Talk:Wave function/Archive 4

Lead edits
It's just my opinion, although recent edits to the lead seem to have gone backwards in clarity.


 * 1) Sbyrnes321 beat me to this at the end of this section. I don't know what a "quantum analyzer" is (a hypothetical or real particle detector?), and it was clearer to get the point across quickly and say "the wavefunction is a function of the all the position coords and time, or momenta and time" etc.
 * 2) A previous concern by Quondum here was mathematical. Is the codomain of the wavefunction always the complex numbers? The statement "However, complex numbers are not necessarily used in all treatments" has been deleted.

It is tempting to revert back to this lead version, but others should input first. M&and;Ŝc2ħεИτlk 17:57, 3 January 2015 (UTC)


 * Input. YohanN7 (talk) 18:49, 3 January 2015 (UTC)
 * To be a bit more precise. Chjoaygame, what is the point of rewriting the lead in an archaic style so that instead of the occasional lay-reader not understanding it, now nobody understands it? This is not the place to invent terminology. Even if Bohr and others might have used it, the language and tone of the early 1900's is inappropriate and not quite penetrable if you want to know what QM is about. It perhaps makes the thing easier to read, but it comes with the price of making it impossible to understand. If we have to compromise between readability and correctness, then correctness will always come out on top. An article about QM that makes you feel good because you understood every single word (in isolation) isn't necessarily good. YohanN7 (talk) 19:05, 3 January 2015 (UTC)


 * It seems, I know what is meant by "analyzers". Just observables. In the same spirit as I wrote in the previous section: 'it is hardly a good idea, to fix once and for all the "right" collection of "physically natural" observables'. Coordinates, yes; momenta, yes; but any other observable, still yes, provided that its spectrum is of multiplicity 1 (the squared momentum does not fit, for instance). Boris Tsirelson (talk) 19:28, 3 January 2015 (UTC)


 * Regardless, the whole lead is not better than it was before. "Quantum analyzer" is not a common term and isn't helpful when you can just say "observable", which is standard and useful terminology.
 * Just to add to my point 2 above: currently the lead now merely states:
 * "Other treatments of quantum physics have been proposed, for example by Louis de Broglie.[6]"
 * which is not helpful since yes there are other formulations of QM, but this is not the point. For this article it should state wavefunctions are not always complex-valued.
 * Also Chjoaygame, could we keep the discussion in this section from this point forward? Thanks, M&and;Ŝc2ħεИτlk 19:40, 3 January 2015 (UTC)


 * "I don't know what a "quantum analyzer" is (a hypothetical or real particle detector?)" It's the device that splits the beam into eigenstates that are sent to their respective detectors, as noted above.


 * I have undone my edit as suggested by Maschen.


 * I was driven to make the now undone edit because I think the lead should make the point that the wave function must be defined with respect to the possible measurements. For example London and Bauer (1939): "The wave function it uses to describe the object no longer depends solely on the object, as was the case in the classical representation, but, above all, states what the observer knows and what, in consequence, are his possibilities for predictions about the evolution of the object. For a given object, this function, consequently, is modified in accordance with the information possessed by the observer." I read the latter as a paraphrase of Bohr's injunction. In mathematical terms that translates to a requirement for the mathematical existence of at least one basis that refers directly to physical existence. London and Bauer's "modification" of the wave function is not an adjustment of its values; it is a restructuring of its functional nature. Also I think that it should be made clear that QM "measurement" is not measurement in the ordinary language sense, as emphasized by Bohr.Chjoaygame (talk) 20:18, 3 January 2015 (UTC)Chjoaygame (talk) 20:47, 3 January 2015 (UTC)


 * For reference here is the reversion.
 * Thanks, we do appreciate the good faith efforts, but writing something along historical lines is not usually helpful, since interpretations, terminology etc. moves on. It's just that overall, the writing was not well-written, right at in beginning paragraph you wrote about "adventures" (...??).
 * About "quantum analyzers", I started this thread at 17:57, you posted in this section at 18:21, so I hadn't seen what you meant. M&and;Ŝc2ħεИτlk 20:54, 3 January 2015 (UTC)


 * For the last point of Maschen, which is really an aside in this thread, there is no need at all to in the lead make the point that wave functions aren't necessarily complex valued under exceptional circumstances. The exceptions fall into two classes. The first class are exceptional circumstances where the imaginary part happens to vanish (Some 1-d problems, Majorana fermions in a certain basis). No mention needed. The second class is when a mathematical framework is used to transform the complex wave functions (and the machinery, including the Schrödinger/Dirac/Whatever equations) to a more complex domain, such as the domain of geometric algebra. (Ironically, this case is covered by complex valued wave functions.) Neither of these exceptions need to be mentioned in the lead. Why? Well, the reader might ask - if the wave function isn't a vector of complex numbers, then what the hell is it? This is not the place to elaborate on such questions. This is making something difficult more difficult. Connoisseurs of geometric algebra might object, but introducing GA in the lead here is overdoing it. The wave function is always an array of complex-valued functions. The fact that you can turn it to something else in a one-one fashion is immaterial. If that wasn't the case, most every lead needs to be rewritten. YohanN7 (talk) 20:35, 3 January 2015 (UTC)


