coordinates or representations, which deals with sets of numbers corresponding to these quantities."<Dirac 1st edition preface p. vi, reproduced on p. viii of the 4th edition.> or "We now assume that each state of a dynamical system at a particular time corresponds to a ket vector, the correspondence being such that if a state results from the superposition of certain other states, its corresponding ket vector is expressible linearly in terms of the corresponding ket vectors of the other states, and conversely."<Dirac 4th edition, p. 16.> I note that there Dirac is not using the term 'wave function'. He does, however, in places use that term, as for example here: "In the accurate mathematical theory each translational state is associated with one of the wave functions of ordinary wave optics, which wave function may describe either a single beam or two or more beams into which one original beam has been split."<Dirac 4th edition, p. 8.> These matters are not child's play.Chjoaygame (talk) 04:12, 30 December 2015 (UTC)

abstraction

A wave function that has not been abstractified by the Dirac scheme is a differentiable function—with domain a distinguished explicitly specified generalized quantum configuration space, and range the field of complex numbers, or, for particles with spin, certain more complicated sets that do not need to be detailed here—that satisfies the relevant Schrödinger equation. The symbolic method of Dirac is abstract because it does not specify the state explicitly as such a function. It designates a state by a corresponding symbol for a vector in a suitable Hilbert space.

A generalized quantum configuration space may be illustrated by some examples. (1) For a system of several particles with spin, the position–spin configuration space is spanned by a position vector for each of the particles and a spin for each of them. (2) The momentum–spin configuration space is spanned by a momentum vector for each of the particles and a spin for each of them. (3) Another permitted generalized quantum configuration space is spanned by position vectors for some of the particles, and momentum vectors for the rest, and a spin for each.

The generalized quantum configuration spaces are like classical configuration spaces, with spins attached. A state vector symbolizes such a function as is mentioned in the immediately foregoing, but does not detail it. The omission of the detail of the configuration space, and of the functional form, is the reason why the symbolic method is said to be abstract.

For brevity one may here call a wave function, that is expressed as a differentiable function on a quantum configuration space, by the name 'explicit wave function'. Then the present version of the article contains only the the abstract expressions of wave functions except for the section headed Examples, which contains only explicit wave functions, not abstract expressions of wave functions, and except for the diagrams.Chjoaygame (talk) 02:13, 6 January 2016 (UTC)

uniqueness

The wave function that corresponds to a state also corresponds to the state vector that symbolizes it, and these correspondences are unique. What is not unique here is how a state can be represented as a superposition of other states. For example, a wave function, for a system consisting of two different particles, on a configuration space of two positions, can be represented as a superposition of wave functions on a configuration space of a position and a momentum, as a superposition of wave functions on a configuration space of a momentum and a position, or as a superposition of wave functions on a configuration space of two momenta. That such different superpositions are possible is the meaning of the non-uniqueness.

pure state

A wave function corresponds to a pure state. This means that there is a distinguished generalized quantum configuration space such that no superposition is needed for the description of the wave function: for that space, the wave function is an eigenfunction for each dimension. The state is pure with respect to the analyzer for each dimension of the distinguished configuration space. The state is mixed with respect to analyzers other than those distinguished ones. It is the existence of such a distinguished set of analyzers that makes the state pure; in fact the pure state is prepared just by passing the beam through all the analyzers of the distinguished set.Chjoaygame (talk) 21:57, 5 January 2016 (UTC)

If there are many readers like you, then it is absolutely necessary to include material about the context of wave functions (precisely the things you dislike) to ward off the kind of misinterpretations that you have demonstrated that you have adopted. YohanN7 (talk) 07:24, 7 February 2015 (UTC)

It puzzles me that you say that I "dislike" wave functions. I am just saying that this article's title is Wave function, which Weinberg and others distinguish from state vector, while the main topic of the body of this article is state vectors.Chjoaygame (talk) 05:47, 28 December 2015 (UTC)

