Talk:Schrödinger equation/Archive 1

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Old messages

I'm a bit concerned that the second paragraph does not make sufficiently clear what role the Hilbert space plays: the elements of the Hilbert space describe instantaneous states of the system. The time-dependent wave function Ψ is therefore not an element of the Hilbert space; it is a function with values in the Hilbert space (and domain time)

Ψ : R -> Hilbert space.

It took me a while to figure this out, and I'm not sure that all physists are aware of it; maybe we should explain it in that paragraph?

Also, the notation Ψ = Ψ(r,t) seems to imply that the Hilbert space is the space of square integrable functions, but that is only true in some special one-particle examples. Often, the Hilbert space is a tensor product of several finite- and infinite-dimensional ones, and the elements of it cannot really be thought of as "wave functions" (which is why I quoted that term in the original article). --AxelBoldt

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I moved Schroedinger equation back here. It hardly makes sense to switch to a new (and no more valid) spelling. We already have everything linking here. Also, the articles on Erwin Schrodinger and Schrodingers cat are already spelled this way. -- CYD

I just realised that you were discussing this issue here.

What is the relevance of "everything linking here"? Everything will link to wherever the redirect sends them. The use of redirects does not harm Wikipedia in the slightest, it is designed to use them. (This is in contrast to double redirects or links to disambiguation pages, which do not work and must be fixed.)

I moved Schroedinger equation because I happened to land on it and decided to improve the spelling. Any individual change from "Schrodinger" to "Schroedinger" will be an improvement (because the latter more precisely reflects the German original), but I wasn't in the midst of a crusade to improve them all; I just improved the one that I happened to be at while I was looking something up. Similarly, I may improve a grammatical construction in an article that I happen to be reading without seeking it out in every article that I can find.

In Talk:Schroedinger equation, you mention consistency. First, spelling in Wikipedia is not required to be consistent. Next, it seems that most mention of the name is already in the form "Schrödinger" (perhaps I am mistaken about this), so consistency demands conforming to that, not to "Schrodinger". And "Schroedinger" is actually closer to "Schrödinger", since "oe" is the accepted representation of "ö" when accents are not available. (Here they are available, but slightly annoying to use.)

"Schrodinger" is acceptable, so is there no need to seek it out and change it in order to ensure Wikipedia's accuracy. But it is deprecated, which is reason to change it when one stumbles across it.

Anyway, I will not move this again, so you may do what you please, but that is what I was thinking.

— Toby 09:59 Oct 18, 2002 (UTC)

PS: I hope that you at least agree that the cat article needs an apostrophe!

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Hey guys (specifically CYD), please please PLEASE use the move page function when you rename pages. Cut-n-paste is a royal pain in the ass for people trying to review a page's edit history down the road, only to discover they have to go across fifteen different spellings and name variations and capitalizations and syntax arrangements, half of which weren't mentioned when they were cut-n-pasted. See Wikipedia:How to rename a page. --Brion 10:19 Oct 18, 2002 (UTC)

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Has the Schrodinger equation been proposed in 1925 or 1926? The text and the references disagree.

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Numerical methods should be included among the solution methods. I nobody is against, I'll do it. Javirl 15:54, 20 July 2005 (UTC)

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Moving to talk page per your request.

If Schrodinger is an acceptable spelling, why did you move the page to Schroedinger's equation? Everything is linked to Schrodingers equation, and nothing is linked to Schroedinger equation; we should just pick one convention and stick to it. -- CYD

(More at Talk:Schrodinger equation)

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I have a problem with the position basis material, in particular

For many (but not all) quantum systems, the state space can be spanned with a "position basis" made out of position eigenkets. For a single-particle system, we write each basis ket as |r>, which is to be interpreted as a state in which the particle is localized at position r.

Assuming that we are talking about a basis in the Hilbert space sense, and not in the Linear algebra sense, then this cannot be correct: the typical Hilbert space L²(R³) of square integrable complex functions of three variables is separable, and therefore each of its Hilbert bases is countable.

I suspect |r> is something akin to the Dirac Delta, but these are of course not elements of the Hilbert space and can therefore not constitute a basis.

It seems as if the position basis material was added in order to get from the bra-ket form of the Schrodinger equation to the wave equation. But that can be done much faster: "in many applications, the underlying Hilbert space is a space of square integrable functions, and the kets are then nothing but such functions." After all, kets are nothing but fancy notations of elements of some Hilbert space, and square integrable functions are also elements of some Hilbert space. AxelBoldt 20:50 Jan 3, 2003 (UTC)

I'm no mathematician, but the course I took on QM included the first chapter of the book Quantum Mechanics by Sakurai, and he did in fact derive the wavefunction using the position basis. He made some other interesting points, including deriving the fact that the representation in the momentum basis is just the Fourier transform of the representation in the position basis. I'm pretty sure he did all this using the position basis as an infinite set of 3D Dirac delta functions, one for each point in $R^{3}$. I have no idea about the mathematical rigor of all of this, but it seemed to work out alright. Edsanville 19:30, 22 Aug 2004 (UTC)

Yeah, Sakurai, along with most QM texts intended for physics students contains many mathematical errors, but like you said everything works out fine physically. The |r> is the dirac delta function, in Sakurai's treatment. Physcists define loosely the spectrum of an operator to be the set of eigenvalues, which we know can be empty for operators on general Hilbert spaces. This sometimes results in language such as "continuous set of eigenvalues". So their consider the dirac delta functions the "eigenfunctions" of the position operator, which is multiplication by x. So to make this formulation work, one needs to do (incorrect)things like "integrating" |r><r| over some "continuous spectrum." As was pointed out, this really makes no sense, since L^2 is separable. To do it right, we use the Borel functional calculus. —This unsigned comment was added by 24.155.72.152 (talk • contribs) .

Well, the bra-ket space is not exactly a seperable Hilbert space since Dirac delta functions aren't functions in the mathematical since. Moreover, the function f(x) = exp(ipx) is not square integrable. However, for all practical cases it is an Hilbert space and it is not so wrong to think of the Delta function as a function. In fact, the Uncertainty Principle gurantees we won't meet a true delta function in our experiments, but only an aproximations of it. For theoretical calculations, the delta function can be used as well as the |x> basis. MathKnight 11:30, 23 Aug 2004 (UTC)

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The first part of this page is unreadable due to superimposed PHP error messages. -- Merphant 02:11 Jan 19, 2003 (UTC)

The culprit appears to have been the following equation:

\int \left| \mathbf{r} \right> \left< \mathbf{r} \right| d^3r = \mathbf{I}

I've taken it out of the article, but now there's a gap where the equation should be, so somebody needs to fix this. It also seems to have made some of the other equations disappear. I wonder if it was just a missing math tag or something... --Camembert