 * Agreed.Chjoaygame (talk) 20:41, 3 January 2015 (UTC)


 * I was trying to not exclude other formulations of QM or mathematical machinery. If people want to cut that part out entirely that's fine, up to them, just write it clear so that no-one will ask the question again in future. M&and;Ŝc2ħεИτlk 20:54, 3 January 2015 (UTC)


 * I know of the previous discussion, and I thought myself the wording was an okay compromise back then. I have cut it out because I no longer think it is okay (too demanding for the reader). The place to reintroduce it is in a new subsection of one of the mathematical sections as just one more formalism. YohanN7 (talk) 21:09, 3 January 2015 (UTC)
 * I also removed the de Broglie-Bohm theory from the lead. It presence seemed to be motivated by that real numbers were used as opposed to complex numbers. (My taking is that de Broglie-Bohm theory is something over and above ordinary QM and has little to do with how it is represented in terms of numbers.) YohanN7 (talk) 21:20, 3 January 2015 (UTC)

Chjoaygame I think some of your edits have been good, just out of place. Some of it (the Stern-Gerlach thing) could, if edited, go elsewhere. But here is another point, "... the wave function must be defined with respect to the possible measurements...". No, this is not so. The interpretation of the wave function might have to be associated with something we can measure (or it wouldn't be accepted as an interpretation), but this is all POV. Granted, it is the POV of some great minds, but it isn't necessary to develop QM to involve measurements. YohanN7 (talk) 21:40, 3 January 2015 (UTC) (Here's my POV: Nature couldn't care less whether we measure it or not. YohanN7 (talk) 21:42, 3 January 2015 (UTC))
 * And mine: Nature measures itself in some situations, and does not in others, and this makes the quantum/classical interface; we are just one form of decoherence makers. Boris Tsirelson (talk) 06:47, 4 January 2015 (UTC)


 * What you say here needs some replies, but not here and now. According to Feynman, nobody understands QM.Chjoaygame (talk) 00:06, 4 January 2015 (UTC)
 * And he was right. YohanN7 (talk) 01:39, 4 January 2015 (UTC)


 * I've not seen the history of edits, and the controversial assertion that measurement is intrinsic to the definition of a wave function clearly cannot be made. To make this assertion would be to imply that the concept would not apply at all in the many-worlds interpretation. This is enough reason to exclude it. —Quondum 16:56, 4 January 2015 (UTC)

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

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

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

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


 * And in that example link where is the wavefunction a function of position and momentum? M&and;Ŝc2ħεИτlk 13:42, 14 January 2015 (UTC)


 * Did you truly expect to find any? I wouldn't have put it as mildly as I did if there were any. YohanN7 (talk) 13:45, 15 January 2015 (UTC)


 * Then what is misleading? Yes, you can use both the position and momentum operators together to solve ladder-like problems like the quantum SHO (an incredibly boring system but nevertheless important and useful), but this has nothing to do with writing a wave function as a function of both position and momentum. M&and;Ŝc2ħεИτlk 14:16, 15 January 2015 (UTC)


 * I think it actually does have something to do with it. It is an example of a canonical transformation. But this is of minor importance, I just thought the formulation was a bit to strong on the emphasis of never mixing $x$ and $p$. The new rule should be to not mix (the classical quantities) $a$ and $a*$. At least I think so, I haven't done the problem in years. The operators $â$ and $â^{&dagger;}$ certainly each commute with themselves. Then again, the virtue of Dirac's method is that the Schrödinger equation (corresponding to the $a$ or $a*$, there should be one since there is one for both $x$ and $p$) doesn't have to be solved. I am admittedly uncertain here, it was a long time ago I looked into this. YohanN7 (talk) 16:25, 15 January 2015 (UTC)


 * "Mixing" $x$ and $p$ is allowed for the operators (obviously e.g. commutation relations, orbital angular momentum). The wave function as a function of both position and momentum is a separate thing. What I mean is the wave function is not a function of the full phase space, but just position (and time, spin etc.) or momentum (and time, spin etc.). M&and;Ŝc2ħεИτlk 18:08, 15 January 2015 (UTC)


 * I know what you mean. The article might put it too strongly. I believe there is a wave function depending on either $a$ or $a*$, for $a = cx + ikp$ (where $c$ and $k$ are constants) describing the dynamics equally well. Do you say this is wrong? YohanN7 (talk) 18:27, 15 January 2015 (UTC)