Examination of possible sources

Dirac According to Dirac<The Principles of Quantum Mechanics, 4th edition> <pp. 11–12> "A state of a system may be defined as an undisturbed motion that is restricted by as many conditions 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 left undisturbed after the preparation. ... The general principle of quantum mechanics of superposition applies to the states of ... any one dynamical system. It requires us to assume that between these states there exist peculiar relationships such that whenever the system is definitely in one state we can consider it as being partly in two or more other states. The original state must be regarded as the result of a kind of superposition of the two or more new states, in a way that cannot be conceived on classical ideas. Any state may be considered as the result of a superposition of two or more other states, and indeed in an infinite number of ways. Conversely, any two or more states may be superposed to give a new state." <pp. 15–16> "It is desirable to have a special name for the vectors which are connected with the states in quantum mechanics ... We shall call them ... kets, and denote a general one of them by a special symbol  |〉. ... We now assume that each state of a dynamical system at a particular time corresponds to a ket vector." <page 80>"The system of notation provided by the wave functions is the one usually used by most authors for calculations in quantum mechanics. ... A wave function is just the representative of a ket expressed as a function of the observables ξ instead of the eigenvalues ξ′ for those observables." <page 81>"The standard ket and bra are defined with respect to a representation." From this I conclude that one could suppose a more abstract ket that represents a class of states of a given species (a species called by Dirac a dynamical system) of quantal entity, each several state being pure with respect to its characteristic quantum analyser. Mixed states are not represented here. These abstract kets are partly involved in the present state of the article, and come from, for example Cohen-Tannoudji, and Weinberg. I think they are not the kets of Dirac, which are always defined with respect to a particular representation (see page 81). Such an abstract ket does not define a unique state. It is therefore a more abstract object than a wave function. Moreover, an abstract ket is not a concrete physical object, for example an electron. It is a class of pure states of a dynamical system such as an electron. On the other hand, a wave function does uniquely correspond to a state pure with respect to its characteristic quantum analyser. A ket contains all possible characteristic information about a dynamical system or quantal entity. A wave function does not directly express all that information, but often can be mathematically transformed so as to provide it. Boundary conditions are relevant here.Chjoaygame (talk) 16:35, 6 February 2015 (UTC) Kramers "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 = hν 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.>Chjoaygame (talk) 16:35, 6 February 2015 (UTC) Messiah "When the system is in a state represented by a wave of type II.34 [Ψ = ψ e−iEt/ħ], it is said to be in a stationary state of energy E; the time-independent wave function ψ is usually called the wave function of the stationary state, although the true wave function differs from the latter by a phase factor exp (−iEt/ħ)." <Messiah, A. (1961). Quantum Mechanics, volume 1, translated by G.M. Temmer from the French Mécanique Quantique, North-Holland, Amsterdam, page 72.> <page 204> "At the initial time t0 one prepares the system by performing on it the simultaneous measurement of a complete set of compatible variables. Its dynamical state is thus completely determined at time t0." <page 243> "The entire interpretation of wave mechanics developed in Chapters IV and V had as its starting point the definition of the probability densities of position and momentum by means of the wave functions Ψ and Φ referring to configuration and momentum space respectively." <page 244> "... just as the functions Φ and Ψ are equivalent representations of one and the same dynamical state ..." This seems odd. Presumably, being wave functions, Φ and Ψ are eigenvectors of observable operators respectively for position and momentum, representing respective physical pure states. <page 244> "One is thus led to build up the entire Quantum Theory by starting directly from the vector concept, without reference to the particular representation which can be made thereof." <page 245> "... we associate with every dynamical state a certain type of vector, which we call, following Dirac, ket vector or ket and which we represent by the symbol  |〉." This sentence suffers from a grammatical error, which makes it perhaps ambiguous. Probably the writer intended "we associate with every dynamical state a vector of a certain type". Is the association intended to be one-to-one?Chjoaygame (talk) 16:35, 6 February 2015 (UTC) Schiff <Schiff, L.I. (1955/1968). Quantum Mechanics, 3rd edition, McGraw–Hill, New York.