Actually, it's just the equation that was immediately above the troublesome one that's disappeared (in the "The Wave Function" section). I don't know why. There's doesn't appear to be anything as obvious as a missing tag, but I don't understand the markup, so can't do anything more, really. Hopefully someone who can, will. --Camembert

Um... I think I've broken it again. Sorry. I tried to revert to Camembert's last version but it didn't seem to work... so... er... um... I'm going to go away now and hide and pretend I had nothing to do with this. -Nommo

Rather weirdly, I seem to have fixed it. The content of the page hasn't actually been changed at all, so it must have been some odd caching error. --Camembert

Idn't that the wrong equation though now? -Nommo

Oh heck. How on earth has that happened? That's not what I pasted in, I pasted in what I originally took from the article, above. Either I'm going insane, or there's gremlins in the system (the two are about equally likely, I think). --Camembert

Well, we're not getting a pageful of errors about it, but it's not rendering the equation either. In any case, I'm leaving a note on Wikipedia:TeX requests, so hopefully somebody who knows what they're doing will help. --Camembert

Ok... I've put the plain old texty version of the equation in. So there's obviously something wrong with the stuff posted up there... I guess... Works now anyway, and makes sense, and is the right equation. Just not in glorious TeXicolor. -Nommo

i think klein-gordon equation describes relativistic systems. -Rahuljp

Why this was removed?

Therefore, if we know the decomposition of |ψ(x,t)> into the energy basis at time t = 0, its value at any subsequent time is given simply by

$|\psi (x,t)\rangle =\sum _{n}e^{-iE_{n}t/\hbar }c_{n}(0)|n(x)\rangle$

More over, if we are given |ψ(x,0)> (initial condition), using orthonormality property we can calculate

$c_{n}(0)=\left\langle n|\psi \right\rangle$

and receive the following expression:

$\psi (x,t)=\sum _{n}n(x)\left\langle n|\psi \right\rangle e^{-iE_{n}t/\hbar }$

The more canonical forms of this expression are

"State vector form"

$|\psi \rangle =\sum _{n}|n\rangle \left\langle n|\psi \right\rangle e^{-iE_{n}t/\hbar }$

"Measurement (projection) form" :

$\langle x|\psi \rangle =\sum _{n}\langle x|n\rangle \left\langle n|\psi \right\rangle e^{-iE_{n}t/\hbar }$

MathKnight 12:39, 11 Sep 2004 (UTC)

Calculating c_n from |ψ> is irrelevant because c_n is defined in terms of |ψ> and the energy basis. As for the expression

$\psi (x,t)=\sum _{n}n(x)\left\langle n|\psi \right\rangle e^{-iE_{n}t/\hbar }$,

it's straightforward to obtain it from the unprojected expression, so I don't see what additional information that imparts. Besides, (i) it might belong in the later "position basis" section, but not the first section, and (ii) the text doesn't define n(x).

As for the stuff in block quotes, it simply repeats an equation that is already there in the text. The position basis version of the Schrodinger equation isn't even introduced until the next section, so it's neither enlightening nor useful to talk about it here. -- CYD

splitting the article

This article is a bit long at the moment I think, perhaps it would be an idea to move the sections Time-independent Schrödinger equation and Schrödinger wave equation into new separate articles, keep the first few lines of text and put this Main article ... at the sections? Passw0rd

Nope. The article is not particularly long as articles go. Please don't split. -- CYD

No. The two aspects should appear together. MathKnight 11:13, 14 Nov 2004 (UTC)

Content

The way I see it, this page doesn't explicitly give several things it should, and is in fact confusingly written. First, it may be helpful/instructive to define the Dirac notation better, and use instead the standard wave function to introduce it. Second, the Heisenberg matrix form of this equation should be placed in the article, since it is of course identical in content and very similar in form. Further, the time-dependent and time-independent forms should be written clearly and well-marked. It may also be instructive to include an intuitive derivation of this equation, perhaps using the historical approach of ${\frac {\partial }{\partial x}}\Psi ={\frac {\partial }{\partial x}}Ae^{({\vec {p}}\cdot {\vec {r}}-Et)/i\hbar }={\frac {1}{i\hbar }}|p_{x}|\Psi$ and so on (excuse the errors in factors of $\hbar$, etc). Also, maybe detail in what situations this equation is accurate (experimentally or theoretically) and what seperates it from some other equations, specifically the Klein-Gordon.

PS. Please do not see this as anything but constructive criticism. I would be happy to undertake this project myself, but posting it here gives someone the chance to listen in the interim before I have time myself to perhaps undertake such a large product. It would just be nice to have more of an instructive page for people perhaps new to the subject. --ub3rm4th 18:48, 17 Feb 2005 (UTC)

I agree, someone not already familiar with QM would have a tough time with this page, as with most other QM related pages. But I wonder if this is the place to bring a new person up to speed? Maybe, as you said, there should be a separate "bring a new user up to speed in QM" page. The Quantum mechanics page does not seem to do it, nor does the Mathematical formulation of quantum mechanics page. Paul Reiser 20:15, 17 Feb 2005 (UTC)

Go ahead and make your edits, if you feel that it will improve the article. Note, however, that from the modern point of view the Schrodinger equation is not a derived equation; rather, it is a fundamental postulate of quantum mechanics. In particular, the equation is exactly correct whenever quantum mechanics holds (i.e. all the time, except when general relativity comes into play.) The Klein-Gordon equation is just a special case of the Schrodinger equation. This is already mentioned in the article, but is worth re-emphasizing. -- CYD

CYD: Although the Schr\"{o}dinger equation is a fundamental postulate, appeals to classical mechanics exist that can help people accept it. Especially simple things as the wave approach I gave above and the simple eigenvalue equation ${\hat {H}}\psi =E\psi$ which obviously implies that the classical kinetic and potential energy functions give directly the energy. In fact, the common method taught for getting quantum operators is to find V(p,x) and replace all the p by iħ d/dx (or obviously whatever operators for your specific representation).

Paul Reiser: I sympathize that maybe Wikipedia shouldn't be a place for teaching people, but what is the point of an article if it is only of use to people already familiar with the material? It seems slightly unnecessary if everyone who might come to the QM pages already knows QM. I often use the math pages on Wikipedia to learn new maths, though it is sometimes difficult. (I just printed off a ton of pages on Category theory, Tensors, and Exterior algebra.) I can see, however, that maybe a page aimed specifically at instructing people might be useful. However, this is not exactly what I was thinking; I want more to just organize this page for clarity and usefulness with myself in mind, and thus other people who actually know some of the material already and who may need to look up some idea they've forgotten or the exact form of an equation. It seems currently to be very thrown together, and even if it reads clearly, I could not find whatever I was looking for the other day (I can't remember what specifically). --ub3rm4th 21:37, 18 Feb 2005 (UTC)

content

I think we should add in the wave equation form of the schrodinger equation, the one in partial differential form as it is far easier to understand at a lower level of mathmatics, and can help people understand the basics of the math in quantum mechanics

-- Cpl.Luke 18:41, 13 Jun 2005 (UTC)

It is there: Schrödinger equation#Non-relativistic Schrödinger wave equation. -- CYD

time-independent SE

That would be H|p> = 0, and the full SE H|p> = E|p>. E, the energy operator is ih\partial_t. This is not what is now in the article. --MarSch 13:15, 22 Jun 2005 (UTC)

Hmm, okay I get it now and will try to clean up a bit.--MarSch 13:19, 22 Jun 2005 (UTC)

Linear eigenfunction operators

Hellow boys! I've studied Math physics and encountered some problems about $\ {L}$(which appears in the chapter of Eingenfunction methods).[1] Wish someone can tell me why they are such.

