Talk:Schrödinger equation/Archive 4

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Archive 4

"Schrödinger Derivation"

Modified the claim that Ballentine presents arguments for the actual derivation of Schrödinger's equation. Ballentine references and reproduces other work that "derives" certain results that are usually derived from analysis of Quantum Mechanics. Symmetry arguments result in commutation relations such as [G, P]=ih (page 76). I see no argument presented in Ballentine for the actual derivation of the equation itself. It is simply postulated that a physical variable is represented by a Hermitian operator, a Hermitian operator necessarily satisfies A|psi>=a|psi>. It is simply postulated that the eigenvalues and vectors of the operator are the only possible physical values. It is simply postulated that |psi> gives the probability. Kevin Aylward 09:06, 5 September 2015 (UTC) — Preceding unsigned comment added by Kevin aylward (talk • contribs)

Thanks for clearing it up. It was most likely the misunderstanding of whoever added the claim. M∧Ŝc²ħεИ_τlk 11:38, 5 September 2015 (UTC)

There is a lot more worse wrong with the whole article. I had one attempt to remove "proves particles takes two paths an once" but this was reverted. I don't have a lot of time to never ending correct all the blatant misconceptions. I am not sure the best way to tackle this comic book view of QM. So few understand that QM is a mathematical structure that predicts probabilistic results, and 99.9% of practitioners don't care a toss about waffle interpretations. All the waffle is a direct result of not accepting that QM is a new axiom of physics. It simply can not be explained by discussing which way some hypothetical entity may or may not travel. Classical reasoning always gets you contradictions, hence is false. Luboš Motl gets this bit spot on. I have a short wordy description here http://www.kevinaylward.co.uk/qm/quantum_mechanics.html, and reference Motl's page with the relevant math, although he is discussing a different problem. Kevin Aylward 14:31, 5 September 2015 (UTC) — Preceding unsigned comment added by Kevin aylward (talk • contribs)

In fact, the standard way of "deriving" the Schrödinger equation is as follows. First, we put some constraints on the time-evolution operator. The fact that norms in the Hilbert space have to be conserved, and that time-evolution leads to a group structure, implies that the time evolution operator has to be unitary. The Hamiltonian is then simply the Hermitian operator that generates this unitary evolution. If we then demand that the dynamics of the system reduce, in the classical limit, to classical dynamics, then it is possible to argue that the quantum Hamiltonian coincides with the classical Hamiltonian up to operator ordering ambiguities, of course. The argument above is quite standard in a graduate treatment. For example, see Sakurai, Modern Quantum Mechanics, p. 71. I'm not sure why this is not present in the article. Is this because editors had some deep philosophical objection to terming this argument a "derivation". Or is it simply not there, because no one put it there. If its the latter, then I'll try and add it to the article later today. Jacob2718 (talk) 16:18, 3 December 2016 (UTC)

It is looking great. Please do not forget to cite your source(s). Nerd271 (talk) 20:34, 13 December 2016 (UTC)

"Non-relativistic" versus "nonrelativistic"

As per this talk page, the article is written in American English, where "nonrelativistic" seems to be more common (the Merriam-Webster dictionary lists the word as "nonrelativistic"). I have thus edited the article accordingly.Penskins (talk) 16:06, 8 January 2017 (UTC)

New section "Formulation without complex numbers"

I reverted this edit, it looks like WP:OR. However it could be of interest, if references are found it could be added back. M∧Ŝc²ħεИ_τlk 15:23, 8 July 2017 (UTC)

One reference which may include splitting the SE into real and imaginary parts is Holland's "Quantum theory of motion", but could be wrong, need to check. M∧Ŝc²ħεИ_τlk 15:43, 8 July 2017 (UTC)

I found errors in the derivation. Geo39geo (talk) 03:57, 19 September 2017 (UTC)

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Units in Derivation Section

I removed the use of hbar in the derivation section, of which was reverted by @Nerd271: due to an argued importance of units in such a derivation.

In response, I would like to note that the derivation is *not* a physical argument but rather a mathematical one that is using arbitrary operators. It isn't until the specific application of the correspondence principle when units can be given any substantive meaning. This is done at the end of the section.

