Talk:Schrödinger equation

From Wikipedia, the free encyclopedia

To-do list for Schrödinger equation:

edit · history · watch · refresh

Here are some tasks you can do:

Expand the following sections:
- Historical background and development
- Free particle Schrödinger equation
- Relativistic generalisations
- Applications

Reduce the maths in the Mathematical forms section by possibly using show/hide boxes.

Write more on the Correspondence principle.

Include more citations and external links.

This article is within the scope of multiple WikiProjects.
Click [show] for further details

	Portal This article is within the scope of WikiProject Physics, which collaborates on articles related to physics.
GA	This article has been rated as GA-Class on the assessment scale.
High	This article is on a subject of high importance within physics.
Help with this template Comments: edit – history – watch – refresh Lead, citation and reference could be improved. Geometry guy 21:33, 17 June 2007 (UTC)

Mathematics Portal

This article is within the scope of WikiProject Mathematics, which collaborates on articles related to mathematics.

Mathematics rating:

B Class

High Priority

Field: Mathematical physics

One of the 500 most frequently viewed mathematics articles.

Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments.
Please add to or update the comments to suggest improvements to the article.
Lead, citation and reference could be improved. Geometry guy 21:33, 17 June 2007 (UTC)

WikiProject Echo has identified Schrödinger equation as a foreign language featured article. You may be able to improve this article with information from the French language Wikipedia.

Schrödinger equation was one of the Natural sciences good articles, but it has been removed from the list. There are suggestions below for improving the article to meet the good article criteria. Once these issues have been addressed, the article can be renominated. Editors may also seek a reassessment of the decision if they believe there was a mistake.

	Article milestones

Date

Process

Result

September 22, 2006

Good article nominee

Listed

February 15, 2008

Good article reassessment

Delisted

Current Status: Delisted good article

Schrödinger equation is included in the 2007 Wikipedia for Schools, or is a candidate for inclusion in future versions. Please maintain high quality standards, and make an extra effort to include free images, because non-free images cannot be used on the CDs.

Archives
*Archive 1 (up to Dec 2006)

[edit] mistake =

shouldnt the formula where kinetic energy + potential have hbar squared instead of hbar? —Preceding unsigned comment added by 62.136.133.147 (talk) 13:57, 17 May 2008 (UTC)

[edit] 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)

[edit] 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)

[edit] 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

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

he looked at a monochromatic wave $exp(i (k x - ω t))$ , 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).

[edit] 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).

[edit] 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)

[edit] Problem with "Historical background and development" section

From artical

and similarly since:
1:
then 
2:
and hence
3:

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

$p_x^2 \psi = p_x p_x \psi = p_x \cdot -i\hbar\frac{\partial}{\partial x} \psi = \hbar k_x \cdot -i\hbar\frac{\partial}{\partial x} \psi = \hbar \frac{\frac{\partial}{\partial x} \psi}{i \psi} \cdot -i\hbar\frac{\partial}{\partial x} \psi = \frac{-\hbar^2\left (\frac{\partial}{\partial x} \psi\right )^2}{\psi} \overset{\underset{\mathrm{?}}{}}{\ne} -\hbar^2\frac{\partial^2}{\partial x^2} \psi$

$\because k_x=\frac{\frac{\partial}{\partial x} \psi}{i \psi} \and \frac{\partial}{\partial x} \psi\cdot\frac{\partial}{\partial x} \psi = \left (\frac{\partial}{\partial x} \psi\right )^2 \overset{\underset{\mathrm{?}}{}}{\ne} \frac{\partial^2}{\partial x^2} \psi$

So how does this get accounted for? (the problem points are denoted $\overset{\underset{\mathrm{?}}{}}{\ne}$ )--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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

