Talk:Schrödinger equation

mistake =

shouldnt the formula where kinetic energy + potential have hbar squared instead of hbar?

SE in spherical symmetric potential

Dan Gluck added a subsection to this article making a few spelling mistakes, which Pfalstad corrected. However, there are also a few minor flaws in the content, e.g., particle mass μ of the electron. No electron was mentioned earlier and mass is indicated by M. Before I (or somebody else) correct(s) these minor things, I like to know whether we all agree that this article is the place for Dan's addition. In the list of analytic solutions we find two articles that treat the spherical symmetric problem. If Dan is not satisfied with these two articles, maybe he should improve those. To me Dan's addition seems fairly arbitrary, he could have added to the present article any solution from the list of analytic solutions. Any comments?--P.wormer 15:24, 22 May 2007 (UTC)

I removed the section, and merged it with particle in a spherically symmetric potential since the additions seem to fit better there. --HappyCamper 16:13, 25 May 2007 (UTC)

Sorry for my speling :-) mistakes, but that's the price English speakers have to pay for their language being the international one. Anyway the merging seems fine to me. Dan Gluck 14:29, 12 June 2007 (UTC)

Solutions of SE

Has it been proved that analytical solutions to the time-independent schrodinger equations of molecular systems are impossible, or is it just the case that solutions have not been found. 212.140.167.98 13:29, 31 July 2007 (UTC)

GA Article?

As a professor in physics from Denmark, I am very disappointed, I am concerned because the English written articles have a tendency to penetrate into all other languages. This article has so many errors/misunderstanding that a GA-status will harm the Wikipedia. Quantum mechanics do not need to be presented as something very complicated, and the use of the Dirac's notation is throughout the article completely wrong. Sincerely j.h.povlsen 80.163.26.74 00:36, 13 August 2007 (UTC)

• I reacted here on this comment.--P.wormer 12:32, 13 August 2007 (UTC)

And I emphasize again, that this article does not present the work of Schroedinger, The same article could as well have been about the equations of Heisenberg. The article starts in it's very first equation with a misinterpretation of the Dirac notation, and then it goes on by describing a Complex Hilbert Space. Let me you remind, that Schroedinger was not aware of any of the above mentioned complexities about the world. He was a physicist, and his equation did not just jump out from nothing! He thought, without a thought on Hilbert space, but on the "quantum mechanics" as he knew it at that time. And the quantum mechanics was the discoveries from Planck, Bohr, Luis de Broglie and Einstein. The Planck/Einstein discovery was, that the energy quantization of light/(Electro-magnetic waves) could be expressed as

$E=\hbar \omega$

while Luis de Broglie discovered a relation between momentum and wavelength

$p=\hbar k$, where k is the wavenumber, and p the momentum.

In connection with that the energy, according to Newton, consists of a kinetic part and potential part as in

1. $E={\frac {p^{2}}{2m}}+V$

he looked at a monochromatic wave $\exp(i(kx-\omega t))$, and realized that the energy could be evaluated as an eigenvalue to

$E<->i\hbar {\frac {\partial }{\partial t}}$

and a momentum component $p_{x}$, similarly could be derived as an eigenvalue to

$p_{x}<->-i\hbar {\frac {\partial }{\partial x}}$

And by inserting this into the Newton energy rule he reached his named equation:

$i\hbar {\frac {\partial }{\partial t}}\psi =[-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V]\psi$

which (in Wikipedia) sadly, seems to become an untold story. I hope some one, some day will tell it. Sincirely j.h.povlsen 80.163.26.74 04:40, 15 August 2007 (UTC)

Thank you, j.h.povlsen, you are correct. The article is unnecessarily opaque. Schroedinger's original derivation couldn't have used Dirac's notation (obviously!); instead it followed directly on from de Broglie's work the year before.

Perhaps the article should have a section entitled "Schroedinger's derivation" or something. --Michael C. Price ^talk 06:20, 15 August 2007 (UTC)

I have added a new "History and development" section. Perhaps some of the details of the next section could be reduced. --Michael C. Price ^talk 12:51, 15 August 2007 (UTC)

Thank you Michael for taking me serious! I know that I some times can seem arrogant!

I still dislike the section Mathematical Formulation, and think it should be omitted.A new section Physical interpretation of the wave function might be an idea? And I also find that the section The Time independent Scrodinger equation has to many (trivial) details, mixed up with a far to "complicated mathematical terminology" and suggest that the section should be reduced and divided into subsections. I would prefer a much stronger focus on the physical implications, without any use the Dirach notation. For instance a discussion on the discrete spectrum and the continuous spectrum. Below I have rewritten The Time independent Scrodinger equation (which now is far to short!!) and also suggested a new section, on the relation to the classical mechanics (here we could also include the Virial thorem).