Placing units before then belittles the important role that the correspondence principle plays in the mathematical description of physical systems. Cjayross (talk) 16:01, 9 August 2019 (UTC)

@Cjayross: I'm afraid that's not how I was taught. Measurements must have units and like I said, exponents must be dimensionless. Moreover, despite its apparently strange appearance, the form of the time-evolution operator does lead to predictions that agree well with experiment, the highest status any scientific theory, hypothesis, or model can have. This is the justification for the appearance of ħ in the starting equation. We know at $H$ must have units of energy. Since it is the time operator in an exponent, we need a constant that has units of action. Nerd271 (talk) 21:42, 12 August 2019 (UTC)

@Nerd271: You are misunderstanding what this derivation is. It is not, in any way, describing measurements; it is just describing the mathematics. The appearance of $H$ is not the Hamiltonian in the context of the derivation: it is just an arbitrary Hermitian operator. It is simply a stand-in for whatever physical interpretation it could be given after the mathematics has been carried out. What is described in the derivation is not the Schrodinger Equation, it is an equation that looks like the Schrodinger Equation before any application of the correspondence principle is applied to attribute physical meaning to its constituents (which is where the units are indeed introduced just as you describe). Perhaps the solution here is to introduce a different symbol for the arbitrary Hermitian operator just as the citation for the section does. Sakurai uses $\Omega$ to denote the operator in the exponent. Although, I would prefer not to do this. Instead, I'd prefer to just change the line where it mentions "...and using appropriate units" to something that references Natural units. Cjayross (talk) 22:16, 23 August 2019 (UTC)

Improving the top animated GIF

Many/most readers will be drawn immediately to the topmost animated GIF -- Wave_packet_(dispersion).gif -- the classic illustration of a quantum wavepacket "traveling freely through empty space". I used it in my attempt to understand the "general Schrödinger equation" immediately to its left. But it made no sense to me for a long time, until I finally discovered (through other websites) that that wavepacket is actually a helical corkscrew in complex space. It was an epiphany moment for me when I finally found this far more explanative GIF: https://i.stack.imgur.com/ycw8Zm.gif . Everything suddenly made sense.

So I would request replacing your GIF with https://i.stack.imgur.com/ycw8Zm.gif (or something similar).

Now, I know that your GIF's caption says "the real part of the wavefunction is plotted here", but this fails to convey just how impossible it is to understand the math vs the GIF without including the imaginary component (i.e. to get a corkscrew). Many readers will gloss over or not fully understand the critical importance of that disclaimer. Just looking at the real component alone is baffling and misleading -- the immediate (but false) visual impression is violation of the Hamiltonian, conservation, etc. So let's not even present such a confusingly incomplete picture in the first place, yes? The 3D version immediately makes visual sense wrt the Hamiltonian.

Also, the red/blue animations further down might be better in 3D -- so that readers don't try to conceptualize the real and imaginary components as separate "overlays" in the same plane, as depicted. Frankly, my brain balks at even attempting to mentally convert the bottom-most one to 3D.

Many apologies if I have posted this section incorrectly in any way -- it's my first time.  :) Damonhastings (talk) 17:05, 18 October 2020 (UTC)

@Damonhastings: Thanks for your opinion. At first It made contradiction to me. But as wikipedia policy works, we can't just put an image from any website (unless it is licensed freely) to show in wiki. If you will to find an free one or make it yourself publish it under free licensed, then It can be replaced. Please read, WP:Copyrights. ~ A. Shohag (ping_me or Talk) 07:14, 19 October 2020 (UTC)

Weird Al

Featured in the White and Nerdy video by Weird Al. whicky1978 ^talk 02:18, 18 March 2021 (UTC)

Use of ket wo bra

Found this confusing at first in the first exposition §§ and think it should named explicitly in the text accompanying the equation boxes in the current §s 1.1,2. 98.4.124.117 (talk) 04:04, 6 June 2018 (UTC)

The Ket notation should be completely removed. (a) It is NOT used in the intro which is correct and well-understandable. (b) It clutters the article with unnecessary notation. (b) BraKet notation may be useful together, but there are almost no Bras. — Preceding unsigned comment added by LMSchmitt (talk • contribs) 04:29, 2 November 2021 (UTC)

The Bra-use together with the Ket-use in the few inner products in this article should be replaced by declaring this <..|..> an inner product which is what it is. There is absolutely no need to introduce Bra-vectors and Ket-vectors here, and clutter the article with the corresponding notation explanation.

I don't think that using the notation that is thoroughly commonplace throughout the physics community is "clutter". The "Preliminaries" section starts using Dirac notation in its second paragraph, which is reasonable for a notation that is standard once one gets past "baby's first quantum mechanics" (~1 semester of undergrad). It says that $|\psi \rangle$ denotes a vector and $\langle \psi |\psi \rangle$ an inner product, and it links to Dirac notation for more details. Up to details of phrasing, that sounds about right for a page that's meant to be an overview of the topic, rather than a textbook building step-by-step from whatever it imagines the student's current level of ignorance to be. XOR'easter (talk) 11:22, 2 November 2021 (UTC)