Now we have a Linear eigenfunction that gives

$\int _{a}^{b}g^{*}(x)L{f(x)}dx=(\int _{a}^{b}g^{*}(x)L{f(x)}dx)^{*}$

$=\int _{a}^{b}f(x)L{g^{*}(x)}dx$

This is first question. Second one is

$\ Ly={\lambda }{\omega }y$

where

$\ \omega$is weightfunction. What roles(or physical meanings) does $\ \omega$ play?

PS:I'm not quite sure what I wrote. If any mistake,correct on me!^^

Standing Wave Math Expression

Transfered to copy onto another web.--HydrogenSu 15:04, 5 February 2006 (UTC)??????????????

Revert by lethe

I was wondering why you reverted my edit of the equation. I find it much easier to read and understand equations with variables defined in a list like that. Paragraph form doesn't work well for math in my POV. And with an equation like the one I was editing, the more clarity the better. Fresheneesz 08:56, 21 April 2006 (UTC)

It is my opinion that paragraphs are the appropriate format and that lists do not look good. Listing the meanings of variables is a standard piece of text for many math articles, and before you set a new precedent different from the one used in all textbooks and wikipedia articles for how to format this bit, I think you should seek some feedback from others. -lethe ^{talk +} 14:33, 21 April 2006 (UTC)

Style. Lethe is following the current style, which is used in thousands of articles. If you want to debate it, I suggest posting to Wikipedia talk:WikiProject Mathematics and or the talk page of the WPMath style guide. linas 14:38, 21 April 2006 (UTC)

I've seen perhaps a hundred or more articles that have the list format - which is why I started emulating it. I'm not trying to change style, I'm simply following in the footsteps of other articles.

Also consider the user that needs a reference rather than the whole article. I'm one of those people, and having variables listed helps me very quickly glean the meaning of the equation - without reading the rest of the article. Its much harder to read an equation in paragraph form. This really isn't a new format. Comments? Fresheneesz 19:07, 21 April 2006 (UTC)

Could you link some of these hundred or more articles? I'd like to see. -lethe ^{talk +} 19:29, 21 April 2006 (UTC)

Yea just looked for a bunch:

1. Force
2. momentum
3. Kinetic energy
4. Ideal gas law
5. Heat engine
6. Mach number
7. Second law of thermodynamics
8. Magnetic field
9. Electric field

If you scrutinize the history, you'll find that some of those I might have implimented the list - but I looked in the history for, mag field, E field, ideal gas, and force - none of those I implimented. I'm not sure about the others, I think i did in mach number and kinetic energy - I can't remember about the rest.

Anyways, its not a new format. Its very helpful for people like me who always skim articles for important info. Fresheneesz 20:01, 21 April 2006 (UTC)

Note that the lists are never in the article intro. It's just not an encyclopedic way to start off an article. (The exception out of your selection, mach number, only has two things to define.) - mako 00:29, 22 April 2006 (UTC)

Whatever the excuse, the long deeply nested list looks hideous in this article. I support lethe's decision to get rid of it. --KSmrq^T 02:32, 22 April 2006 (UTC)

Fresheneesz is right, this class of topics are often presented in a bulletted style. Also, a quick review of Wikipedia:Manual of Style (mathematics) seems to indicate that in fact we have no written policy on this (unless I skimmed over it). I'll proclaim "neutral opinion" on this; if the argument gets out of hand, I strongly suggest talking it to the talk page of the style manual, where you will find a more appropriate place to discuss this. linas 03:59, 22 April 2006 (UTC)

It may be true that lists aren't usually in the article intro. In that case, I would be perfectly happy to move the equation, along with its list of variable, down under the TOC. What do you think? Fresheneesz 22:27, 22 April 2006 (UTC)

Ok.... I'm going to try to reimpliment it. Fresheneesz 09:26, 25 April 2006 (UTC)

Alternative list

I made this as an alternative, cause it makes the list a bit shorter

$H(t)\left|\psi (t)\right\rangle =i\hbar {\partial \over \partial t}\left|\psi (t)\right\rangle$

where

i is the imaginary unit, t is time, $\hbar$ is the reduced Planck's constant, $\psi (t)$ is the wave function of the particle,

${\partial \over \partial t}$ is the partial derivative with respect to t, and H(t) is the Hamiltonian - a self-adjoint operator acting on the state space.

Lower level equations

This article needs lower level equations like this:

One dimensional time-independant:

$-{\frac {\hbar ^{2}}{2m}}{\frac {\mathbf {d^{2}} \psi (x)}{(\mathbf {d} x)^{2}}}+U(x)\psi (x)=E\psi (x)$

3-dimensional time-independant:

$-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi (r)+U(x)\psi (r)=E\psi (r)$

source: [2]

[3]

Not everyone is familiar with bra-ket notation - and the page on bra-ket notation isn't very helpful in learning it. Fresheneesz 09:45, 25 April 2006 (UTC)

I agree with this 100%. The first section presented to the reader should contain the above. The following sections can then delve into the bra-ket notational version. Ed Sanville 10:21, 25 April 2006 (UTC)

Illustrations of solutions

It would be really instructive to include figures illustrating solutions to the equation, answering questions about how the wave function of that particle that just flew by really looks like, etc. Bromskloss 20:11, 28 April 2006 (UTC)

Template for special case Schrödinger equation solutions

In order to organize the many different articles about special case Schrödinger equation solutions, perhaps a template could be created that could be included at the bottom of each of the articles that links to other special case solutions. For example, it could be organized as follows (but with better formatting):

• One dimension:
  • free particle
  • infinite potential well (particle in a box)
  • finite potential well
  • delta potential well
  • finite potential barrier/square potential
  • delta potential barrier
  • quantum harmonic oscillator
  • particle in a one-dimensional lattice (periodic potential)
  • particle in a ring/ring wave guide
• Three dimensions:
  • particle in a spherically symmetric potential
  • hydrogen atom or hydrogen-like atom

I would make the template myself, but I do not really have much experience with making/using templates like this; also, it would be good to get additional input regarding such a template.--GregRM 14:56, 20 August 2006 (UTC)

