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

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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 <math>R^3<math>. 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)

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 over the Delta function as a function. In fact, the Incertainty 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

Contents

1 Why this was removed? 2 splitting the article 3 Content 4 content 5 time-independent SE

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

<math>|\psi(x,t)\rang = \sum_n e^{-iE_nt/\hbar} c_n(0) |n(x)\rang <math>

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

<math> c_n(0) = \left\langle n | \psi \right\rangle <math>

and receive the following expression:

<math>\psi(x,t)= \sum_n n(x) \left\langle n | \psi \right\rangle e^{-iE_nt/\hbar} <math>

The more canonical forms of this expression are

"State vector form"

<math>|\psi\rang = \sum_n |n\rang \left\langle n | \psi \right\rangle e^{-iE_nt/\hbar} <math>

"Measurement (projection) form" :

<math>\lang x|\psi\rang = \sum_n \lang x|n\rang \left\langle n | \psi \right\rangle e^{-iE_nt/\hbar} <math>

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

<math>\psi(x,t)= \sum_n n(x) \left\langle n | \psi \right\rangle e^{-iE_nt/\hbar} <math>,

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 <math>\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<math> and so on (excuse the errors in factors of <math>\hbar<math>, 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 <math>\hat{H}\psi=E\psi<math> 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)