Talk:Wave function

Normalized, or not?

It seems, position-space and momentum-space wave functions are normalized, but others are not; and every linear combination of wave functions is a wave function... a mess? Boris Tsirelson (talk) 14:26, 30 April 2016 (UTC)

It says in the inner product section when a wave function is normalized, orthogonal, orthonormal, or not. M∧Ŝc²ħεИ_τlk 14:30, 30 April 2016 (UTC)

Yes, and still... even in that section, the interpretation of the inner product is given before introducing normalization. Does this interpretation of the inner product apply to non-normalized wave functions? Boris Tsirelson (talk) 15:45, 30 April 2016 (UTC)

"...this general requirement a wave function must satisfy is called the normalization condition..." So, really, must satisfy? Throughout the article? Or, depending on the context, sometimes? How does the reader know, when it must? Boris Tsirelson (talk) 15:49, 30 April 2016 (UTC)

Where does the article say "this general requirement a wave function must satisfy is called the normalization condition...". I can't find it anywhere. I thought the wave function must be normalizable for the probability interpretation to work, and collapse assumes normalized wave functions because the formula given is for the transition probability. M∧Ŝc²ħεИ_τlk 16:26, 30 April 2016 (UTC)

The lead, the paragraph before the last:

In Born's statistical interpretation,[8][9][10] the squared modulus of the wave function, | ψ |2, is a real number interpreted as the probability density of measuring a particle's being detected at a given place, or having a given momentum, at a given time, and possibly having definite values for discrete degrees of freedom. The integral of this quantity, over all the system's degrees of freedom, must be 1 in accordance with the probability interpretation, this general requirement a wave function must satisfy is called the normalization condition. Since the wave function is complex valued, only its relative phase and relative magnitude can be measured. Its value does not in isolation tell anything about the magnitudes or directions of measurable observables; one has to apply quantum operators, whose eigenvalues correspond to sets of possible results of measurements, to the wave function ψ and calculate the statistical distributions for measurable quantities.

Boris Tsirelson (talk) 16:47, 30 April 2016 (UTC)

The problem is, that the article is for now the mix of statements that are true when "wave function" means "normalized" (and wrong otherwise), and statements that are true when "wave function" means "not just normalized" (and wrong otherwise). Boris Tsirelson (talk) 16:51, 30 April 2016 (UTC)

Ack, careless me, missed the lead. Just tried rearranging more sections in the article to make the flow easier to follow. It would be helpful to separate off all content on the probability interpretations in its own section, and maybe rearrange statements when normalization is necessary and unnecessary. Maybe you would like to try? M∧Ŝc²ħεИ_τlk 17:10, 30 April 2016 (UTC)

Hmmm... let us think. The lead uses the probability interpretation, and this is necessary, since otherwise it is just some math. First of all, a wave function describes a quantum state, and here it must be normalized. But later, probably, it appears that physicists do a lot of calculations, postponing interpretation (till the end of calculation); and in this process they are, somehow, temporary, effectively, mathematicians... and tolerate non-normalized, and even non-normalizable, functions... and do not hesitate to call them wave functions... right? A kind of abuse of language. Surely, the superposition principle does not mean that a linear combination of wave functions is a wave function even if it is identically zero (which really could happen)! But, who cares... Really, not many centuries ago, mathematicians got true equalities after hard calculations with divergent series, without bothering too much... These intermediate divergent series were probably called "functions", but they were not...

But no, I am reluctant to edit physical articles myself. Since my text smells of math, inevitably, I know. And another, no less important reason: I never read elementary textbooks on physics, and so, I do not know, in which form all that is written there. Boris Tsirelson (talk) 17:58, 30 April 2016 (UTC)

The observation

...physicists do a lot of calculations, postponing interpretation (till the end of calculation); and in this process they are, somehow, temporary, effectively, mathematicians... and tolerate non-normalized, and even non-normalizable, functions...

is absolutely correct. This does not provide an excuse for us to do the same in this article, at least not before we tell the reader what is about to happen. I'll try to do my bit in due time. YohanN7 (talk) 16:43, 2 May 2016 (UTC)

Well, we must follow the physics that stretches outside Wikipedia... that is, "do the same"; but indeed, we should tell the reader what is about to happen. At least, it is done this way in math; for instance, "Baire set". Boris Tsirelson (talk) 18:12, 2 May 2016 (UTC)

On a side note: Gieres, F. (2000). "Mathematical surprises and Dirac's formalism in quantum mechanics". Rep. Prog. Phys. 63: 1893–1931. arXiv:quant-ph/9907069. YohanN7 (talk) 16:54, 2 May 2016 (UTC)

Woooooow! Boris Tsirelson (talk) 18:12, 2 May 2016 (UTC)

Rearrangement of sections

On one hand, it is nice to pin down all mathematical and physical properties/requirements first thing (after the history).