GA Re-Review and In-line citations

Members of the Wikipedia:WikiProject Good articles are in the process of doing a re-review of current Good Article listings to ensure compliance with the standards of the Good Article Criteria. (Discussion of the changes and re-review can be found here). A significant change to the GA criteria is the mandatory use of some sort of in-line citation (In accordance to WP:CITE) to be used in order for an article to pass the verification and reference criteria. Currently this article does not include in-line citations. It is recommended that the article's editors take a look at the inclusion of in-line citations as well as how the article stacks up against the rest of the Good Article criteria. GA reviewers will give you at least a week's time from the date of this notice to work on the in-line citations before doing a full re-review and deciding if the article still merits being considered a Good Article or would need to be de-listed. If you have any questions, please don't hesitate to contact us on the Good Article project talk page or you may contact me personally. On behalf of the Good Articles Project, I want to thank you for all the time and effort that you have put into working on this article and improving the overall quality of the Wikipedia project. Agne 00:11, 26 September 2006 (UTC)

Can we have less of Maths?

I have been increasingly noting how entries in WIKIpedia are becoming more heady and obscure instead of being clear as an encyclopedic entry should in fact be.

Contrast the entry here with the following entry I found elsewhere.

http://www.physlink.com/education/askexperts/ae329.cfm

I think Mathematicians have taken over WIKI and will kill it.

I agree, the link you provided is the kind of introduction needed for this article. We don't need less math, we need an article that does not begin with a purely mathematical introduction. The heavy lifting should come later. Mathematicians are not a bunch of territorial ogres trying to obscure everything to everyone but themselves. People write what they know, and many people who know the mathematics find it difficult to write a simple introduction because they have taken many steps to get to their knowledge, and its hard to then backtrack and lead someone else along that path. PAR 03:38, 28 September 2006 (UTC)

Order in which equtions are shown.

I am reading the article "Improving Student's Understanding of Quantum Mechanics" in the August 2006 Physics Today (which was under some other papers). The first example of misconceptions is that the time independent Schrödinger equation is true for all states. To protect against that, maybe this page could be organizes so the time deponent equation appears, without bras and kets, before the time deponent equation does. That way those who do not know much linear algebra would still see the general form before the stationary state form. David R. Ingham 20:55, 21 October 2006 (UTC)

Question:

In the mathematical formulation of quantum mechanics, each system is associated with a complex Hilbert space such that each instantaneous state of the system is described by a unit vector in that space.

Since "instantaneous state" of a system extended in space is meaningless in relativity (even special relativity), does this mean tha QM is inherently a non-relativistic theory? —The preceding unsigned comment was added by Pmurray bigpond.com (talk • contribs) 00:29, 5 December 2006 (UTC).

Heisenberg picture versus Schrödinger picture

Currently, the subsection Schrödinger equation#Matrix mechanics and the Schrödinger picture makes it appear wrongly that the equation

$i\hbar {d{\hat {A}} \over dt}=[{\hat {A}},{\hat {H}}]={\hat {A}}{\hat {H}}-{\hat {H}}{\hat {A}}$

is universally true and consistent with the Schrödinger equation. On the contrary, that equation is only true in the Heisenberg picture (and then only when the partial derivative of A wrt time is zero) while the Schrödinger equation is only true in the Schrödinger picture. Although the two pictures give the same physical results, they are mathematically inconsistent with each other. (They use different bases for the Hilbert space as a function of time.) So any calculation should be done entirely in just one of the two pictures. This article should state at the outset that the Schrödinger equation is part of the Schrödinger picture, not the Heisenberg picture. JRSpriggs (talk) 09:34, 15 March 2012 (UTC)

I just moved this to Matrix mechanics without realizing this comment - sorry. -- F=q(E+v×B) ⇄ ∑_ic_i 09:21, 17 March 2012 (UTC)

Changes have been made here. I'm still in the process of cleaning up formatting and may re-locate it somewhere else in the article. That section (corrected) is better suited to the Matrix mech article than here.-- F=q(E+v×B) ⇄ ∑_ic_i 09:28, 17 March 2012 (UTC)

Atomic weight versus atomic number

Could you explain to me why atomic number Z is used in the equation in section Schrödinger equation#Two-electron atoms or ions in the kinetic energy term (and the formula for the reduced mass) rather than mass number A (or, more precisely, the mass of the nucleus as a whole including neutrons as well as protons divided by the mass of a proton)? Of course, Z is appropriate in the potential energy term. JRSpriggs (talk) 20:07, 3 April 2012 (UTC)

Zm_p is the mass of the nucleus, m_p is the mass of the proton. For 2-electron atoms mentioned there would be Z protons and... some neutrons. You're right. Lets replace with just M for the mass of the nucleus. F = q(E+v×B) ⇄ ∑_ic_i 20:18, 3 April 2012 (UTC)

Fixed. Happy? Sorry for not thinking clearly again (2nd time it happened today, see here)... =( F = q(E+v×B) ⇄ ∑_ic_i 20:22, 3 April 2012 (UTC)

Out of interest, isn't the reduced mass usually mu $\mu \,\!$, not the obscure $m_{ep}\,\!$? I have yet to see that in a source and arn't we supposed to be using the most common notation? Maschen (talk) 21:14, 3 April 2012 (UTC)

Don't remind me of that link - It’s been a bad day today. Does it matter? Greek $\mu \,\!$ is overused, if it annoys you so much why don't you fix it?? F = q(E+v×B) ⇄ ∑_ic_i 21:16, 3 April 2012 (UTC)

I will. Maschen (talk) 21:18, 3 April 2012 (UTC)

Introductory Preamble

I wanted to say something like:-

"Like Newton's second law Schrödinger's equation for a quantum system has not been proved or derived from more elementary principles. It is used because it works and has predicted a multitude of reactions time and time again - the only proof is in nature or natural." Blueawr (talk) 07:55, 1 September 2012 (UTC)

You could put that in somewhere, but it is the kind of statement that needs to be backed up with a good secondary source. RockMagnetist (talk) 16:53, 1 September 2012 (UTC)

Separate article on exact solutions?

I have proposed before (in passing) that we make an article on the exact solutions (and forgot, till now), such as Schrödinger equation (exact solutions), because;

• it will trim the article (currently 81.527 kB, too big)
• reduces the amount of maths in this article, allowing for more qualitative content future editors may add,
• parallels the nice idea of Exact solutions in general relativity from Einstein field equations,
• helpful for those just introduced to the equation, since there will not be many realistic systems it can be solved for exactly, and serves as a springboard to perturbation theory (quantum mechanics), variational method (this article is for QM), ladder operators...

written summary style for detailed articles on specific systems (step potential, delta potential, Harmonic oscillator (quantum), hydrogen atom etc.). Any objections? Maschen (talk) 09:59, 11 November 2012 (UTC)

Update: I'm drafting this in a pre-article (in namespace), it will not be created if people desire to re-merge in the future... Maschen (talk) 18:11, 1 December 2012 (UTC)

"Incorrect - why?"

