Talk:Schrödinger equation

Wave Function is a vector

Is a wave function a vector?

"A wave function can be an eigenvector of an observable" 88.111.117.83 (talk) 18:28, 31 August 2023 (UTC)

Yes, it's a vector on a Hilbert space. Jähmefyysikko (talk) 19:46, 31 August 2023 (UTC)

Possible typo under "Separation of variables" section.

I'm no expert so maybe I'm just confused, but under the section "Separation of variables", the first sentence after the equation, it says, "The operator on the right side depends only on time; the one on the left side depends only on space." but the left operator is i\hbar\frac{\partial}{\partial t} and the right operator contains the \nabla^2. Then, below the next equation, the next sentence says, "Substituting this expression for \Psi into the time dependent left hand side shows that...".

Am I confused? The two places seem to contradict each other. It seems to me the first sentence is wrong. It should read:

"The operator on the right side depends only on space; the one on the left side depends only on time."

That is to say, should the words "time" and "space" be swapped in that sentence?

(P.S. Due to my lack of experience editing wikis, if my observation is correct, could someone make that change other than me. I get this deep only once a decade or so, so it would be better for everyone if I just watch. Thanks.) BornRightTheFirstTime (talk) 18:20, 8 November 2023 (UTC)

Fixed, but next time just take a chance. Give an edit-description and if you are not comfortable add a topic just like you did but post-fix: "Fixed possible typo..." for some one to check. Johnjbarton (talk) 18:28, 8 November 2023 (UTC)

Thank you and will do. See ya around 2033. :-) BornRightTheFirstTime (talk) 16:27, 9 November 2023 (UTC)

Less not more math.

@EditingPencil Thanks for your recent edits. I want to encourage more qualitative physics and less math.

I think we should expect readers of this article to include interested first physics undergrads. For example, in my opinion the section on "Probability current" should be describing what the heck probability current is and why it is related to Schrodinger's equation. It should be a summary only; they should not be faced with a proof. There are lots of other articles for details, eg Probability current Johnjbarton (talk) 19:19, 14 December 2023 (UTC)

Hello. Yeah, I agree. I have moved it. But I'm not sure if the bit about phase can stay or go. I don't mind either way so I'll let anyone here decide. EditingPencil (talk) 19:50, 14 December 2023 (UTC)

Mysterious GIF

@Rolancito recently added this GIF:

The caption says:

• The time-independent Schrödinger equation for a harmonic oscillator has two independent solutions, one even and one odd, for any value of the energy E. Only the normalisable solutions are valid in quantum mechanics, while the rest are discarded.

But I can't figure out what it means. I assume that the graph plots 1D solution amplitude $\psi (x)$ vs x. What is "40"? Why is the graph flopping around? We are told there are two solutions, but only some are valid: are these the valid ones? If yes, why are you telling us about the discarded ones? What is the significance of the seemingly infinite amplitude for large values of x?

Finally, and most important, which reference can I consult to verify this image? Johnjbarton (talk) 16:16, 20 April 2024 (UTC)

Hi, I placed the GIF next to the reference I used, namely the equation in the section about the harmonic oscillator of this article. I scanned the energy in steps of 1/40 because this value gave me a smooth-enough animation. Rolancito (talk) 20:44, 20 April 2024 (UTC)

At present the GIF is positioned at the top of Schrödinger_equation#Harmonic_oscillator which has 4 equations. None of these equations seem likely to produce the graph in my opinion. The most likely one

$$\psi _{n}(x)={\sqrt {\frac {1}{2^{n}\,n!}}}\ \left({\frac {m\omega }{\pi \hbar }}\right)^{1/4}\ e^{::-{\frac {m\omega x^{2}}{2\hbar }}}\ {\mathcal {H}}_{n}\left({\sqrt {\frac {m\omega }{\hbar }}}x\right),$$

has an exponential factor which I guess eventually dominate at large |x|, unlike the graphic.

The legend says "even" and "odd" but the equation says "n".

The text in the article claims the solutions are eigenstates, meaning only discrete values of E. The animation seems continuous to me. Johnjbarton (talk) 21:27, 20 April 2024 (UTC)

As per the caption of the GIF: "The time-independent Schrödinger equation for the 1D harmonic oscillator...". The equation that was solved to generate these plots is $$E\psi =-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}\psi +{\frac {1}{2}}m\omega ^{2}x^{2}\psi ,$$ which is the first equation in the section and referred to as "The Schrödinger equation for this [1D QHO] situation ..." Rolancito (talk) 22:09, 20 April 2024 (UTC)

Plugging in values of $E$ that are not actually eigenvalues of the harmonic oscillator Hamiltonian will give meaningless "solutions". There's no point in plotting unphysical solutions in this context. This article has suffered enough over the years from people shoving in every calculation using the SE that they find cute. It doesn't need a movie, the only meaningful frames of which go past in the blink of an eye. XOR'easter (talk) 23:16, 20 April 2024 (UTC)