The Time independent Scrodinger equation

We can find stationary solutions to the Schrodinger by looking at solutions separable in time and space as:$\psi \left(x,y,z,t\right)=\phi \left(x,y,z\right)g\left(t\right)$ which inserted into the time dependent Schoedinger equation reveal solutions on the form $\psi \left(x,y,z,t\right)=\phi \left(x,y,z\right)\exp \left(-i{\frac {E}{\hbar }}t\right)$ with the time independent part being an eigenfunction to:

$\left[-{\frac {\bigtriangledown ^{2}}{2m}}+V\left(x\right)\right]\phi \left(x,y,z\right)=E\phi \left(x,y,z\right)$

where $E$ is interpreted as the energy. This equation can be analytically solved for a number of very physical important cases such as the Coulomb potential (orbitals in the hydrogen atom), and the harmonic oscillator (lowest order approximation of arbitrary potential functions $V\left(x,y,z\right)$ around a minimum).

Connection with classical mechanics

The quantum mechanics need in a proper formulation to include the classical mechanics in it's macroscopic limit, and the Shroedinger equation does indeed that, as realized by Ehrenfest. Ehrenfest showed from the time dependent Schroedinger equation that the the expectation value $\left\langle A\right\rangle$, defined as

$\left\langle A\right\rangle \equiv {\frac {\int \int \int \left(\psi ^{*}A\psi \right)dxdydz}{\int \int \int \left(\psi ^{*}\psi \right)dxdydz}}$

of a pysical operator $A$ (i.e. a Hermetian operator) evolves in time as

${\frac {\partial }{\partial t}}\left\langle A\right\rangle ={\frac {i}{\hbar }}\left\langle \left[A;H\right]\right\rangle$

where $H$, the Hamiltonian is $\left[-{\frac {\bigtriangledown ^{2}}{2m}}+V\left(x\right)\right]$ and $\left[A;H\right]$ denotes the commutator defined as

$\left[A;H\right]=AH-HA$

and by considering the posiotion operator ${\overrightarrow {r}}=\left(x,y,z\right)$ and the momentum operator ${\overrightarrow {p}}=\left(p_{x},p_{y},p_{z}\right)$ he derived the correspondance equations

${\frac {d}{dt}}\left\langle {\overrightarrow {r}}\right\rangle ={\frac {\left\langle {\overrightarrow {p}}\right\rangle }{m}}$

${\frac {d}{dt}}\left\langle {\overrightarrow {p}}\right\rangle =-\left\langle {\overrightarrow {\bigtriangledown }}V\right\rangle$

In agreement with Newtons second law.

Sincirely 80.163.26.74 00:01, 17 August 2007 (UTC)j.h.povlsen

Thanks --I'm glad you liked it (I thought you would).

I agree we could probably lose the entire "Mathematical Formulation" section (and trim the other sections), but I'd rather not delete it immediately -- let's wait for a consensus to develop one way or the other.

I don't see the need for a classical limit section here, since it is not part of Schroedinger's work (certainly not his equation), but more Ehrenfest's, which why it is explained at Ehrenfest theorem and classical limit.--Michael C. Price ^talk 03:06, 17 August 2007 (UTC)

Problem with "Historical background and development" section

From artical

and similarly since:
1:${\frac {\partial }{\partial x}}\psi =ik_{x}\psi$
then 
2:$p_{x}\psi =\hbar k_{x}\psi =-i\hbar {\frac {\partial }{\partial x}}\psi$
and hence
3:$p_{x}^{2}\psi =-\hbar ^{2}{\frac {\partial ^{2}}{\partial x^{2}}}\psi$

I agree with formula 1 and 2 as currently derived however I derive a different answer for formula 3:

$p_{x}^{2}\psi =p_{x}p_{x}\psi =p_{x}\cdot -i\hbar {\frac {\partial }{\partial x}}\psi =\hbar k_{x}\cdot -i\hbar {\frac {\partial }{\partial x}}\psi =\hbar {\frac {{\frac {\partial }{\partial x}}\psi }{i\psi }}\cdot -i\hbar {\frac {\partial }{\partial x}}\psi ={\frac {-\hbar ^{2}\left({\frac {\partial }{\partial x}}\psi \right)^{2}}{\psi }}{\overset {\underset {\mathrm {?} }{}}{\neq }}-\hbar ^{2}{\frac {\partial ^{2}}{\partial x^{2}}}\psi$

$\because k_{x}={\frac {{\frac {\partial }{\partial x}}\psi }{i\psi }}\land {\frac {\partial }{\partial x}}\psi \cdot {\frac {\partial }{\partial x}}\psi =\left({\frac {\partial }{\partial x}}\psi \right)^{2}{\overset {\underset {\mathrm {?} }{}}{\neq }}{\frac {\partial ^{2}}{\partial x^{2}}}\psi$

So how does this get accounted for? (the problem points are denoted ${\overset {\underset {\mathrm {?} }{}}{\neq }}$)--ANONYMOUS COWARD0xC0DE 23:51, 4 September 2007 (UTC)

The formula are valid for the plane wave solution. More complex solutions are built up by superposition / fourier analysis. --Michael C. Price ^talk 06:32, 5 September 2007 (UTC)

No information regarding the problem was conveyed to me in those two sentences. Please respond to my problem in particular. --ANONYMOUS COWARD0xC0DE 22:19, 5 September 2007 (UTC)

In the plane wave example $p_{x}=\hbar k_{x}$ is not a function of x.