Incorrect in that it doesn't detail the whole situation which would require something to be going faster when it is lower than when it is higher. Consider that a SR71 could be higher and be going faster (than I am), and you'll see why the statement is wrong. I think removal of the line is appropriate. --Izno (talk) 00:48, 6 December 2012 (UTC)

We're talking about the sentence "For example, a frictionless roller coaster has constant total energy; therefore it travels slower (low kinetic energy) when it is high off the ground (high gravitational potential energy) and vice versa." Right?

I read this as a statement about frictionless roller coasters, and I believe it is a correct statement about frictionless roller coasters. It seems to me that readers are unlikely to read this sentence and get the impression that it is true in any context besides frictionless roller coasters.

But maybe I'm wrong. So here is an idea:

• "For example, a free, frictionless roller coaster has constant total energy; therefore this roller coaster will travel slower (low kinetic energy) when it is high off the ground (high gravitational potential energy) and vice versa." --Steve (talk) 16:20, 6 December 2012 (UTC)

I read it for what it was, and that it was trying to convey an example of conservation of energy. Your change makes it slightly better. But connecting a roller coaster to the law needs more; the assumptions are not laid out which would make the statement precisely true. On top of this, it still does not illustrate why it is the case that the roller coaster will travel faster in a different place. What is needed, quite frankly, is a link to the conservation of energy article. I still favor deletion of the line and would favor insertion of a link to the law of conservation of energy, which is quite oddly lacking for the first time that implications are discussed in the article. (Perhaps because it is used as a part of the assumptions section?) --Izno (talk) 17:11, 6 December 2012 (UTC)

In other words, it is not clear to me the connection. "It goes fast here and slow here" doesn't make it clear to a reader why that is the case, only that it is the case. --Izno (talk) 17:24, 6 December 2012 (UTC)

As the one who reverted, it should be clear that potential energy and kinetic energy interchange and the sum of these is constant assuming no dissipative effects (e.g. friction, heating). The worded equation is given there, no?. If the PE increases (for gravity, increase in altitude is one way) then the KE decreases (speed decreases by the square root), and vice versa. The aeroplane you linked to has nothing to do with roller coasters and frictionless systems so I have no clue why you point to that... Maschen (talk) 18:00, 6 December 2012 (UTC)

According to archive 3, the roller coaster example was added to analogize a familiar system (or at least one that is easy to understand) with a particle in a potential, i.e. the roller coaster tends to move in directions of decreasing potential, and there are stable/unstable equilibrium points. If you're finding this example confusing - maybe there is no harm in removing it... but it's not complicated... Maschen (talk) 18:07, 6 December 2012 (UTC)

I added the link conservation of energy in the first sentence of that section. Better? Maschen (talk) 20:23, 6 December 2012 (UTC)

Naturally, but this is only true for an object in motion with only one force (and conservative at that) acting upon it (or if there is a second force, it is a normal force and does not counteract completely the first force). I point to an airplane because it is a system which goes high and fast, which makes it a problematic counterexample to our example of a roller coaster. In effect, the comment of "high and slow" is so basic as to be useless, because it does not lay out the requisite assumptions.

If it is our desire to comment on the fact that systems act toward states with a lower potential, then the example certainly does not convey that.

The thought I had was to switch it with a pendulum, or a vibrating string. It's not uncommon to see either of the two in context of wave equations (in general). (I'll note that a pendulum is probably better for this example.) Even so, switching the type of system would not sufficiently fix the problem of the original statement. --Izno (talk) 20:38, 6 December 2012 (UTC)

Now I see what you are trying to say (except for when you said the example was wrong when it isn't, but we're past that now). Apologies... Perhaps it is too short to be fully understandable. The pendulum (or the like) sounds like a nice idea. Let's see what others think... Maschen (talk) 21:06, 6 December 2012 (UTC)

Actually (you'll like this), maybe the example could just be deleted since it only serves as an example of classical energy conservation in a specific case, suppose there is no loss of continuity. Up to others from this point on, I'll stay out of this... Maschen (talk) 21:16, 6 December 2012 (UTC)

Partial waves method

Something should be mentioned concerning the use of the equation to scattering problems by partial wave analysis.--188.26.22.131 (talk) 12:31, 2 August 2013 (UTC)

• I don't agree. Of cause it is true that the Schroedinger equation is the basis of scattering theory. But this is true for basically any quantum mechanical problem. There is no point in listing all problems that build on this equation. Ciao. --Falktan (talk) 16:52, 26 August 2013 (UTC)

Article gives impression that the relativistic forms are also called the Schrödinger equation

I have not read through this thoroughly, but noticed the following that might confuse a reader:

• The lead gives no indication that the domain of applicability of the Schrödinger equation is strictly the nonrelativistic domain. In fact, it says that it can be transformed into the Feynman path integral formulation, and since the latter is relativistic (I'm not even aware that it has any nonrelativistic form), I would go so far as to suggest that this is seriously misleading.
• The article discusses both nonrelativistic and relativistic wave equations, giving the impression that the name "Schrödinger equation" encompasses them all. My understanding is that it refers strictly to the nonrelativistic equations given early on, and that the Dirac equation and the Klein–Gordon equation are not considered to be examples of the Schrödinger equation.
• The subsections on time-dependent and time-independent equations would be better named for the respective forms of the Hamiltonian, not the resultant form of the equation, e.g. General Hamiltonian and Time-independent Hamiltonian. They could also be written to emphasise that with a time-independent Hamiltonian (which in itself is generally inherently an approximation) the equation can take another form

My involvement, time and expertise in this sort of article is not sufficient to make such edits, but I hope any who are suitably interested would like to use these pointers as inputs to their edits. — Quondum 06:30, 23 September 2013 (UTC)

The Schro eqn is a fundamental postulate of QM, and the article states the general equation can be used in relativistic and non-relativistic context (yes, it can, all that's needed is the correct Hamiltonian), if this were not the case then how can non-relativistic QM and RQM be consistent with the postulates of QM?

Feynman path integrals are equivalent to the Schrodinger equation, and can be relativistic or non-relativistic (there is a connection between non-relativistic QM and classical mechanics... the correspondence principle). There is plenty literature on Feynman path integrals, but I don't have any to hand right now. For a WP pointer, this section in the path integral formulation article uses the classical kinetic energy expression.

A "general Hamiltonian" is time-dependent in general anyway, and is more meaningful than "general" and provides contrast with "time-independent".