"Modern psuedo-history" is very important in science pedagogy. Yes, it should not be misrepresented as actual-history, but it hasn't been here, so no worries. A "modern derivation" is not possible because, strictly, it is impossible to theoretically "derive" any new theory. Actual-history is also not very useful for a variety of reasons: different language/notation; correct conclusions chanced upon despite incorrect reasoning; different context of thought. Cesiumfrog (talk) —Preceding comment was added at 03:24, 31 May 2008 (UTC)

[edit] 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)

My problem with the longer "derivation" is that much of the material is irrelevant to the derivation. We don't have to know about Hamiltonian's equations or conservation of momentum or the "short-wave limit" in order to derive Schrodindger's equation from the classical E = KE + PE. What are described are attributes or properties of the SE which should be described else where in the article.--Michael C. Price ^talk 16:50, 18 May 2008 (UTC)

As I commented below, you are absolutely right on this if the goal is only to justify the mathematical form of the equation, and you might even be right from a pedagogical point of view. But I wanted to make it clear, from a physical point of view, that when Schrodinger sat down to find his equation, he wasn't faced with a problem that had too few constraints, it was the exact opposite. Students, I think, have the impression that it's like a guessing game, where there's a lot of freedom to choose. But the requirement that Hamilton's equations come out right for wavepackets is so strong that there is more information there than what is strictly necessary to get at the form of the equation, and the rest of the stuff is a consistency check which allows you to be sure that it is correct as physics. If Hamilton's equations weren't just so, there would have been no wave equation. Of course, the modern point of view on this is the other way around--- that classical mechanics has a Hamiltonian description ultimately because quantum mechanics is there underneath. Anyway, I hope you feel the article is improved. Cheers.Likebox (talk) 21:13, 18 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)

I undid most of the reworking, so that it goes more like it did before. If you still hate it, by all means restore the original. But I found a way to make a nice quick way to argue the second of Hamilton's equations.Likebox (talk) 20:25, 17 May 2008 (UTC)

Ok, maybe it's no good, but give it a chance before you revert, because the earlier discussion was both a little verbose and misrepresented Schrodinger's contribution.Likebox (talk) 20:48, 17 May 2008 (UTC)

I'm not going to revert your additions, but I still think there is a place for the original heuristic derivation in the article that someone without a physics degree will have a chance of following. BTW the claim that Schroedinger accepted the CI is incorrect.--Michael C. Price ^talk 11:40, 18 May 2008 (UTC)

(deindent) Maybe you're right, I am not so great with psychology. But I think that the previous argument was essentially identical, except it was done with more detail to make the algebraic steps more explicit. The issue I had with that (and I might be psychologically totally off base here) is that I think it makes the reader lose the forest for the trees. It had many equations whose only purpose was to restate the equivalence of E and \omega and p and k. By doing this, a larger fraction of the "steps" are easier to follow, but only because the number of steps is made larger by including more algebraic restatements of p=hk. So, it would say "p=hk" and later "k=i d/dx \psi" and later "p=i\hbar d/dx". This sort of thing bothered me when I was a student (At least I think I remember that it did) because it would obscure the essentially new insights--- in this case the substitution of derivative operators in the energy equation--- by hiding them among a list of algebraic manipulations which should be internalized by the reader first, before doing anything else. But perhaps its the exact opposite. Maybe if there are more familiar identities, there is a feeling of familiarity which aids the process of understanding.

The group-velocity stuff might be an unnecessary hurdle for a first read, and it might need to be removed. You might also be right that a careful restatement of p=hk multiple times will reinforce understanding, because different people might "click" on different restatements. I have no confidence that the current version is any good at all. The previous presentation might be significantly better. Probably this is best figured out by asking some students who have just finished learning quantum mechanics.