 * I don't know. Maybe it is possible somehow using canonical transformations? I have done them in the Hamiltonian formulation of classical mechanics, but never in quantum mechanics. It just doesn't make sense for the wave function to depend on x and p because of the uncertainty principle, and what becomes of the relation between position and momentum space wavefunctions (Fourier transform)? M&and;Ŝc2ħεИτlk 18:51, 15 January 2015 (UTC)


 * Yes, canonical transformations is what I'm talking about, see post four levels up. It is a wave function of $x$ and $p$ in a sense. It is a wave function of the single variable $cx + ikp$, much like an analytical function is a function of $x$ and $y$, constrained to $z = x + iy$, not a function of $x$ and $y$ varying independently, but still a function of $x$ and $y$. Again, I don't say the formulation in the article is wrong, it just might be misleading, ruling out what we are discussing here. I have also not seen the motivation (in the article) that it is the HUP that would rule out $Ψ(x, p, t)$. If I'm right here, you can have $Ψ(x, p, t)$, it must just be constrained to be of the form $Ψ(cx + ikp)$ which does depend on $x$ and $p$. YohanN7 (talk) 19:21, 15 January 2015 (UTC)


 * For now, best to delete the phase space section. It can be re-introduced later. There is no motivation for the exclusion in this article since the section is just a short digression, but how could you have a function of two observables which do not commute (in this case x and p)? If you know all the position coordinates how do you know the momenta? What meaning does $Ψ(cx + ikp)$ have then? I don't know. If you have a source it may be interesting to add this to the article. M&and;Ŝc2ħεИτlk 19:39, 15 January 2015 (UTC)


 * The supposed wave function is not a function of variables whose canonically associated operators do not commute. From above, the operators $â$ and $â^{&dagger;}$ certainly each commute with themselves (because every operator commutes with itself). No wave function $Ψ(a, a*)$ exists (probably because $â$ and $â^{&dagger;}$ do not commute). Few things in QM have a sensible interpretation, this is quite general.
 * This is not intended for the article. Do you see now why the section of (the previous version of) the article could be misunderstood, provided I am right? YohanN7 (talk) 20:37, 15 January 2015 (UTC)


 * Maybe it is misleading, but I can't see the point in debating this further. M&and;Ŝc2ħεИτlk 21:16, 15 January 2015 (UTC)


 * Then don't reply is a manner sneezing me off as your final comment. YohanN7 (talk) 21:48, 15 January 2015 (UTC)

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


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


 * The tensor product section should definitely precede the multi-particle sections, and possibly the spin section as well. YohanN7 (talk) 14:11, 16 January 2015 (UTC)

Still not right
This formula,
 * $$\begin{align} \Psi(\mathbf{r},t,s_z) & = \psi_{-s}(\mathbf{r},t)\xi_{-s}(s_z) + \psi_{-s+1}(\mathbf{r},t)\xi_{-s+1}(s_z) + \cdots \\

& + \psi_{s-1}(\mathbf{r},t)\xi_{s-1}(s_z) + \psi_{s}(\mathbf{r},t)\xi_{s}(s_z) \,, \end{align}$$ is really saying nothing. The $(2s + 1)$ spin functions referred to in the above formula are a complete set of basis spin functions,
 * $$\xi_{s_z}(s'_z) = \delta_{s_z,s'_z}.$$

The formula then essentially says
 * $$\Psi(\mathbf{r},t,s_z) = \psi_{s_z}(\mathbf{r},t),$$

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


 * Based on this and this I was tempted to suggest the Kronecker delta expression above but held off since then you'd say that was wrong or too restrictive since the spin functions are complex valued. M&and;Ŝc2ħεИτlk 13:42, 14 January 2015 (UTC)


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

What makes you think I'd automatically say you're wrong? Not a habit of mine. And I wouldn't object very loudly to the statement $ℝ ⊂ ℂ$. The ground level is this:
 * $$\Psi : \mathbb{R}^4\times\{-s,-s+1,\ldots,s-1,s\} \rightarrow \mathbb{C}$$
 * $$\Psi(\mathbf{r},t,s_z) \equiv \psi_{s_z}(\mathbf{r},t) \equiv \xi_{\mathbf{r}, t}(s_z)$$
 * $$\psi_{s_z} : \mathbb{R}^4 \rightarrow \mathbb{C}$$
 * $$\xi_{\mathbf{r}, t} : \{-s,-s+1,\ldots,s-1,s\} \rightarrow \mathbb{C}$$

[On occasion this factors,
 * $$\Psi(\mathbf{r},t,s_z) = \psi_(\mathbf{r},t)\xi(s_z) = \phi_(\mathbf{r})\zeta(s_z, t),$$

and spin dynamics can be studied in isolation.]