On the other hand, it would be easier for typical readers to just see what the wave function is with examples, and the mathematical details later. Also, since the probability interpretation is what most sources present that section could be moved further up. The Hydrogen atom content in the function spaces section has been moved to the section of that example.

I tried to rearrange to see how it looks, and will not make further edits. Anyone is free to revert if they disagree. M∧Ŝc²ħεИ_τlk 06:50, 3 May 2016 (UTC)

Well, this happened. But anyone can revert to this version. M∧Ŝc²ħεИ_τlk 12:37, 3 May 2016 (UTC)

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one word or two?

Is wavefunction (one word) now a word? (I note that the spell checker/corrector on my browser believes it is two words.) I note that there is a redirect from the one word form, but, even more, it is used in many articles. Gah4 (talk) 23:40, 17 January 2017 (UTC)

As far as I can tell from [1] it is two words. I believe that the redirect is fine, but articles should use the two word form until the OED says it is one word. Gah4 (talk) 00:09, 18 January 2017 (UTC)

Particle in a box

Subsection Wave function#Particle in a box is created by Maschen in 2014 by moving the content from "normalization example". However, the "normalization example" was just about normalization, it did not mention any potential, while Particle in a box is about potential infinite outside the box, and stipulates the corresponding boundary conditions (leading to discrete spectrum). So, this subsection is unsatisfactory (since 2014 till now). Boris Tsirelson (talk) 17:07, 2 May 2017 (UTC)

We could just link to standing wave where this is all explained in the general context of waves. Gah4 (talk) 20:48, 2 May 2017 (UTC)

No. The same problem: a sum of two terms there versus a single term (running wave) here. Also, you never have continuous spectrum in a bounded region. Boris Tsirelson (talk) 05:29, 3 May 2017 (UTC)

And, ridiculously, without any relation between $k$ and $\omega$ "our wavefunction" violates the Schrodinger equation. Thus, it is an example on the normalization only, with no other relation to physics. Boris Tsirelson (talk) 05:56, 3 May 2017 (UTC)

I agree. It would be easy to extract a correct version from any introductory text. YohanN7 (talk) 08:38, 3 May 2017 (UTC)

Deleted. Boris Tsirelson (talk) 05:20, 4 May 2017 (UTC)

There is something called "Box Normalization" that is frequently used, and has little (but still something) to do with "Particle in Box" (no potential). The effect is quantization of momentum, and a normalization is possible within the (very large) box. YohanN7 (talk) 08:05, 28 June 2017 (UTC) Discussed here. YohanN7 (talk) 08:09, 28 June 2017 (UTC)

How to visualize wave functions

The phase of a wave function can be represented by a continuum of different colors. The probability density can be represented by the intensity of the color. In this way we can give full information about any 2D wave function. There is a problem of contrast. If the probability density of one part of the image is too little than the one on another part, it won't be seen. But there is a simple solution : the intensity of the color shall depend logarithmically, not linearly, on the probability density. This is a mistake I made in my previous animations (linear dependence, not logarithmic). I just thought about it. I will make use of this thought in my future animations. I hope also that others will make use of this idea. With it we can visualize many quantum wave functions.TD (talk) 08:51, 16 August 2023 (UTC)

Use of proper vectors

@EditingPencil I'm not sure if I agree with the use of proper vectors and improper vectors. They are effectively eigenfunctions according to the source you cite (on page 67), but with respect to (what the author calls) a "physical" Hilbert space. Apart from the fact that this type of Hilber space hasn't been introduced, I also think it's better to refer to them as eigenfunctions as that's much more conventional. The subsection Eigenvalues and eigenfunctions of Hermitian operators clarifies its use in continuous settings and cites a good source as well. Roffaduft (talk) 07:02, 16 November 2023 (UTC)

Hello!

We can refer to that as a vector following the Dirac notation, also that entire section is dedicated to introduce states as vectors and it already introduces it as such. Functions are of a different form and structure: ${\textstyle \psi (\mathbf {x} ,t)=\langle \mathbf {x} |\psi (t)\rangle }$.

If you find a good spot to edit in the part about improper or proper vectors, go for it. I'm not so sure where it would fit here though.

I will add main article links to that section though.

PS. "physical" Hilbert space comes from a postulate of QM, here. EditingPencil (talk) 09:41, 16 November 2023 (UTC)

""physical" Hilbert space comes from a postulate of QM, here"

@EditingPencil No it doesn't. I see you've cited Shankar, you can look up the distinction on page 67 of his book.

The point is that a proper vector is just an eigenfunction; in bra-ket notation normalized by the Kronecker delta $\delta _{ij}$. The improper vector are normalized by the Dirac delta $\delta (x'-x)$, which is not a function and does not belong to the Hilbert space.

