Can schrodinger equation be proved

[dextercioby]

The laughing smilies are after your (user)name...

Daniel.

P.S.I don't find "marlon" to be that funny...However,"de gustibus..."

 

[marlon]

Prof Dextercioby


regards
marlon

 

[masudr]

Guys, guys, chill out. I can't tell if you guys are being antagonistic for the hell of it or because you really are getting annoyed. Anyway, it's been a while since I posted anything of notable worth on this forum, so here we are.

The "proper" Schrodinger equation describes the infinitesimal time-evolution of the quantum state as being proportional (with a complex proportionality factor) the Hamiltonian operator acting on the state.

I would say this is more general than specifying Laplacians etc. as the Hamiltonian varies for systems in general. Symbolically,

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

(I know this is the same one as what Dextercioby put before).

Now, as far as I know, this equation has no "proof" (it'd be akin to proving Newton's second law); if it did have one, I would like to see it. The equation has strong empirical support (obviously). One may say, of course, that physical equations can be "proved" but what they really mean is that it can be described/explained as a facet of some other more correct/coherent (with other physical theories)/fundamental description of the universe, which itself couldn't be proved without recourse to some other more "fundamental" description.

Masud.

 

[hellfire]

My impression (which may be wrong, of course) is that the principles dextercioby refers to (which are described in the first pages of the second chapter of Sakurai’s Modern QM), can be considered to be more fundamental than the equation itself. One can derive similar equations based on that principles (e.g. the ‘Schrödinger equation’ in momentum space, or any equation based on an unitary operator). To me it seams that this makes this a bit different to Newton’s second law.

 

[Kane O'Donnell]

Actually there are quite a few difference sets of postulates one can use to get equivalent formulations of QM, but the TDSE is often postulated as the equation that describes the time evolution of the wavefunction.

I would really like to say at this point that the whole argument is essentially a moot point - depending on the strength of the *other* postulates of QM, the time-evolution axiom may or may not be independent of the others, but in physics we don't worry so much about the independence of postulates as in mathematics.

The point is that for undergraduates, the TDSE is often a postulate, but at the same time a rough outline of why it is reasonable is usually given, either from the expected form of a free-particle plane wave solution, or by quantising the classical Hamiltonian using Dirac's commutator postulate and then using Stone's theorem to link the infinitessimal generator H to the time-evolution operator, which must be unitary because we expect conservation of probabilities.

None of these are 'proofs', but the latter especially is a very elegant and powerful way of showing how the dynamics of QM arise.

Cheerio,

Kane

 

[marlon]

I can only agree with masudr and Kane. I also agree with the content of Dextercioby's posts and i am not arguing the fact that in the SE can be seen as a postulate. However, i find it difficult to believe that you just say well the SE is a postulate and that is it. This claim gives the impression to me that the SE "just fell out of the sky"

I am just asking for a justification for the actual equation of this formula and this is independent of what equivalent representation you write down. I gave a possible way out in my very first post and i would like to ask you all how YOU would justify the SE.

I have read somewhere the argument that you cannot prove the second equation of Newtion F=ma. I disagree because this was not just postulated by Newton. He did experiments and then he realized that this connection between mass and acceleration and force existed empirically. So, whether the justification is based upon theory or experiment is equally good in my opinion. But, in the end i do remain convinced of the fact that there is more to say on the SE then just : "it is a postulate"

i hope you all see my point.

regards
marlon

 

[gptejms]

I haven't read all the posts here,but answering the first post,Schrodinger equation can be justified very well if not derived.After all it didn't drop from the sky.Try a solution like cos(kx-wt),try satisfying E=p^2/2m--you can't.Try exp(i(kx-wt)--you can,you know the eqn.

Besides,any eqn. like \del^2 \psi/\del x^2 = (k^2/w^2) \del^2 \psi/\del t^2 is not a good candidate because it involves k,w in the equation--so does not admit superposing plane waves of different k(i.e. a wavepacket which De Broglie showed mimicked a particle).

 

[marlon]

I haven't read all the posts here,but answering the first post,Schrodinger equation can be justified very well if not derived.After all it didn't drop from the sky.Try a solution like cos(kx-wt),try satisfying E=p^2/2m--you can't.Try exp(i(kx-wt)--you can,you know the eqn.

Besides,any eqn. like \del^2 \psi/\del x^2 = (k^2/w^2) \del^2 \psi/\del t^2 is not a good candidate because it involves k,w in the equation--so does not admit superposing plane waves of different k(i.e. a wavepacket which De Broglie showed mimicked a particle).

All this was the content of my very first post in this thread. This is also how i see the SE-"justification"

regards
marlon

 

[BlackBaron]

I have read somewhere the argument that you cannot prove the second equation of Newtion F=ma. I disagree because this was not just postulated by Newton. He did experiments and then he realized that this connection between mass and acceleration and force existed empirically.

I somehow disagree, but I think our disagreement is just a matter of vocabulary. To me, you "demostrate" something from a more fundamental set of axioms, if you can't you "postulate" it and then you "confirm" it's validity with experiments. In other words, "demostrations", to me, are purely theoretical, so experiments don't "demostrate", the "confirm".
With that in mind, to me, Newton's law is just a perfect example of a "postulate" (except it stopped being a postulate the moment it could be derived from a more fundamental set of axioms, but I discussed that already).
Anyway, that's just my opinion, please, don't start an argument if you only disagree with my definiton of "demostration".


.

I haven't read all the posts here,but answering the first post,Schrodinger equation can be justified very well if not derived.After all it didn't drop from the sky.Try a solution like cos(kx-wt),try satisfying E=p^2/2m--you can't.Try exp(i(kx-wt)--you can,you know the eqn.

Then you have to postulate de Broglie relations

 

[marlon]

Anyway, that's just my opinion, please, don't start an argument if you only disagree with my definiton of "demostration".


.

I most certainly disagree with your statement, but i will respect your wishes...for once ...

regards
marlon

 

[gptejms]

Then you have to postulate de Broglie relations

Of course you have to-----so?