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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.
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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.
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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
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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
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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
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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
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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,
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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
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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}
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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
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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
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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
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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
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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?
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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?
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1380:. 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.
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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 :
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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.
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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
753:(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.)
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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.
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In the first equation after "THE HYDROGEN ATOM" should it be r^2 instead of r. For a learner like me it would be good to have the units of the parameters described eg I assume the units for epsi are
Coulombs per cubic metre.
1142:{\displaystyle \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),}
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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...".
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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.
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No, that history of suffering is the inevitable consequence of how
Knowledge (XXG) pages naturally grow by accumulation, needing to be pared back when their contents become too
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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?
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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
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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.
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The text in the article claims the solutions are eigenstates, meaning only discrete values of E. The animation seems continuous to me.
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to go into steps between writing the equation and presenting its solutions, because it's a top-level overview. This page is
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Am I confused? The two places seem to contradict each other. It seems to me the first sentence is wrong. It should read:
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1289:{\displaystyle E\psi =-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}\psi +{\frac {1}{2}}m\omega ^{2}x^{2}\psi ,}
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which has 4 equations. None of these equations seem likely to produce the graph in my opinion. The most likely one
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The double-slit experiment and wave-particle duality as an argument for describing particles with a wave function.
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which is the first equation in the section and referred to as "The Schrödinger equation for this situation ..."
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on
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Knowledge (XXG). If you would like to participate, please visit the project page, where you can join
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Knowledge (XXG). If you would like to participate, please visit the project page, where you can join
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Classical mechanics predicts the time dependence of properties like position, momentum and energy.
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has an exponential factor which I guess eventually dominate at large |x|, unlike the graphic.
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Thanks for your recent edits. I want to encourage more qualitative physics and less math.
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doesn't need a movie, the only meaningful frames of which go past in the blink of an eye.
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But I can't figure out what it means. I assume that the graph plots 1D solution amplitude
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should not be faced with a proof. There are lots of other articles for details, eg
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Finally, and most important, which reference can I consult to verify this image?
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That is to say, should the words "time" and "space" be swapped in that sentence?
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I agree, this paragraph and the one before and after it are not
Preliminaries.
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The following paragraph in the "definition" of the SE is irrelevant:
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being plotted. It mentioned normalization but in a confusing way.
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Quantum mechanics predicts the time evolution of probabilities.
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The legend says "even" and "odd" but the equation says "n".
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Possible typo under "Separation of variables" section.
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of the decision if they believe there was a mistake.
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1522:Johnjbarton
1458:Johnjbarton
1159:Johnjbarton
908:Johnjbarton
817:Johnjbarton
770:Johnjbarton
584:High school
574:Outline of
386:Mathematics
377:mathematics
333:Mathematics
90:renominated
28:written in
1539:Categories
1382:XOR'easter
1347:XOR'easter
1333:XOR'easter
869:discarded.
636:Archives:
534:To-do list
1502:Rolancito
1442:Rolancito
1405:Rolancito
1352:Rolancito
1298:Rolancito
924:Rolancito
852:Rolancito
475:Chemistry
466:chemistry
438:Chemistry
197:is rated
674:365 days
156:Delisted
43:traveled
562:refresh
550:history
502:on the
413:on the
308:on the
281:Physics
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228:Physics
199:B-class
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137:Listed
118:Result
556:watch
186:This
35:color
1526:talk
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544:edit
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