Then one might want to define
 * $$\xi_{s_z} : \{-s,-s+1,\ldots,s-1,s\} \rightarrow \mathbb{C}; \quad \xi_{s_z}(s'_z) = \delta_{s_z,s'_z},$$

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

Any spin

 * Assuming you still mean:
 * $$\begin{align} \Psi(\mathbf{r},t,s_z) \equiv \psi_{s_z}(\mathbf{r},t) \equiv \xi_{\mathbf{r}, t}(s_z) & = \psi_{-s}(\mathbf{r},t)\xi_{-s}(s_z) + \psi_{-s+1}(\mathbf{r},t)\xi_{-s+1}(s_z) + \cdots \\

& + \psi_{s-1}(\mathbf{r},t)\xi_{s-1}(s_z) + \psi_{s}(\mathbf{r},t)\xi_{s}(s_z) \,, \end{align}$$
 * then the article will be updated later. Before that, I'd like to provide a transition from the "scalar form" to the vector form using an analogy the reader may understand and based on what you just said. The links above (especially the first) actually seem to describe this better than even the best books like LL. M&and;Ŝc2ħεИτlk 20:01, 14 January 2015 (UTC)


 * Link looks good. Suggestion: Associate to each function $ξ_{s_{z}}|undefined$ a vector $χ_{s_{z}}|undefined$ (tne image of $ξ_{s_{z}} ⊂ ℂ^{2s + 1}|undefined$) to make the transition to the vector notation. (I still don't see the benefit of the long expansion of $Ψ$ before introducing the $ξ_{s_{z}}|undefined$, since only one term survives.) YohanN7 (talk) 20:36, 14 January 2015 (UTC)


 * Why would we introduce (a particular choice of) basis spin functions without referring to the expression that includes them? Why include the long expression at all if only one term survives? Because it parallels the linear combination of the column vector after it. The expression should really be done for an arbitrary direction and the choice to the z-component be made. Correct me if wrong, but the spin functions are not the same for all directions and neither are the eigenvectors of the spin operator in other directions. Feel free to make edits. M&and;Ŝc2ħεИτlk 21:04, 14 January 2015 (UTC)
 * Ok! YohanN7 (talk) 21:10, 14 January 2015 (UTC)


 * If the "Ok!" refers to you editing (and no doubt sharpening/correcting), then I'll temporarily stay out of the way so we don't edit conflict, but I will come back to this soon. M&and;Ŝc2ħεИτlk 21:22, 14 January 2015 (UTC)
 * No, not editing, ok means I understand your explanation. (I better not edit because I'm too tired for that. Besides, it looks good on the surface of things at least.) YohanN7 (talk) 21:37, 14 January 2015 (UTC)
 * Ok! (as in I'll carry on editing). M&and;Ŝc2ħεИτlk 21:51, 14 January 2015 (UTC)

The wave function for a particle with spin can be expressed as a single complex number with space, time, and spin dependence, or an array of complex numbers with space and time dependence only. The space in either case may be referred to as "position-spin space".

The spin dependence is different to the space and time dependence, which are continuous variables. All three spatial coordinates $r = (x, y, z)$ can be known simultaneously, but not all of the components of the spin vector $S = (S_{x}, S_{y}, S_{z})$ can be known simultaneously, only one component and the square of its magnitude $|S|^{2}$ can. (This is determined by the commutation relations of the position and spin operators). This means all three position coordinates can be placed as variables in the wave function, but only one spin direction can be a variable.

For the case of spin, we choose a direction and project the spin along that direction. A conventional (but arbitrary) choice is the z-direction, which will be considered below. Other directions can be obtained from the z-component of spin if the wave function is transformed appropriately. The spin projection quantum number along the $z$ axis is denoted $s_{z}$. The $s_{z}$ parameter, unlike $r$ and $t$, is a discrete variable. In general, for spin $s$, the $2s + 1$ allowed values of $s_{z}$ are $s, s − 1, ..., −s + 1, −s$ and no other values. For example, for a spin-1/2 particle, $s_{z}$ can only be the two values $+1/2$ or $−1/2$. The spin quantum number is also very frequently used as an index, but this tends to obscure its place as a discrete variable.

