Talk:Schrödinger equation

Old discussion available at Talk:Schrödinger equation/From Talk:Schrodinger equation

Unheadered stuff at the top

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).

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)

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)