OK: You don't think that using the notation is "clutter". Me, who has published quantum research [Applied Physics A: Materials Science & Processing 1999 / 05 Vol. 68; Iss. 5] thinks so. Wikipedia is for the babies, I am often told. We dont need the rasmatasm (expert-jargon) which the experts are aware of. There is even a WP-rule. ;-) Whether you use a simple (greek) letter, ket-notation for vectors, bold notation, or an arrow over the symbol, it's your choice as long as you let the reader know. ---- In this case, to understand the Schroedinger equation (SE), no ket-notation is needed at all as the very first version of the SE in the article shows. In the whole of the presentation not a single ket is needed. ---- I enjoy when I read expert non-sense in WP: the 2nd paragraph uses bra-ket notation.? The <..|..> is standard notation for an inner product. (It is also part of the Dirac formalism.) Otherwise, one would have to explain why it's not <..||..>, even though that notation convention is not difficult to guess. There is no serious computation involving bra-ket in the 2nd paragraph. And nothing presented (recall: for the babies with usually at most highschool math) using ket contributes to understanding what the SE is all about. ---- Clutter. ---- And the rule fetishism [[WP:NOTTEXTBOOK|, beautiful.

One should also ask why the article repeatedly jumps between ket and noKet notation for solutions of the SE. This is inconsistent and confusing. — Preceding unsigned comment added by LMSchmitt (talk • contribs) 13:30, 2 November 2021 (UTC)

I've published physics papers too, not that it matters because Wikipedia doesn't operate on credentials. WP:NOTTEXTBOOK is policy; following it is not "rule fetishism". The ket notation $|\psi \rangle$ is introduced in the first sentence of the second paragraph. The first paragraph is explicitly phrased in terms of the position basis, which makes sense as an introduction, since it's typically how Quantum 101 does the Schrödinger equation. The third paragraph then goes on to give the Born rule in terms of bra-ket notation. Generally, the article moves to the more sophisticated notation when the discussion becomes more advanced, which is sensible and unremarkable. State vectors are written as kets, while the projections of state vectors into bases are written as functions, which is also pretty standard. XOR'easter (talk) 13:46, 2 November 2021 (UTC)

In my opinion, WP has degenerated to a rule-cult. Sorry. ---- From your rule: A Wikipedia article should not be presented on the assumption that the reader is well-versed in the topic's field. So, even a standard notation which is according to the article (first eq.) not needed to represent a Schroedinger Eq., and is according to you (XOR'easter) beyond the baby-stage, but the formulation-method of university-level beyond 1rst semester physics, violates (in my judgement) your very WP-rule. We can disagree on this rule, or whether high-level, notation is useful. I find it useful, but not at all in this case. However you have brought forward nothing but formal points. Not anything that shows something that's needed to understand the SE. ---- To illustrate my point: IN can be described by counting {1,2,3...}, even though some WP-snowflakes see their safe space threatened when one doesnt start counting with 0. One doesnt need to introduce and use the Peano axioms for 90% of the WP article about IN. The latter is much better structured in my opinion.

PS You seemingly have recognized the inconsistency in the article, since you have repaired linearity. Well, sorry to say, you have missed some. For example, the (countable) linear combination [superposition] in the same section. There are more. ---- Peace. — Preceding unsigned comment added by LMSchmitt (talk • contribs) 20:36, 2 November 2021 (UTC)

The countable linear combination later in that section is explictly in terms of position-basis wave functions, rather than kets. XOR'easter (talk) 05:11, 3 November 2021 (UTC)

Regarding recent changes

There seems to have been a spat of changes since I last looked at this page, most of which took it in a poor direction in my opinion. For example, this edit added an opaque "It can be show that" with multiple inline formulas, ending with "the concept of probability can be used". What does it mean to say "the concept of probability can be used"? Two paragraphs later, we explain the Born rule in detail, so the addition is redundant (in addition to putting undue emphasis on the position-space representation, I'd say). This addition is not so bad in principle, but it seems the wrong level of detail when we have the page Dirac equation for such things. This edit confuses the general class of Schrödinger equations with the introductory example thereof and seems to be undue promotion of a paper published just last week. I could go on, but all in all, they're not changes I can get behind. XOR'easter (talk) 01:10, 14 January 2022 (UTC)

There's probably something worth saying about covariantly quantized theories, where the SE reads $H|\psi \rangle =0$ and the role of the Hamiltonian is to annihilate physical states. I'm not sure how much time needs to be spent on that, though. One reason this article needed a major overhaul a year ago was because every little fact about every instance of a Schrödinger equation ended up included. XOR'easter (talk) 06:59, 14 January 2022 (UTC)

Thanks for taking care of that. The problematic editor is also working on several other articles, so I think it's more fruitful to centralize discussion in WT:PHYSICS, I made a post there. Tercer (talk) 11:13, 14 January 2022 (UTC)

Invoking Galilean invariance as in this edit is another step that I don't think is suitable (at least in that location), since wave functions are not straight-up Galilean scalars: to get it right one has to compensate the boost by transforming away a phase, and that gets us back into finicky details and the "everybody's favorite fact" problem that plagued the article before. XOR'easter (talk) 18:38, 16 January 2022 (UTC)