As a single complex number, it is a linear combination of space and basis spin functions:


 * $$ \Psi(\mathbf{r},t,s_z) = \psi_{-s}(\mathbf{r},t)\xi_{-s}(s_z) + \psi_{-s+1}(\mathbf{r},t)\xi_{-s+1}(s_z) + \cdots + \psi_{s-1}(\mathbf{r},t)\xi_{s-1}(s_z) + \psi_{s}(\mathbf{r},t)\xi_{s}(s_z) \,.$$

The functions $ψ_{−s}, ψ_{−s + 1}, ..., ψ_{s − 1}, ψ_{s}$ are space functions corresponding to individual, definite values of $s_{z}$, which take in space coordinates and time and return a complex number. The functions $ξ_{−s}, ξ_{−s + 1}, ..., ξ_{s − 1}, ξ_{s}$ are basis spin functions which take in the spin number and return a complex number. They must be linearly independent and form a complete basis. For the case of the z-projection they can be defined simply by the Kronecker delta:


 * $$\xi_{s_z}(s'_z) = \delta_{s_z s'_z}$$

which ensures the basis spin functions are then orthonormal and complete. Substitution of any allowed spin number yields the particular component of the entire wave function for that spin number. This motivates the notation:


 * $$\Psi(\mathbf{r},t,s_z) \equiv \psi_{s_z}(\mathbf{r},t)$$

which may be misleading since the spin number is a variable, not just an index.

Often, the complex values of the wave function for all the spin numbers are arranged into a column vector, in which there are as many entries in the column vector as there are allowed values of $s_{z}$. In this case, the spin dependence is placed in indexing the entries and the wave function is a complex vector-valued function of space and time only:


 * $$\Psi(\mathbf{r},t) = \begin{bmatrix} \Psi(\mathbf{r},t,s) \\ \Psi(\mathbf{r},t,s-1) \\ \vdots \\ \Psi(\mathbf{r},t,-(s-1)) \\ \Psi(\mathbf{r},t,-s) \\ \end{bmatrix}$$

To see the connection to the scalar case, write this as a linear combination using the eigenvectors of the z-component spin operator for a basis:


 * $$\Psi(\mathbf{r},t) = \Psi(\mathbf{r},t,s) \chi_s + \Psi(\mathbf{r},t,s-1) \chi_{s-1} + \cdots + \Psi(\mathbf{r},t,-(s-1)) \chi_{-(s-1)}+ \Psi(\mathbf{r},t,-s) \chi_{-s}$$

where the entries of the column $χ_{sz}$ are given by the spin basis function above,


 * $$ \chi_{s_z} = \begin{bmatrix} \xi_{s_z}(s) \\ \xi_{s_z}(s-1) \\ \vdots \\ \xi_{s_z}(-(s-1)) \\ \xi_{s_z}(-s) \\ \end{bmatrix} \quad \leftrightarrow \quad [\chi_{s_z}]_{s'_z} = \xi_{s_z}(s_z') = \delta_{s_z s'_z} \,, $$

explicitly they amount to


 * $$ \chi_s = \begin{bmatrix} 1 \\ 0 \\ \vdots \\ 0 \\ 0 \\ \end{bmatrix} \,,\quad \chi_{s-1} = \begin{bmatrix} 0 \\ 1 \\ \vdots \\ 0 \\ 0 \\ \end{bmatrix} \,,\quad \ldots \,, \chi_{-(s-1)} = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 1 \\ 0 \\ \end{bmatrix} \,,\quad \chi_{-s} = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \\ 1 \\ \end{bmatrix} \,.$$

The action of the z-component of the spin operator on these eigenvectors $χ_{sz}$ yields the corresponding eigenvalues $ħs_{z}$.

The process can be repeated for the x and y directions, although the basis spin functions are less trivial to determine than those for the z-direction, since the eigenvectors for the spin matrices of higher spins are needed. The same applies for the more general case of projection along any direction, defined by a spatial unit vector $n(θ, φ)$ using standard spherical coordinate angles, in which case the basis spin functions will depend on those angles.

In some situations, the wave function factors into a product of a space function and a spin function, and the time dependence could be placed in either function:


 * $$\Psi(\mathbf{r},t,s_z) = \psi(\mathbf{r},t)\xi(s_z) = \phi(\mathbf{r})\zeta(s_z, t)\,.$$

The dynamics of each factor can be studied in isolation. This is not possible for certain interactions, when an external field or any space-dependent quantity couples to the spin. Mathematically, this may appear as the dot product of the field or quantity with the spin operator in the Hamiltonian operator of the Schrödinger equation. For example, a particle in a magnetic field $B$ is influenced by the field because of the magnetic moment corresponding to the spin, the interaction term is $B · S$. Another example is spin-orbit coupling, the orbital angular momentum $L$ couples to the spin in the term $L · S$. These terms prevent factorization because the position coordinates are mixed into the spin operators, which are matrices that multiply the column matrix wave functions above.