This distinction can be made mathematically rigorous by introducing Rigged Hilbert space, though this might be beyond the scope of the subsection. Regardless, you may want to look at the Dirac delta function#Quantum mechanics page for the more conventional terminology. Roffaduft (talk) 10:44, 16 November 2023 (UTC)

_{If I'm interpreting it correctly, page 67 mentions he considers "physical" Hilbert spaces to be only the space acted upon by Hermitian operators. That postulate requires the same. I assume this also justifies taking the basis of Hilbert space as (orthogonal set) ${\textstyle |x\rangle }$ which are eigenvectors of some Hermitian operator. I could've fumbled this but that isn't the topic anyway.}

I just don't see why you want to call something written in Dirac notation as a function. Referring to this specific symbol: ${\textstyle |x\rangle }$ as eigen-"function" is not standard practice and I recommend you use alternative term, eigenvector.

This is a shot in the dark but I'm not arguing that eigenvectors (or eigenfunctions) of continuous eigenvalues of Hermitian operators can't be improper vectors (or functions). In fact they strictly have to be! I just don't see why the converse has to hold. I guess your point is that improper vector shouldn't be defined without some physically relevant Hermitian. You can edit that if you find sources and an appropriate place to add it in the article.

I'm uninterested in this topic though, sorry. EditingPencil (talk) 12:00, 16 November 2023 (UTC)

The Dirac Delta function is neither a proper vector nor an eigenfunction.

I agree with you that the Dirac notation is not appropriate here, but that is besides the point. You used the term proper vector without any introduction. A proper vector in this context is just the eigenfunction (according to your own source). Roffaduft (talk) 12:11, 16 November 2023 (UTC)

Ah. I get what you mean. This bookmark talks about this issue, although I'm too lazy to find better sources.

But given that almost every textbook uses the same notation and terminology, I don't think we need to do rocket science here. I think that introduction is enough as well, it is known by context that those vectors are position eigenvectors.

Anyway, that's that ig. EditingPencil (talk) 12:20, 16 November 2023 (UTC)

Any Quantum Physics textbook perhaps, but almost all of them place proper vectors and improper vectors in context of eigenstates and spectral theory. So while it might be clear to physicists, it is much less clear for mathematicians.

Therefore it is much better to refer to proper vectors as either eigenstates or eigenvectors; including a wiki link for reference, and instead of improper vectors using the same argumentation as in Dirac delta function#Quantum mechanics.

If we can agree on that, I will change it today or tomorrow. If not, I'd like to know why. Roffaduft (talk) 13:01, 16 November 2023 (UTC)

No worries, you can do that. We have this place to discuss the edits anyway. EditingPencil (talk) 16:20, 16 November 2023 (UTC)

PS. I see you reverted part of an edit I've made. A Hilbert space is a generalization of a finite-dimensional complete inner product space. In other words: a finite-dimensional complete inner product space can be regarded as "a subset" of Hilbert spaces. It has therefore nothing to do with being "modern". Roffaduft (talk) 10:51, 16 November 2023 (UTC)

PPS. According to you own source: However, the term is often used nowadays, as in these notes, in a way that includes finite-dimensional spaces, namely: finite-dimensional Hilbert space.

Ergo, the classification "modern" does not refer to the finite-dimensionality, but to use of the term 'finite-dimensional Hilbert space' Roffaduft (talk) 11:09, 16 November 2023 (UTC)

Hey, I see you removed the original meaning of a sentence I wrote.

I will just paraphrase the sentence from the source. Given that you didn't remove the citation, I assume that you don't have issue with the source nor my edit. EditingPencil (talk) 13:20, 17 November 2023 (UTC)

I undid your revision because it was simply incorrect and was not in line with the rest of the text. Your source is a very limited summary, so that is not sufficient. I suggest you read up on the meaning of a Hilbert space (e.g. on wikipedia) in the context of QM before you make another change. Roffaduft (talk) 16:46, 17 November 2023 (UTC)

PS. If you want to make another edit about this, maybe discus it here first. I don’t want to end up edit warring over it. What about the current edit is incorrect according to you? Roffaduft (talk) 17:11, 17 November 2023 (UTC)

I won't do such a thing. We can leave this issue alone.

_{I have the same criticism about your own edit, I think it avoids talking about historical usage of the term for no reason whatsoever but I can live with that. Also you reverted a paraphrased edit and kept the citation anyway. Also I don't think mathematical definitions have anything to do with historical use of terms and neither do I think I misinterpreted anything out of the 4 or so words that wasn't directly in text. All in all, your 5 words was fine too, so I won't change those.}

_{PS, I've seen people add "Reverted good faith edits" in the edit summary to avoid edit wars. Just passing along this information if you have similar interactions with someone else. cheers!} EditingPencil (talk) 17:28, 17 November 2023 (UTC)

Alright, I clarified what it meant. Please edit a better word for "modern" if possible, I can't think of anything right now though. EditingPencil (talk) 12:10, 16 November 2023 (UTC)