Hence

$p_{x}^{2}\psi =p_{x}(\hbar k_{x}\psi )=\hbar k_{x}(p_{x}\psi )=(\hbar k_{x})^{2}\psi =(-i\hbar {\frac {\partial }{\partial x}})^{2}\psi =-\hbar ^{2}{\frac {\partial ^{2}}{\partial x^{2}}}\psi$

and hence:

$p_{x}^{2}\psi =-\hbar ^{2}{\frac {\partial ^{2}}{\partial x^{2}}}\psi$

--Michael C. Price ^talk 09:18, 7 November 2007 (UTC)

Unheadered stuff at the top

Someone should go back through this article and change the sloppy notation for the wavefunction Psi. Psi is a function of x and t, while psi is one a function of x. This can wildly confuse a physicist or student looking for mathemetical expressions described by the Schrodinger Equation. Perhaps putting the wavefunction in the form of its variables Psi(x,t) and psi(x) can alleviate this confusion. —Preceding unsigned comment added by GaiaMind (talk • contribs) 05:29, 28 October 2007 (UTC)

Removed "one dimensional"

Referring to the state vector as "one dimensional" is misleading; it is typically infinite dimensional Peter1c 07:29, 7 November 2007 (UTC)

Delisted from GA

In order to uphold the quality of Wikipedia:Good articles, all articles listed as Good articles are being reviewed against the GA criteria as part of the GA project quality task force. While all the hard work that has gone into this article is appreciated, unfortunately, as of February 15, 2008, this article fails to satisfy the criteria, as detailed below. For that reason, the article has been delisted from WP:GA. However, if improvements are made bringing the article up to standards, the article may be nominated at WP:GAN. If you feel this decision has been made in error, you may seek remediation at WP:GAR.

I've had to delist this article from GA status as part of the good article quality control sweeps. It lacks inline references which became a good article requirement in 2006, possibly after this article was passed. I've listed this article at our unreferenced good article task force. Once adequate inline sources have been added, hopefully the article can easily reattain GA status. --jwanders^Talk 12:26, 15 February 2008 (UTC)

My sandbox version

I am in the process of cleaning up the article via my sandbox version. Feel free to comment on it. Thanks. MP ^{(talk•contribs)} 12:14, 17 February 2008 (UTC)

Rewrite

Upon request, I've just rewritten the article with my sandbox version. A lot more work still needs to be done though. MP ^{(talk•contribs)} 17:09, 8 March 2008 (UTC)

Great. Hopefully many hands will make light work. --Michael C. Price ^talk 17:51, 8 March 2008 (UTC)

Removal of huge chunk of text

I decided to remove the subsections Schrodinger wave equation and wave function as I think there is no point in repeating what's already been said in the article and a lot of the stuff will not help to actually understand the Schrodinger equation per se. Hope this is ok; comments/criticisms welcome. Thanks. MP ^{(talk•contribs)} 06:53, 14 March 2008 (UTC)

OK with me. --Michael C. Price ^talk 09:39, 14 March 2008 (UTC)

Notation

I've always seen the time-dependent wavefunction written with a capital psi and the time-independent function with a lowercase one, e.g. $\Psi (\mathbf {x} ,t)=\psi (\mathbf {x} )e^{-iEt/\hbar }$. It took me a moment to work out what the article was talking about as a result. Is there a particular notation used by most physicists, or was my physics textbook just using an unconventional convention (so to speak)? — Xaonon (Talk) 19:37, 27 March 2008 (UTC)

This article

Is it really necessary to add 'citation needed' 3-4 times per sentence? It's unreadable. Put a sentence in the beginning to give a conceptual (useless) description, then give the math for people who want to know. To paraphrase Lord Kelvin, if you cannot quantify it, you don't know what you're talking about. —Preceding unsigned comment added by 70.249.215.163 (talk) 22:58, 27 March 2008 (UTC)

too complex

this article is too complex to be comprehended by general public. this article presume the reader to have an advanced understading and knowledge in the subject. a less mathematical approach, a more conceptual approach is required for people who have limited knowledge in physics and mathematics. I suggests to remove the more complex parts from this page into a new article. So that it would be possible for general reader to understand this subject. —The preceding unsigned comment was added by 202.152.240.246 (talk) 15:22, 2 May 2007 (UTC).