This is analogous to a vector in 3d Euclidean space, using a Cartesian orthonormal basis in the language of index notation. Take a vector:


 * $$\mathbf{a} = a_1 \mathbf{e}_1 + a_2 \mathbf{e}_2 + a_3 \mathbf{e}_3$$

which may be cast in the alternative matrix notation:


 * $$\mathbf{a} = \begin{pmatrix} a_x \\ a_y \\ a_z \end{pmatrix} = a_x \begin{pmatrix}1 \\ 0 \\ 0 \end{pmatrix} + a_y \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} + a_z \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} $$

The ith component of the vector is:


 * $$\begin{align} \left[ \mathbf{a} \right]_i & = a_1 [\mathbf{e}_1]_i + a_2 [\mathbf{e}_2]_i + a_3 [\mathbf{e}_3]_i \\

& = a_1 \delta_{1i} + a_2 \delta_{2i} + a_3 \delta_{3i} \\ \end{align} $$

where the Kronecker deltas are the entries of the matrices, not the basis vectors themselves.


 * M&and;Ŝc2ħεИτlk 20:01, 14 January 2015 (UTC)


 * Begins to look pretty good. YohanN7 (talk) 21:06, 14 January 2015 (UTC)

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

Elementary particles have the property of a "built-in" angular momentum, called spin. The wave function for a particle with spin can be expressed as a single complex number with dependence on space, time, and spin, or an array of complex numbers with dependence on space and time only. The space in either case may be referred to as "position-spin space".

The wave function has a different dependence on spin than it does for space and time. For one thing, all three position coordinates of the particle $r = (x, y, z)$ can be known simultaneously, but not so for all of the components of the particle's spin vector $S = (S_{x}, S_{y}, S_{z})$, only one component and the square of its magnitude $|S|^{2}$ can. (These facts follows from the commutation relations of the position and spin operators). This means all three position coordinates can be placed as variables in the wave function, but only one spin direction can be a variable. For the case of spin, we choose a direction and project the spin along that direction. A conventional, but arbitrary, choice is the z-direction, which will be considered in detail first. Other directions can be obtained from the z-component of spin if the wave function is transformed appropriately.

For another thing, unlike $r$ and $t$ which are continuous variables, spin is a discrete variable, for a particle of spin $s$ there are $2s + 1$ values along any direction. (The exception is in 2d space, where particles called anyons can have continuous spin, but these are not considered here).

The spin projection quantum number along the $z$ axis is denoted $s_{z}$, and the allowed values are $s, s − 1, ..., −s + 1, −s$ only, and no other values. For example, for a spin-1/2 particle, $s_{z}$ can only be the two values $+1/2$ or $−1/2$. The case of spin-1/2 will be exemplified below for simplicity, concreteness, and practical interest - all the leptons and quarks which constitute matter are elementary particles with spin-1/2.

As a single complex number, it is a linear combination of space functions and basis spin functions:


 * $$ \Psi(\mathbf{r},t,s_z) = \psi_{-1/2}(\mathbf{r},t)\xi^z_{-1/2}(s_z) + \psi_{1/2}(\mathbf{r},t)\xi^z_{1/2}(s_z) \,.$$

The space functions corresponding to individual, definite values of $s_{z}$ are $ψ_{−1/2}$ and $ψ_{1/2}$, which take in space coordinates and time and return a complex number. The basis spin functions $ξ_{−1/2}^{z}$ and $ξ_{1/2}^{z}$ take in the spin number and return a complex number. For the case of the z-projection they can be defined simply by the Kronecker delta:


 * $$\xi^z_{s_z}(s'_z) = \delta_{s_z s'_z}$$

which ensures the basis spin functions are then orthonormal and complete. Substitution of any allowed spin number yields the particular component of the entire wave function for that spin number. This motivates the notation:


 * $$\Psi(\mathbf{r},t,s_z) \equiv \psi_{s_z}(\mathbf{r},t)$$

which may be misleading since the spin number is a variable, not just an index.

Often, the complex values of the wave function for all the spin numbers are arranged into a column vector, in which there are as many entries in the column vector as there are allowed values of $s_{z}$. In this case, the spin dependence is placed in indexing the entries and the wave function is a complex vector-valued function of space and time only:


 * $$\Psi(\mathbf{r},t) = \begin{bmatrix} \Psi(\mathbf{r},t,1/2) \\ \Psi(\mathbf{r},t,-1/2) \end{bmatrix}$$

The connection between the "scalar-valued" and "vector-valued" wave functions is the following. As with all discrete observables in quantum mechanics, the eigenstate $Ψ(r, t)$ of the z-component spin operator can be expanded as a linear combination of the eigenvectors $χ_{s_{z}}^{z}|undefined$ of the operator, in other words the $χ_{s_{z}}^{z}|undefined$ form a basis and the functions $Ψ(r, t, s_{z})$ are the complex components of the vector:


 * $$\Psi(\mathbf{r},t) = \Psi(\mathbf{r},t,-1/2) \chi^z_{-1/2} + \Psi(\mathbf{r},t,1/2) \chi^z_{1/2}$$

where the eigenvectors have entries given by the spin basis function above,


 * $$ \left[\chi_{s_z}\right]_{{s'}_z} = \xi_{s_z}({s'}_z) \,, $$

in full


 * $$\chi^z_{1/2} = \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} \xi^z_{1/2}(1/2) \\ \xi^z_{1/2}(-1/2) \end{bmatrix} $$
 * $$\chi^z_{-1/2} = \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} \xi^z_{-1/2}(1/2) \\ \xi^z_{-1/2}(-1/2) \end{bmatrix} $$

The action of the z-component of the spin operator on these eigenvectors $χ_{s_{z}}^{z}|undefined$ yields the corresponding eigenvalues $ħs_{z}$.