> <page 171> "... we wish to reserve the term "representation" for designation of the choice of axes in Hilbert space, or, equivalently, for the choice of the complete orthonormal set of functions, with respect to which the states and dynamical variables are specified."Chjoaygame (talk) 16:35, 6 February 2015 (UTC) Bransden & Jochain 1983 <Bransden, B.H., Jochain, C.J. (1983). Physics of Atoms and Molecules, Longman, Harlow, UK, ISBN 0-582-44401-2.> <page 69> "Once a given basis — also called a representation — has been chosen, Ψ is completely specified by its components cj ." Like Dirac, Bransden & Jochain expect to be given a representation. Like Cohen-Tannoudji et al., B & J refer the reader to Schiff. B & J also refer the reader to Messiah.Chjoaygame (talk) 16:35, 6 February 2015 (UTC) Cohen-Tannoudji et al. Volume 1, 2nd edition <page 4> "This avoids the useless repetition which results from presenting the more general bra-ket formalism only after having developed wave mechanics uniquely in terms of wave functions." <page 19> "The quantum state of a particle such as the [spinless] electron is characterised by a wave function ψ(r,t), which contains all the information it is possible to obtain about the particle." <page 109>    "      ψ(r) ←→ |ψ〉" <page 147>    "〈r|ψ〉 ←→ ψ(r) 〈p|ψ〉 ←→ ${\bar {\psi }}(\mathbf {p} )$." Evidently, there is a correspondence of a kind between wave functions and kets as defined by Cohen-Tannoudji, but they are different things, and many wave functions correspond to one ket as defined by Cohen-Tannoudji et al.. It is here evident that there are two radically different kinds of transformation in this. One kind is a transformation from one quantum configuration space to another (e.g. |ψ〉 → ${\bar {\psi }}$   and   ${\bar {\psi }}$ → |ψ〉, Fourier transforms of functions); the other is a coordinate transform within a fixed quantum configuration space, a geometrical transform, say $\mathbf {p}$ → $\mathbf {p} ^{\prime }$. The transform of quantum configuration spaces represents a change of experimental instrumentation in the laboratory. The transform of coordinates occurs in the physicist's notebook. A ket of Cohen-Tannoudji describes a class of physical states proper for a species of particle, while a wave function defines a particular physical state for the species. This text refers the reader to Schiff, and to Eisberg & Resnick.Chjoaygame (talk) 16:35, 6 February 2015 (UTC) Zettili 2nd edition <Zettili, N. (2009). Quantum Mechanics, 2nd edition, Wiley, Chichester UK, ISBN 978-0-470-02678-6.> <page 85> "Remark: In wave mechanics we deal with wave functions $\psi ({\vec {r}},t)$, but in the more general formalism of quantum mechanics we deal with abstract kets   |ψ〉. Wave functions, like kets, are elements of a Hilbert space. We should note that, like a wave function, a ket represents a system completely, and hence knowing   |ψ〉 means knowing its amplitudes in all possible representations. As mentioned above, kets are independent of any particular representation." Evidently thus, these kets of Zettili are not the kets of Dirac. These kets are the abstract vectors of Weinberg. Importantly for the present concerns, Zettili distinguishes between wave functions and abstract vectors. A wave function has its proper representation, but an abstract vector doesn't. This kind of abstract vector is different from a ket of Dirac, about which he writes <page 81> "The standard ket and bra are defined with respect to a representation." <page 166> "Postulate 1: State of a system The state of any physical system is specified, at each time t by a state vector  |ψ(t)〉in a Hilbert space ${\mathcal {H}}$ ;   |ψ(t)〉 contains (and serves as the basis to extract) all the needed information about the system. Any superposition of state vectors is also a state vector." Read in the light of Zettili's immediately foregoing statement of his page 85, this statement contradicts the view of Dirac (his page 81) and other classical quantum mechanics authors, who attach a wave function to a state that is pure with respect to its proper quantum configuration space and quantum analyser. This can be expressed by saying that Zettili's and Weinberg's and Cohen-Tannoudji's abstract vectors are classes of traditional quantum states. It seems to me that Zettili's statement of his Postulate 1 is contrary to the traditional idea of a quantum state.Chjoaygame (talk) 16:35, 6 February 2015 (UTC)