About Schrodinger and Copenhagen--- I read conflicting stuff on this. Everybody agrees that at first, he hated the Copenhagen interpretation, and thought \psi was a physical charge density or something, but after Born's paper he (and Einstein and everyone else) came to agree that, given the formalism, a statistical interpretation is the only sensible one. The question that remains is whether he held out hope that this was a statistical approximation to an underlying determinisic DeBroglie-Bohm theory, or if he had some Wigneresque proto-Everettist ideas. This I don't know for sure. In the new Einstein biography by Isaacson, the author states that Schrodinger was sympathetic to Einstein's view at least up to EPR, but in the 1940s goes on a rant to someone or other about how Einstein was a stupid old man when it came to quantum mechanics. That's why I assumed he had come to accept that the Copenhagen interpretation is correct. If you have better information, please fix it. I don't have any insight into Schrodinger's thinking.Likebox (talk) 20:14, 18 May 2008 (UTC)

I noticed you put back the old discussion--- that's a good solution. Let the reader have a choice. I would put the reference to Isaacson for Schrodinger, but it's not really conclusive at all.Likebox (talk) 20:52, 18 May 2008 (UTC)

"Schrödinger tried unsuccessfully to interpret it as a charge density." This is a very interesting claim! In the history of relativistic QM, it is well known that the Klein-Gordon equation was originally discarded because its wavefunction failed to fit the probability interpretation. BUT! In modern QFT, the Klein-Gordon equation is actually correct -- the wavefunction just has to be interpreted as charge density rather than probability. So if Schroedinger really started out interpreting the wavefunction as charge density, and considering that he also preceded Klein and Gordon by trying that equation before trying the one he is now known for, surely there is more interesting history worth telling here (or at least, the Klein-Gordon page needs a better history section). Cesiumfrog (talk) 05:01, 31 May 2008 (UTC)

[edit] Full Copenhagen vs. Statistical interpretation

There is a difference between believing that the Copenhagen interpretation is incorrect or incomplete and believing that the statistical interpretation of the wavefunction is incorrect. While I don't know for sure whether Schrodinger believed that the Copenhagen interpretation was a final statement on the nature of physical reality, I am sure that, like Einstein and everybody else, he accepted Born's analysis as correct and realized that the wavefunction could only be interpreted as a probability. This was a universal conclusion, everyone agreed with Born's interpretation. For example, Einstein is directly quoted in Isaacson as saying that the only possible interpretation of the wavefunction is statistical, that this is an unavoidable conclusion. This was (and still is) such a clear fact that I can't imagine Schrodinger rejected it. He certainly understood, although I'm sure with distaste, that the probability is proportional to the wavefunction squared (although perhaps with a philosophy radically different than those that were codified by Bohr). His often quoted dislike of quantum discontinuities should be properly read as a sign that he initially wanted to avoid all quantum jumps with a continuous wave formalism, but failed. I don't have any insight into whether he thought that quantum mechanics was a statistical approximation to a deeper underlying theory.Likebox (talk) 00:17, 20 May 2008 (UTC)

You may be sure that Schrodinger "accepted Born's analysis as correct and realized that the wavefunction could only be interpreted as a probability." but I am not; in fact I'm sure he didn't. Can you find a statement by S to that effect? --Michael C. Price ^talk 03:22, 20 May 2008 (UTC)

Honestly, if you are not I am not so sure anymore--- since I am pretty sure that you've probably read as much about it as I have. But please verify it, because I am sure that Einstein accepted Born's interpretation, and Schrodinger and Einstein were pretty much on the same page. The reason I am sure about Einstein is Isaacson's biography.Likebox (talk) 07:51, 20 May 2008 (UTC)

I do see the distinction you're making between Born's probability amplitude and the CI (although note that Born was a committed CIist)). And I know Einstein accepted that Born's probability amplitude was empirically correct to mid 20th C levels of measurement. However I'm not so sure that Schrodinger went along with this (although, as you say, Einstein and Schrodinger normally agreed on such matters). I've got Moore's bio of S so I'll double-check.--Michael C. Price ^talk 08:42, 20 May 2008 (UTC)

Thanks for clearing that up.Likebox (talk) 21:45, 22 May 2008 (UTC)