And, more fundamentally, merely incorporating Lorentz invariance into a single-particle wave equation isn't enough to account for all the effects of relativity, so emphasizing the Galilean/Lorentzian difference up front seems to put the emphasis in the wrong place. So, this edit, though adequately sourced, is in my view a step in the wrong direction. (Since the given source does go into detail about the compensatory modification of the state vector, the added text isn't a good summary of it.) XOR'easter (talk) 19:49, 16 January 2022 (UTC)

I could be convinced that the "Unitarity" section might benefit from some rearrangement, but this cut out important information, IMO. And tricks like modeling particle decay by hacking non-Hermiticity into a "Hamiltonian" are sufficiently niche that language like "for a Hermitian Hamiltonian" is more confusing than clarifying, particularly when it contradicts the second paragraph of the whole article. XOR'easter (talk) 22:51, 17 January 2022 (UTC)

Review

This article is very poor for the purpose of Wikipedia because a lay reader can learn nothing from it. The concept needs to be explained in an understandable way, assuming that the reader has not studied advanced physics and math. If you can’t explain it simply, you don’t understand it well enough.

It is fine to have hairy math deeper in the article, but to confront the reader with Hamiltonians in the lede when they do not know what that means, it will lose the vast majority of the audience. Jehochman ^Talk 12:25, 14 August 2022 (UTC)

The lede is supposed to summarize the body of the article, which is full of math, and so the lede will necessarily contain at least a few fancy words. I'm sure the text we have now can be improved upon, but some lay readers will start tuning out as soon as they hit the word "equation", and there's not much we can do about that. XOR'easter (talk) 13:13, 14 August 2022 (UTC)

Possible reassessment for Good article nomination

Does anyone think enough/a sufficient amount of content has been added since the last assessment in February 2008 to warrant a re-nomination of this article for Good Article status? HapHaxion (talk) 04:20, 22 March 2017 (UTC)

I agree ! Its authors obviously wishes the readers to "get it" (in words, explaining each variable and also through good illustrations), I would say. They have managed to make a rather understandable article here. Especially compared to the complexity of the issue. That in itself ought to be awarded. I also guess it's comprehensive enough. Boeing720 (talk) 19:59, 13 December 2017 (UTC)

In my opinion, it's badly written. It uses notation for the sake of notation. For example, Dirac Notation <..|..>=1 is introduced at the beginning where to say length 1 is clearly enough. Any Hilbert space has a naturally defined length. No-one needs the ket-notation to understand the Schroedinger Equation. It is, consequently, repeatedly dropped in the article, and then, for the enlightenment of another author only, miraculously resurrected. That's inconsistent. Also, it uses alt_facts (meaning: dead wrong). The vector space of square integrable functions L2(C) is NOT a Hilbert space. L2(C)/(functions_with_support_on_a_null_set) is. If one talks about integrable, then one should specify the measure (here the Lebesque measure on IR). In addition, the whole discussion about unitary is a mess. The citation [11] which should illuminate the WLOG argument about a parametrization of unitaries does not mention parametrization, but only draws conclusions from the norm=1 requirement of states for quantum mechanical time-evolution. That the generator of a unitary one-parameter group can be an unbounded, densely defined, self-adjoint linear operator should be mentioned in order to make clear that all this is not so simple, but not explained in order not to threaten anybody's safe space. And referenced within WP. — Preceding unsigned comment added by LMSchmitt (talk • contribs) 21:26, 2 November 2021 (UTC)

Completely disagree. Nowhere - at all - does the author actually explain Schrodinger's Equation! It dives immediately from a superficial explanation of the equation to then talking about Dirac's generalised formalism, including using unexpalined concepts such as Hilber Space, eigenvalues, and so forth.

This article is, to me, posturing; showing off a niche and hard earned knowledge, with scant attempt to keep to the Wikipedia spirit of offering explanations and insight to a wide readership. There needs to be:

1. More definitions of complex terms.

2. Less mixing of the various formalisms of quantum mechanics

3. A better text based explanations of the equation, and of what the wave function is. Knolan7799 (talk) 15:10, 4 January 2023 (UTC)

Missing a simple definition

Maybe the definitions are too complicated for most people, since Schrödinger's epitaph (in the photo) is simpler and sufficed for that important purpose.

I think it's fine to keep the existing ones. Maybe the epitaph form goes in the intro. 2600:4041:5976:4400:79B1:6365:9ED7:4A70 (talk) 20:52, 5 January 2023 (UTC)

This was me.. didn't realize I wasn't logged in Pablo Mayrgundter (talk) 20:54, 5 January 2023 (UTC)