M∧Ŝc²ħεИ_τlk 05:59, 24 September 2013 (UTC)

The article says "The general form of the Schrödinger equation is still applicable, but the Hamiltonian operators are much less obvious, and more complicated." Such an equation strikes me as highly contrived; for example, the Hamiltonian needed to obtain the Klein–Gordon equation would be really strange, and the Dirac equation cannot be considered to be a Schrödinger equation, as the Dirac equation is in terms of matrices, and the Schrödinger equation in terms of real values. Is this really in sources?

I'll accept what you say about Feynman path integrals. — Quondum 01:23, 25 September 2013 (UTC)

Yes, the Dirac equation can be considered a Schro equation, Penrose says this in the chapter on the Dirac equation (page 621 in my copy "the Dirac equation can be written in the form of a Schro equation ... Of course the singling out of the time derivative is not relativistically invariant but the entire Dirac equation is"). Particle physics books (many are listed in the RQM article references, the one I have to hand now is Particle physics by B.R. Martin and G. Shaw) usually say something like "Dirac proposed a Hamiltonian of the form:

${\hat {H}}={\boldsymbol {\alpha }}\cdot {\hat {\mathbf {p} }}+\beta mc$

where a = α₁, α₂, α₃ and β are to be found subject to the constraints..."

While non-relativistic quantum Hamiltonians are just functions of position, momentum, and time (as in classical Hamiltonian mechanics), the relativistic quantum Hamiltonians are also functions of position, momentum, time, and spin matrices. The Schro equation does not restrict the wavefunction to be a scalar, it could be anything that can describe quantum mechanics.

You have a point about the KG Hamiltonian, most people don't derive it from the Hamiltonian since it can be derived directly from the energy-momentum relation or viewed as the square of the Dirac equation (you knew that), but I'm sure it's possible one way or another. I'll look for sources.

M∧Ŝc²ħεИ_τlk 07:37, 25 September 2013 (UTC)

By the way, a good faith edit was made to the RQM section of this article saying QM is formulated to be consistent with SR. As YohanN7 confirmed a while back (see talk:relativistic quantum mechanics#Suggestions): RQM = SR + QM, i.e. special relativity and quantum mechanics applied together. The statement "Quantum mechanics formulated to be consistent with special relativity" may have the implication that QM is modified in some way (or may not...). Quantum mechanics doesn't actually change, all pictures of QM are already applicable with SR, but the outcome is that new predictions and mathematical objects appear, which did not from QM alone. M∧Ŝc²ħεИ_τlk 07:56, 25 September 2013 (UTC)

I suppose the Schrödinger equation evolved rather than being replaced. This is in accordance with the physicist's natural approach of "if it works, use it", in contrast with the mathematician's more rigorous approach. Given that the Schrödinger equation can be adapted to both Galilean and special relativity (and no doubt anything else), that the wavefunction and the underlying abstract algebra is regarded as being free in type, and that the Hamiltonian is unconstrained (it presumably becomes only loosely representative of energy), the Schrödinger equation is essentially not falsifiable, which is not a pretty state of affairs. It asserts that there exists some wavefunction, which when subject to some operator (which is expressed as the difference between the "Hamiltonian" and the partial derivative with respect to time multiplied by a constant), yields zero. More strictly interpreted, it says that the wavefunction is the solution of a homogenous differential equation, or equivalently that it obeys the superposition principle - and nothing else whatsoever. Beyond this, anything to be added is freely chosen in the choice of abstract algebra and Hamiltonian.

On my edit, I have reverted it since as you suggest, there should not be a suggestion of reformulation of QM, only that its form is constrained to be relativistically invariant. I am not particularly comfortable with "applied together", due to what I perceive as a semantic difference between "applied together" and "simultaneously apply". The former is an active form, suggesting a process in which one takes QM, and then applies SR, getting something new. The latter is a passive form, indicating that they both are part of the framework, i.e. they act as constraints. Would this change make more sense? — Quondum 09:53, 25 September 2013 (UTC)

The second change, for the RQM section, seems fine with me.

Remember the Schrodinger equation is a postulate of quantum mechanics, assumed to be true, so it could be falsified (just like the postulates of special/general relativity) if there was some experimental phenomenon which the SE cannot describe (the Hamiltonian for the phenomenon is impossible to obtain or just pathologically unphysical).

I'm not sure what the issue is with the wavefunction, that in itself is another, separate, postulate. The SE is the evolution equation for it. It was constructed to be linear so that the superposition of states applies. What about it? M∧Ŝc²ħεИ_τlk 10:18, 25 September 2013 (UTC)

The article states:

This is the equation of motion for the quantum state. In the most general form, it is written:

$i\hbar {\frac {\partial }{\partial t}}\Psi ={\hat {H}}\Psi .$

With no constraint on the Hamiltonian other than it be a linear operator field over time and space, or on the underlying algebra other than that it be compatible with complex scalar multiplication, the general Schrödinger equation can be rewritten as

${\hat {P}}\Psi =0$

for an arbitrary linear operator ${\hat {P}}$.

This is why I said what I said before: with no further constraints, one can deduce linearity, but nothing else. The only way of falsifying this equation is if linearity does not apply, and I don't think that linearity is in contention. If there are some constraints on ${\hat {P}}$ (equivalently the Hamiltonian), for example that it obeys some conditions (e.g. locality, which would manifest as the operator being expressible in terms of partial derivatives with respect to the space and time coordinates), the equation would assert more than superposition. For the Galilean and special relativity cases, further constraints would apply. Even so, it is still saying very little until one adds yet further constraints, such as conservation of energy and momentum. So, I hope you can see that pretty much everything that can be "deduced" (including that it is an equation of continuity/evolution) must be built into the form of (or constraints on) the Hamiltonian. — Quondum 12:12, 25 September 2013 (UTC)

Yes, I do know that constraints are needed on the Hamiltonian in the context to construct it using physical principles, but I'm still not 100% sure what you're trying to get at improving the article...

OK... This article already gives the construction for the non-relativistic Hamiltonian, for one and n particles in any potential, we can agree on that. Are you saying to discuss the Hamiltonian (with constraints) for other situations in this article, like for the Pauli equation, and relativistic wave equations in general? We could always rewrite the article in some places (maybe have a section on this after some reorganizing), linking to the remaining details in the Hamiltonian operator article.

I apologize for misunderstanding... M∧Ŝc²ħεИ_τlk 19:33, 25 September 2013 (UTC)

I need to review the article in more detail to make specific suggestions, which I will be able to do more sensibly now that you have disabused me of some of my preconceptions. I should then be able to make more specific suggestions. — Quondum 21:43, 25 September 2013 (UTC)

Split off the non-relativistic content to another article(s)?

In response to the above thread...

I proposed before (more than once now) to transfer the exact solutions of the Schro eqn to a separate article Schrödinger equation (exact solutions). Note also the main article quantum mechanics has overlap on exact solutions to the non-relativistic Schro equation. Would cutting out the "exact solution" sections in this article and the QM article be useful?