The basis spin functions for the x and y directions can be found in a similar procedure to the above.

For any direction in the direction of the spatial unit vector, using spherical coordinates $θ$ for the polar angle from the z-axis and $φ$ for the azimuthal angle in the xy-plane from the x-axis:


 * $$\hat{\mathbf{n}} = \sin\theta(\cos\phi\mathbf{e}_x + \sin\phi\mathbf{e}_y) + \cos\theta\mathbf{e}_z$$

the eigenvectors of the spin operator in this direction are


 * $$\chi^\hat{\mathbf{n}}_{1/2} = \begin{bmatrix} \cos(\theta/2) \\ e^{i\phi}\sin(\theta/2) \end{bmatrix} = \begin{bmatrix} \xi^\hat{\mathbf{n}}_{1/2}(1/2) \\ \xi^\hat{\mathbf{n}}_{1/2}(-1/2) \end{bmatrix} $$
 * $$\chi^\hat{\mathbf{n}}_{-1/2} = \begin{bmatrix} e^{-i\phi}\sin(\theta/2) \\ -\cos(\theta/2) \end{bmatrix} = \begin{bmatrix} \xi^\hat{\mathbf{n}}_{-1/2}(1/2) \\ \xi^\hat{\mathbf{n}}_{-1/2}(-1/2) \end{bmatrix} $$

and the spin functions depend on the angles,


 * $$\xi^\hat{\mathbf{n}}_{\pm 1/2}(\pm 1/2) = \pm \cos(\theta/2) \,,\quad \xi^\hat{\mathbf{n}}_{\pm 1/2}(\mp 1/2) = e^{\pm i\phi} \sin(\theta/2) \,. $$

The relation from the z-projection basis $χ_{s_{z}}^{z}|undefined$ to the $n$-projection basis $χ_{s_{n}}^{n}|undefined$ is a change of basis.

The extension to the general case of a particle with higher spin is straightforward in principle, but finding the basis spin functions, for arbitrary spin, is nontrivial for directions other than the z-axis. The linear combinations are summed over all the allowed spin quantum numbers, in the z-direction:


 * $$ \Psi(\mathbf{r},t,s_z) = \psi_{-s}(\mathbf{r},t)\xi^z_{-s}(s_z) + \psi_{-s+1}(\mathbf{r},t)\xi^z_{-s+1}(s_z) + \cdots + \psi_{s-1}(\mathbf{r},t)\xi^z_{s-1}(s_z) + \psi_{s}(\mathbf{r},t)\xi^z_{s}(s_z) \,, $$

and column matrices are indexed by the allowed spin quantum numbers:


 * $$\Psi(\mathbf{r},t) = \begin{bmatrix} \Psi(\mathbf{r},t,s) \\ \Psi(\mathbf{r},t,s-1) \\ \vdots \\ \Psi(\mathbf{r},t,-(s-1)) \\ \Psi(\mathbf{r},t,-s) \\ \end{bmatrix}\,.$$

In some situations, the wave function factors into a product of a space function and a spin function, and the time dependence could be placed in either function:


 * $$\Psi(\mathbf{r},t,s_z) = \psi(\mathbf{r},t)\xi(s_z) = \phi(\mathbf{r})\zeta(s_z, t)\,.$$

The dynamics of each factor can be studied in isolation. This factorization is always possible for potentials or interactions which do not depend on the spin of the particle, the simplest case is the free particle. This is not possible for certain interactions, when an external field or any space-dependent quantity couples to the spin. Mathematically, this may appear as the dot product of the field or quantity with the spin operator in the Hamiltonian operator of the Schrödinger equation. For example, a particle in a magnetic field $B$ is influenced by the field because of the magnetic moment corresponding to the spin, the interaction term is $B · S$. Another example is spin-orbit coupling, the orbital angular momentum $L$ couples to the spin in the term $L · S$. These terms prevent factorization because the position coordinates are mixed into the spin operators, which are matrices that multiply the column matrix wave functions above.


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

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


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

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


 * The wave functions are just the solutions of the Schrödinger equation, determined by the quantum mechanical Hamiltonian that defines the system. The Schrödinger equation is mathematically a type of wave equation. This explains the name "wave function", and gives rise to wave–particle duality. The wave of the wave function is not a wave in physical space; it is a wave in an abstract mathematical "space", and in this respect it differs fundamentally from water waves or waves on a string.