Inner product

There is a world of understanding (and a Nobel prize) between you and Weinberg. The second to last paragraph above pinpoints exactly the misunderstanding of yours. You do not need to express a state in terms of a particular observable to make it a "particular" state. All information abut a state is contained in the state vector so it is already "particular". An electron doesn't give a damned whether you express it's state abstractly or as a wave function of momentum or as a wave function of position.

Besides, regarding your first paragraph. The inner product on Hilbert space (any which one, the abstract state space or a Hilbert space of wave functions) is modeled on physical reality. The fact that two states (or wave functions if you want) are orthogonal has physical interpretation. YohanN7 (talk) 06:35, 7 February 2015 (UTC)

Might also add here that the inner product is between vectors, not between vectors and dual vectors as you believe. See Wave function#More on wave functions and abstract state space. YohanN7 (talk) 12:26, 15 March 2015 (UTC)

According to Gottfreid

According to Kurt Gottfried:

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

In a later edition, we read:

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

Is Gottfried a reliable source on this point?Chjoaygame (talk) 06:53, 13 September 2015 (UTC)Chjoaygame (talk) 15:27, 30 September 2015 (UTC)

Never heard of him. Perhaps reliable, but decidedly wrong. It is a standard mistake made by many, and this is pointed out strongly in some math texts, e. g. Lie Groups, Lie Algebras, and their representations by Brian C. Hall. See also Inner product. Please do not push this. As I recall, the article explains well why one gets away with confusing the concepts. YohanN7 (talk) 12:26, 29 September 2015 (UTC)

That is to say, the left argument in

$\langle \Psi _{1}|\Psi _{2}\rangle ,$

is a member of the space, not the dual. This is so even if

$\langle \Psi _{1}|$

is a member of the dual. Confusing perhaps, but decidedly true (and explained in the article). Perhaps not well enough. YohanN7 (talk) 12:34, 29 September 2015 (UTC)

I am puzzled by your request "Please do not push this." I thought it trivial until you drew attention to it, I suppose because it is significant.

Gottfried's being decidedly wrong on this point would seem to imply that he is not a Wikipedia-reliable source on this point.

You say it is perhaps confusing. Perhaps, therefore, you may very kindly be willing to help me along here. In the article, I find this:

The state space is postulated to have an inner product, denoted by

$\langle \Psi _{1}|\Psi _{2}\rangle ,$

that is (usually, this differs) linear in the first argument and antilinear in the second argument. The dual vectors are denoted as "bras", ⟨Ψ|. These are linear functionals, elements of the dual space to the state space. The inner product, once chosen, can be used to define a unique map from state space to its dual, see Riesz representation theorem. this map is antilinear. One has

$\langle \Psi |=a^{*}\langle \psi |+b^{*}\langle \phi |\leftrightarrow a|\psi \rangle +b|\phi \rangle =|\Psi \rangle ,$

where the asterisk denotes the complex conjugate. For this reason one has under this map

$\langle \Phi |\Psi \rangle =\langle \Phi |(|\Psi \rangle ),$

and one may, as a practical consequence, at least notation-wise in this formalism, ignore that bra's are dual vectors.

Is this one of the passages in the article to which you intend to draw my attention? Or should I be looking elsewhere? And perhaps you may also very kindly be willing to say what, if any, is the significance of this?Chjoaygame (talk) 15:45, 29 September 2015 (UTC)

Yes, that is the passage. First off, the whole issue is notational, and it is peculiar to the Dirac bra-ket notation. The significance is that many (including physicists) gets tricked into believing that the inner product marries a bra and a ket. That is, they think that we really have the action of a linear functional on a vector. In the case of the inner product, we do not. If that were the case, we wouldn't have a Hilbert space to begin with (postulate one of QM would be ruined).