Or instead, we could just transfer all of the non-relativistic formalism to another article: Schrödinger equation (non-relativistic) (by all means correct me on the dashes, this is just a suggestion...). This may have been proposed by someone before also...

Maybe both? In Schrödinger equation (non-relativistic), we have a discussion of the general ideas of the non-relativistic case, leading onto the mathematical details (such as those in the "properties" section of this article), without the exact solutions, then linking to the Schrödinger equation (non-relativistic) article.

Any thoughts? M∧Ŝc²ħεИ_τlk 20:36, 25 September 2013 (UTC)

Update: see also this section. M∧Ŝc²ħεИ_τlk 12:51, 27 September 2013 (UTC)

I just realized there is the article List of quantum-mechanical systems with analytical solutions. So there is no need for a new article. M∧Ŝc²ħεИ_τlk 09:16, 27 January 2014 (UTC)

Axiomatic construction using time-evolution operator

I guess the article is missing the key concept of axiomatic derivation of the Schrodinger's equation using the time evolution operator. This is the most natural and modern approach to understanding the equation (although, most likely, not originally proposed by Schrodinger). The details can be found, for example, in "Modern Quantum Mechanics" by J. J. Sakurai or this lecture by Prof. Susskind. - Subh83 (talk | contribs) 22:07, 21 February 2014 (UTC)

Distance and potential confused in Constant Potential section

http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation#Constant_potential

I was going to cut part of a sentence, but I decided to associate it with the following sentence so it'd flow right and not cause the misapprehension that made me thik of cutting it. 68.2.82.231 (talk) 17:54, 10 May 2014 (UTC)

Actually, now that I've edited it, I think the text doesn't really go with the picture. It sounds like it's describing a particle in a box, not a particle impinging on a single wall. I'll think more and maybe improve on it later. If there's an equally good animation of the particle in a box, this paragraph could be used with it in a new section (maybe collapsed a bit since it's somewhat redundant). 68.2.82.231 (talk) 18:01, 10 May 2014 (UTC)

"Derived from symmetry principles"??

I never noticed this untill now. Someone in the lead has made the claim:

"But Schrödinger's equation, although often presented as a postulate, can in fact be derived from symmetry principles."

with the citation to the book:

Ballentine, Leslie (1998), Quantum Mechanics: A Modern Development, World Scientific Publishing Co., ISBN 9810241054</ref>

So apparently the SE is not a postulate but can be "derived". Is this a misunderstanding of the editor who added the citation? Is the author of the book making a non-mainstream claim? Why would so many other sources treat the SE as a postulate while this random textbook says it can be derived?

I suggest removing it, could be WP:OR. M∧Ŝc²ħεИ_τlk 10:12, 14 September 2014 (UTC)

...or at least rewriting it, saying that some authors say it is a postulate while others consider it to be derivable, and for the latter case what the foundations are to "derive" it from symmetry principles (most likely it is justified by symmetry principles, but I could be wrong). I may be very picky - but this is an important point to make clear about the equation. M∧Ŝc²ħεИ_τlk 21:41, 15 September 2014 (UTC)

When one works backwards and takes the symmetries of an equation and adds the necessary constraints, you can derive almost anything. In this case the required symmetry is presumably Galilean relativity, and a constraint to a scalar field, conservation of energy and a perhaps a few others ... I don't think that it belongs in the lead. —Quondum 01:17, 16 September 2014 (UTC)

In the lead we need to state at the outset whether or not the equation is a postulate or a derivable equation. These are two different things. There have been a number of confused edits to the article (and likely other QM articles) which assert the SE can be derived. First I'll wait for more opinions before making changes. M∧Ŝc²ħεИ_τlk 20:46, 26 September 2014 (UTC)

Well, actually I made the changes anyway, along my own proposal since no one has objected, but when possible input from others would still be very helpful. M∧Ŝc²ħεИ_τlk 23:47, 26 September 2014 (UTC)

Fails to Specify Variables

In a lot of the equations it doesn't specify the meanings of all the variables. This isn't "quantum physics experts we already know all this" wiki. Every variable should be specified as to what it means.35.8.84.29 (talk) 19:52, 22 October 2013 (UTC)

Specifically the backwards 6s.35.8.84.29 (talk) 19:59, 22 October 2013 (UTC)

For almost every equation the variables are specified, and there are links to the related articles like De Broglie relations and partial differential equations. Nevertheless, I agree there are some notational gaps (such as partial derivatives - reverse "6"), and will try to fill them in. M∧Ŝc²ħεИ_τlk 15:27, 23 October 2013 (UTC)

Done. Please point out any specific equations which lack definitions of symbols. Thanks, M∧Ŝc²ħεИ_τlk 15:43, 23 October 2013 (UTC)

When it talks about time-reversal symmetry, shouldn't it be that psi(x, t) = psi*(x, -t) where it is the complex conjugate on the right? Gronteam (talk) 18:43, 14 November 2014 (UTC)

Schrödinger field theory vs Graviton

Schrödinger field theory handles particles as part of the same wavefunction, and time degrades distance in order the spin of the particles is maintained. Particles are treated as "sacks of noise spots" a discrete mathematics approach. No extra particles needed, but in order the Schrödinger Equation describes all particles as parts of the same wider and complex wavefunction, particles should be considered as "noise sacks". Discrete mathematics and combinatorics needed to express the "entropic degradation of distance due to time" and the overall urge of achieving a "mean inner time perception of the particles of a gravitational system" — Preceding unsigned comment added by 2.84.209.134 (talk) 07:17, 30 May 2015 (UTC)

"Simultaneous Positions"

Removed the metaphysical waffle of "...can be in several different locations at the same time...". That sort of description has no basis in standard quantum mechanics. Any experiment that can only be interpreted as a particle in two positions, would constitute an actual measurement of a particle having two eigenvalues at the same time. However, such a measurement would falsify a basic postulate of QM that requires all measurements to result in a single eigenvalue. The basic issue here is that QM is quantum, not classical. A classical argument might well result in such nonsense, but its not part of QM. — Preceding unsigned comment added by Kevin aylward (talk • contribs) 10:24, 30 August 2015 (UTC)

Again removed the line stating that states "...proves particle went through both slits at once." after my edit was reverted. It "proves" diddly squat. There are many interpretations of Quantum Mechanics that can "explain" such behavior without such an assumption, for example Bohmian Mechanics. Secondly, again, such an assumption contradicts QM, despite claims by otherwise qualified individuals to the contrary. The standard mathematical formulation of QM requires that the probability of a system exhibiting two simultaneous eigenvalues is zero. Period. There is no disagreement or controversy on this. Unfortunately, many "experts" appear to forget what a Hilbert space demands when they have their descriptive hat on rather than their mathematical one. Claims as to what values are before measurement are vacuous metaphysics and not part of, nor relevant to, the standard formalization of Quantum Mechanics. Kevin Aylward 15:14, 5 September 2015 (UTC) — Preceding unsigned comment added by Kevin aylward (talk • contribs)