• Why should every Wikipedia article be for the general reader (and who is the "general reader": College graduate? High school graduate with science or without science? High school dropout?) This article starts with a reference to the article Introduction to quantum mechanics#Schrödinger wave equation that is meant to be as easy as is possible for an abstract subject as quantum mechanics. Read this if you really want to know something about the Schrödinger equation and you lack the mathematical background. It is useless to keep on starting new articles because somebody out there takes him/herself as the absolute standard of comprehension. Personally, I skip articles about Kantian philosophy and such things, but if philosophers add advanced articles about it, I applaud it. It will make Wikipedia the better for it. Nobody forces me to read these articles, but they are there if I ever develop an interest in Immanuel Kant. The computer disks are patient, specialized articles are in nobody's way and hurt nobody. If you don't understand them, ignore them.--P.wormer 16:03, 2 May 2007 (UTC)

• • You make a good point that leads me to illustrate a general rule you seem to suggest. Let me repeat your question - who is the "general reader"? Indeed, the only thing we may presuppose about a reader of this article is that they are interested in the subject matter of the page in question. They either searched directly for the article, or followed a link from a related stub. Who knows what level of scientific or mathematical understanding they possess? In fact, we cannot really be assured that they have a basic level of English literacy. Some concepts are unable to be presented in a completely lowest-common-denominator fashion. Since we cannot make any assumptions about the reader, we ought to engage the subject on _the_terms_of_the_subject_, in absence of a clearly defined target audience. There exists a direct link at the top of the article to an introductory article outlining some of the basic concepts of QM, and that really ought to be enough. It makes no sense to attempt to simplify the subject, especially when much of it is in the realm of multivariable function math, with a variety of very specific structures which are already as condensed and fundamental as they can be expressed. Any attempt to simplify this page will only lead to confusion. In fact, I find the page to be lacking in specific definition of the particular constraints wherein S.E. is discussed. I would like to see a more technically apt page, that begins by discussing S.E. at a more general level, rather than the specific constraints applied to this page (which appears to be aimed at the particular problem of solving S.E. under assumed constraints on the _actual_ original works). Not that I pretend to be an expert on the matter, QM is considerably difficult both in concept and mathematical construction. So, although I don't believe in _simplifying_ the article, I do support the expansion of the article to give a clearer definition that is both More General in context, and which provides more detailed explanations of some of the mathematical constructs used. I admit my own contributions need reworking by an expert in the field. Wernhervonbraun (talk) 11:57, 24 April 2008 (UTC)

Microscopic particles

I dissagree with the wording that an electron is a microscopic paricle, its smaller than that! Noosentaal (talk) 17:53, 7 April 2008 (UTC)

Historical Accuracy

I think that one should not present pseudo-history as history. It is possible to give modern mathematical derivations, but in a historical discussion I think it is good to stick to the actual events, if not the exact formulas.Likebox (talk) 02:44, 15 May 2008 (UTC)

I don't know what you're referring to, but the previous heuristic derivation is now as clear as mud. --Michael C. Price ^talk 19:03, 16 May 2008 (UTC)

Pedagogical Note

Too much detail is just as bad as too little. The person who is reading this page should be able to reproduce elementary algebraic manipulations.Likebox (talk) 21:58, 15 May 2008 (UTC)

I don't know what you're referring to, but the previous heuristic derivation is now as clear as mud. --Michael C. Price ^talk 19:03, 16 May 2008 (UTC)

Is it? I'll try to fix it. Maybe I don't have the knack.Likebox (talk) 20:13, 16 May 2008 (UTC)

The reason that I edited the heuristic derivation is because I remembered some of my confusions when I was learning this: The original derivation substituted operators for the energy and momentum for plane waves where it is obvious, and then made it seem clear that the potential will just add to the kinetic energy. This is best justified by the short-wavelength limit, where wavepackets have sharp trajectories with the k changing from place to place as the wavepacket moves while the frequency stays fixed. It takes a little thinking to see that this produces wavepacket trajectories which have the right acceleration.

What does not take much effort is to see that wavepackets have the classical velocity, or that the energy is conserved. In one dimension, if you have a wavepacket moving with the classical velocity and and you know that the total energy is conserved, it follows that the acceleration is the classically correct one, since the change in momentum is determined by the change in the potential.

but in two dimensions or three, only the change in the magnitude of the momentum is guaranteed to come out right from energy conservation. To see that the wavepackets change direction correctly according to the classical force requires thinking about the way in which wavefronts shift about when the wavenumber is slowly varying.Likebox (talk) 21:51, 16 May 2008 (UTC)

Do you have any objection the to heuristic derivation being restored alongside the current reworked version? --Michael C. Price ^talk 12:27, 17 May 2008 (UTC)