The point of the passage is that the misconception, while regrettable, is harmless. You would never have the same issue if you used another notation for the inner product, say like

$(\Phi ,\Psi ),$

(like e.g. Weinberg) because no elements of the dual space can be suspected here. I hope this helps. Let me know otherwise. (I'll not be very much at the computer the nearest future.) YohanN7 (talk) 13:10, 30 September 2015 (UTC)

Aha! I have looked closer at Gottfried's second statement:

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

He is falling into the trap just like I thought. He gets the order wrong. The thing is this. Without an inner product, there is no way of canonically identifying kets and bras. You must have an inner product first.. Then you can make the identification. It is done precisely as described in "the passage". When this has been done, there is a canonical identification between bras and kets. Thereafter, any misconception about what the left argument is in

$\langle \Psi _{1}|\Psi _{2}\rangle ,$

really is is completely harmless (because you can think of it either way under the identification) - but regrettable (e.g. his book spreads the misconception). YohanN7 (talk) 13:25, 30 September 2015 (UTC)

Thank you for this.Chjoaygame (talk) 16:17, 30 September 2015 (UTC)

Just a small point: I think Gottfried is not an eccentric citation. Having said LL is a good book, J.S. Bell writes:

The second good book that we will look at here is that of Kurt Gottfried (Quantum Mechanics Benjamin 1966). Again I can give three reasons for this choice:

      (i) It is indeed a good book. The CERN library had four copies. Two have been stolen - already a good sign. The two that remain are falling apart from much use. ......

Not that I think very highly of Bell.Chjoaygame (talk) 13:22, 23 January 2016 (UTC)

Yes, I probably shouldn't be vetting Gottfried. It wasn't really my intention, but it reads that way now. It is entirely possible that he provides enough context making his own definitions to make things entirely correct and internally consistent. This is the big danger with quotes. They come without context. YohanN7 (talk) 12:51, 25 January 2016 (UTC)

Thank you for this and your other recent reply, on the Born rule. Since writing the above, I have learnt a little more of Gottfried. I can't say anything useful about it so far, but it is interesting/puzzling. As for your careful comments on the Born rule, again I haven't much useful to say, hence am silent for now. I would like to continue on my talk page some time. Bell's 1990 paper cited above is nearly a copy of his presentation at Erice, published in a 1989 report edited by A.I. Miller with title 'Sixty-two years of Uncertainty', Plenum Press.Chjoaygame (talk)

According to Dirac

As I read Dirac, his view is like that of Gottfried. They both speak of "the scalar product", not the inner product. It is perhaps not too important, but one likes to try to get appropriately sourced terminology.

Halmos for finite dimensional vector spaces presents the concept of duality on page 20. He derives the natural correspondence between a vector space and its dual on page 25, without speaking of orthogonality or inner product, though orthogonality is in a sense implicit in what he writes there. He introduces orthogonality and inner products on page 118, and then links them back to duality. I don't know how far that would work for infinite-dimensional vector spaces.

Dirac does not speak of inner products as distinct from the action of a dual vector upon a vector. He writes in § 6 of Chapter 1 of the scalar product as the action of a bra on a ket, and thereby defines orthogonality.

According to others

Like those of Dirac and of Gottfried, Messiah's systematic text also speaks of the scalar product, and not of the inner product. So also do the Lectures of Weinberg.

It may be reasonable for the present purposes to speak of the scalar product.Chjoaygame (talk) 15:00, 27 December 2015 (UTC)Chjoaygame (talk) 05:18, 28 December 2015 (UTC)Chjoaygame (talk) 05:34, 28 December 2015 (UTC)

state or species ?

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

"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

"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

"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

"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

"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 = hν 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

"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

"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

"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

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, S_z) and Ψ(p, S_y), 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)

This edit significantly addresses the concerns that prompted my just foregoing remarks. I am still thinking this over.Chjoaygame (talk) 18:35, 2 February 2015 (UTC)

Thinking it over, I find the above comment on "basic states" is hardly a direct response to the concern I raised in this section. So I will not go further with it in this section. My concern, expressed in this section, is still alive, and I have addressed and extended it in a new section below, for ease of editing.Chjoaygame (talk) 16:43, 6 February 2015 (UTC)