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

Historical information
I have given historical information here. At that time, 1905, Planck did not admit the notion of a photon. The eponym "Planck–Einstein relation" occurs in a very few otherwise reliable sources, but currency of it is largely an invention of Wikipedia editors which we do not need to copy here. It is hardly necessary to give de Broglie's name to Einstein's recognition of the momentum of the photon.Chjoaygame (talk) 19:49, 17 January 2015 (UTC)
 * Are you sure about the last point? (The "Planck–Einstein relation" is of course nonsense). De Broglie is attributed the glory of having suggested the de Broglie wavelength for massive particles in practically every reference there is. YohanN7 (talk) 20:07, 17 January 2015 (UTC)
 * Hm. The name "Planck–Einstein relation" might on second thought be appropriate to mention. (Hey, YOU named the article that way) Planck didn't suggest photon quanta (while he did find his own constant), so it should be attributed to Einstein. The de Broglie wavelength is the definite name in the massive case. I think attribution should go to him as well. YohanN7 (talk) 20:19, 17 January 2015 (UTC)


 * Yes, it's horrible. I did it with reluctance and distress, to try to escape something even worse. People did a Google search and found some popularizing sources that called it the 'Planck relation', and were insistent for it. Very few reliable sources use that one, and I hoped to avoid Wikipedia giving it currency by circular quotation. Planck had a quantum for emission that belonged to a purely heuristic virtual simple harmonic oscillator. More he thought ε = nhν, not just ε = hν. He did not think of photons till years later, after Einstein's ideas. Amongst reliable sources, I found scarce consistency, but two used the eponym 'Planck–Einstein', so I used it reluctantly as the least worst reliably sourced escape option.


 * As for de Broglie, since you think it matters, what about putting it in a sentence of its own?


 * Say we add 'From Einstein's idea for the photon, in 1923 Louis de Broglie generalized to a 'wavelength' for massive particles'?Chjoaygame (talk) 21:27, 17 January 2015 (UTC)
 * I have edited to that effect (or something like it). YohanN7 (talk) 21:33, 17 January 2015 (UTC)


 * Ok. Date fix.Chjoaygame (talk) 21:37, 17 January 2015 (UTC)


 * On closer reading, I am suggesting some changes.Chjoaygame (talk) 21:56, 17 January 2015 (UTC)

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

Instead, I would prefer:
 * It is a central entity in quantum mechanics, important in all modern quantum theories, including the quantum theory of fields.

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


 * Seems fine with me. M&and;Ŝc2ħεИτlk 10:06, 18 January 2015 (UTC)

Basis
Removed this for now:
 * Since linear combinations of wave functions obtain more wave functions, the set of all wave functions $I$ is an infinite dimensional vector space over the field of complex numbers. To form a vector space basis $W = {Ψ(x, t)}$, we need a maximal set of wave functions $B$ in $ψ_{1}, ψ_{2}, ...$ which are linearly independent: each one of them is not a linear combination of the others, for example $W$ and $ψ_{1} ≠ z_{2}ψ_{2} + z_{3}ψ_{3} + ...$, etc., for any complex numbers $ψ_{2} ≠ z_{1}ψ_{1} + z_{3}ψ_{3} + ...$ and every function in $z_{n}$ is a linear combination of functions in $W$. This linear independence allows a linear combination of $B$ to uniquely construct an arbitrary wave function in $ψ_{1}, ψ_{2}, ...$:


 * $$ \Psi(x,t) = \sum_n a_n \psi_n(x,t) $$


 * In this way, $W$ can be viewed as an infinite dimensional vector, where the complex-valued coefficients $Ψ(x, t)$ are the components of the vector. The choice of which wave functions to use as a basis is not unique, but if a change of basis is made, the components $a_{n}$ need to change to compensate.

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

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


 * Belated reply - it may be shaky and does not explicate anything about free particle solutions, but the idea was to pitch to the reader, in the simplest possible way, the idea that wave functions form vector spaces (linear combinations of wave functions are wave functions). I'm neutral on the deletion, since now it is in the lead. M&and;Ŝc2ħεИτlk 10:04, 18 January 2015 (UTC)


 * It needs a rewrite, not permanent deletion. YohanN7 (talk) 10:13, 18 January 2015 (UTC)


 * In the literature the x and p (and others) "bases" are usually called "representations", not sure if that is misleading though. M&and;Ŝc2ħεИτlk 10:43, 18 January 2015 (UTC)


 * That solves the problem. Thanks for pointing out. Sometimes it is just too obvious to see for oneself. YohanN7 (talk) 23:41, 18 January 2015 (UTC)