"Only One Wave Equation"

Added a Ballentine sourced quote that illustrates the fundamental issue with claiming that particles are really waves. It should be noted that there is absolutely no disagreement in standard quantum mechanics that there is only one wave equation for a multi particle system. For example, the Debroglie wavelength of a molecule is a single number, irrespective of the fact that may be several sub particles. This wavelength may be either much larger or much smaller than the size of the molecule. Quantum mechanics simply don't work by imagining little wave packets for each particle. dah...Kevin Aylward 08:23, 6 September 2015 (UTC)

I'm not a physicist

Hey guys. I was curious about learning more about the cat in a box I heard and this page is really thick. You need a knowledge of quantum physics just to read the article. Any chance we can dumb it down for the lay person? — Preceding unsigned comment added by 216.106.18.70 (talk) 13:08, 12 August 2013 (UTC)

Maybe it needs something, somewhere, but this article is about physics. The "cat in a box business" was intended to make a serious point, but Schrödinger was a person who wrote ironically and sometimes sarcastically. His writings are frequently not to be taken exactly at face value. It may help to know that he disliked cats. It appears that his initial idea was to ridicule some of the early and emerging conclusions of the Copenhagen group by saying, in effect, "If you believe this bleep then you would have to accept the idea of a cat smooshed out in some state neither alive nor dead but in some sense both." Then, as the months and years rolled on, it became clear that electrons turned in both counterclockwise and clockwise rotation at the same time, and many other kinds of "superposition" could be demonstrated. Then, going back to the cat example, if you really did try to do the experiment, what would that mean? It turns out that the original quip left out a lot of detail that has been funneled together into the idea of decoherence. The original idea was that if nobody observes the geiger counter that is linked to a release mechcanism for the poison gas, then the geiger counter doesn't do anything, so the cat and everything in the box is suspended until a human being (delusions of grandeur here) opens the box and makes an observation. Who says that the cat cannot be aware of the geiger counter either clicking or not clicking? Who says that "observation" has to mean the report of stimuli received in human eyes or ears to brain tissue that processes the report somehow and concludes "the geiger counter has (not yet) clicked." Why won't a rolling movie camera in with the cat record the flickering needle of the geiger counter just as well? Why won't anything else that is done by being connected to the needle or the guts of the geiger counter serve just as well as an observation? And blah, blah, blah. You can't boil it down to a paragraph without making it into a dogmatic assertion that anybody with good sense should be suspicious of. The best you could do would be to say that Dr. Big explained/defined it that way ex cathedra. But it is, at heart, a big, complicated discussion. P0M (talk) 18:39, 12 August 2013 (UTC)

The "cat in a box" was supposed to illustrate the concept that two "impossible states" simultaneously occur (i.e.wavefunction) before one or the other does happen (as in wavefunction collapse).

Unfortunately, it seems one does need some exposure to the very basics of QM before the SE is introduced in most courses/textbooks I've seen, mainly in wave-particle duality which the article already does include, but given that it's a crucial aspect of QM (indeed a fundamental postulate) maybe we need to alter the background QM. For now, not sure how rewrite... M∧Ŝc²ħεИ_τlk 19:09, 12 August 2013 (UTC)

Note that there is an article on Schrödinger's cat. RockMagnetist (talk) 02:12, 13 August 2013 (UTC)

Regarding the above comment that 'it became clear that electrons could turn clockwise and anti-clockwise at the same time'. This is not true. Quantum mechanics expressly prohibits simultaneous values in its fundamental postulates. The common confusion on this point is that a quantum state represented by say |up> is one to one with the probability P(up) not with the actual value 'up'. Additional, the '+' sign is not an addition, it is a probability 'OR' operator. The confusion lies in that classical mechanics deals and writes expression/relations of the variables directly, such as position x and momentum p. Quantum mechanics only deals and writes expressions, effectively, dealing only with probabilities such as P(x) and P(p). One has to know what the symbols in say, |psi> = a|up> + b|dn> mean. It means a|up> OR b|dn>. For example, in Boolean logic 1 + 1 is still 1 not two. A state in QM is a probability distribution of a position, not an actual position. Kevin Aylward 17:01, 11 September 2015 (UTC) — Preceding unsigned comment added by Kevin aylward (talk • contribs)

r in the equations goes undefined

Noticed that r is not defined, I am making the assumption that r is the radius along the axis of wave propagation, but formally r should be defined. PB666 ^yap 12:22, 25 March 2016 (UTC)

Equation boxes

To me, the equation boxes in Schrödinger_equation#Equation seem redundant. I could see the value of an equation box to highlight the result if there is a derivation with several equations (as in Angular_momentum#Collection_of_particles), but here there is a subsection each for the time-dependent and time-independent versions, and both equations in each subsection are highlighted. Shall we remove the boxes? RockMagnetist(talk) 16:12, 12 April 2016 (UTC)

Boxes are not limited to the results of a derivation, they make the main equation(s)/definition(s) of a subject stand out, titles are for reference.

For this article, I don't mind if the boxes are kept, or deleted to reduce clutter and redundancy. If they are kept, it would be preferable to just use the minimalistic black/white theme, but people keep reinstating the blue/green colours.

As a related aside, I have used them a lot in Lagrangian mechanics and Lorentz transformations with a minimalistic black/white theme, because those are equation-heavy articles (coloured boxes have been added by other editors), the titles are for self-contained reference if someone wanted to name the equation in a box. M∧Ŝc²ħεИ_τlk 21:10, 12 April 2016 (UTC)

Yes, derivations are just the first example I thought of for equation-heavy articles or sections where the boxes might make sense. RockMagnetist(talk) 01:44, 13 April 2016 (UTC)

Multiverse

In Dublin in 1952 Erwin Schrödinger gave a lecture in which at on point he jocularly warned his audience that what he was about to say might "seem lunatic". It was that, when his equations seem to be describing several different histories, they are "not alternatives but all really happen simultaneously". This is the earliest known reference to the multiverse (David Deutsch, The Beginning of infinity, page 310). Kartasto (talk) 11:22, 16 April 2016 (UTC)

Consistency with QFT etc.

Apart from inserting a derivation of the equation (see above), I have also removed the incorrect statement that Schrödinger's equation is inconsistent with quantum field theory. Perhaps the editors who inserted this meant that it is impossible to write down a Hamiltonian for a single particle consistent with special relativity. However, quantum field theory is still based on Schrödinger's equation, except that the wave-function must be defined on the Fock space. Alternately, one may consider wave-functions on field configurations. In fact, even in extensions of quantum field theory like string theory, Schrödinger's equation is not modified. Jacob2718 (talk) 09:07, 4 December 2016 (UTC)