@XOR'easter That history of suffering is an indication that something is clearly missing. This whole article is lacking clarity in the step where the SE equation is solved as a differential equation, and "solved" by those physical solutions that are cherry-picked by the normalisation postulate. I can make the physical solutions last longer, no problem. You can give it context if you want, but clearly there is a jump between equations 1 and 2 in this section that is made too abrupt and unclear. A movie is the best way to illustrate this step without going into the mathematical derivation, just like the movie below in this same section plays a movie of the time evolution without solving the time-dependent SE. Rolancito (talk) 23:45, 20 April 2024 (UTC)

No, that history of suffering is the inevitable consequence of how Wikipedia pages naturally grow by accumulation, needing to be pared back when their contents become too indiscriminate. A top-level overview of the Schrödinger equation doesn't need to go into steps between writing the equation and presenting its solutions, because it's a top-level overview. This page is not a textbook. (Nor is the one about the quantum harmonic oscillator specifically, for that matter.) Indeed, a solution-by-animation verges on OR. While I get the impulse to illustrate the article, I don't see how this particular illustration reduces the abstruseness: the reader can only use a visualization of this form to pick out the physical solutions if they already know the subject matter. XOR'easter (talk) 23:59, 20 April 2024 (UTC)

Thanks. It seems your previous experience is biasing against my submission. Each (legitimate) contribution should be review on a case-by-case basis, regardless of previous traumas. Now, this particular step of picking solutions of the Schrödinger differential equation is crucial because it is what gives quantum mechanics its "quantumness", that is, its discrete nature. Because of this, it is justified to be included in a top-level article like this one. The best way is to show an animation, that speaks for more than 1000 words (or equations). What I did is also not original research, a simple Google search results in plenty of more-or-less detailed/accessible information, for example here.

Look, I simply took an equation that already existed on the article, solved it with a standard ODE solver, and used its plot to illustrate how the normalisation postulate leads to the choice of physical solutions, a postulate that also gets mentioned in the article but does not get enough attention, even though it is what puts the "quantum" in quantum mechanics. I don't see it out of context, low-level or original research. Why am I getting indiscriminate resistance? Rolancito (talk) 21:39, 21 April 2024 (UTC)

I find the picture interesting and I did learn something while pondering about it. But the idea it is illustrating is quite technical (a specific numerical solution procedure), and not suitable for this article, which is supposed to give a high-level overview of the SE. Perhaps place it in Shooting method#Eigenvalue problem which seems to discuss the exact method?

To improve this article, it might be appropriate to add some discussion about numerical solutions, which do not seem to be discussed at all. Perhaps a single paragraph would be enough, with few well-chosen links to more specific articles. Jähmefyysikko (talk) 05:27, 22 April 2024 (UTC)

@Rolancito

• "Why am I getting indiscriminate resistance?"

Speaking for myself, I don't believe the image as inserted conveyed the ideas you think it did. The caption have no hints about "quantum", but rather focused our attention on "even and odd" and on the "flapping wings" effect of the graph. It did not mention ODE, numerical solution, or even what was being plotted. It mentioned normalization but in a confusing way.

If the math formulae can be included in the article, a graph of the formula can be included. I think an image illustrating the impact of the normalization constraint might succeed if included in a separate section "Normalization constraint" and if the image demonstrated it, eg by showing unnormalized functions in gray and normalized ones in red. Johnjbarton (talk) 15:26, 22 April 2024 (UTC)

Irrelevant paragraph?

The following paragraph in the "definition" of the SE is irrelevant:

• Physical quantities of interest – position, momentum, energy, spin – are represented by "observables", which are Hermitian (more precisely, self-adjoint) linear operators acting on the Hilbert space. A wave function can be an eigenvector of an observable, in which case it is called an eigenstate, and the associated eigenvalue corresponds to the value of the observable in that eigenstate. More generally, a quantum state will be a linear combination of the eigenstates, known as a quantum superposition. When an observable is measured, the result will be one of its eigenvalues with probability given by the Born rule: in the simplest case the eigenvalue is non-degenerate and the probability is given by , where is its associated eigenvector. More generally, the eigenvalue is degenerate and the probability is given by , where is the projector onto its associated eigenspace.

The paragraph above goes off-topic as it talks about the measurement postulate and observables, not about Schrödinger's equation. I propose to move it to a page about measurement in quantum mechanics. This is a major change, just wanted to ask what was the purpose of this paragraph? Rolancito (talk) 19:51, 7 May 2024 (UTC)

I agree, this paragraph and the one before and after it are not Preliminaries.

I suggest moving this paragraph and the one following it into the section called "Properties" naming it "Observables". The paragraph directly before, beginning "Broadening beyond this simple case..." should also be moved, maybe under Properties > Normalization? The first sentence is history. Johnjbarton (talk) 21:57, 7 May 2024 (UTC)