Schroedinger Equation and Hamiltonian

[jeremyfiennes]

Thanks. I did. But what I want to know is how Schroedinger derived it. Given that no-one at the time really knew what the equation represented till the probability interpretation came along. One derives an equation for something. What was that "something" in Schroedinger's mind?

 

[bhobba]

Thanks. I did. But what I want to know is how Schroedinger derived it. Given that no-one at the time really knew what the equation represented till the probability interpretation came along. One derives an equation for something. What was that "something" in Schroedinger's mind?

See section 8 - it is explained clearly how he did it and the mistake he made that just happened to be corrected by another mistake.

For example on page 26:
To derive the time-dependent equation [7] Schrodinger makes the ansatz: ψ = u(x)[exp(±iEt/h)], which, when used to eliminate E from the previously derived time-independent equation, yields (4.8) on choosing the minus sign in the complex exponential. The ansatz used (except that a sign ambiguity devoid of any physical significance is retained) is exactly the time dependence of the path amplitude phase (c.f. Eq. (6.18)) specified by Feynman’s postulate II, in the case of a time-independent Hamiltonian. It is then not surprising that Feynman was able to derive the Schrodinger equation from his postulates!

The paper also makes other observations on his derivation such as the mistake he made I mentioned previously.

Of course it makes use of math not really suitable for a beginner, and I should have mentioned how he did it can not be fully understood at the beginner level. But hopefully you will get a gist. If not, then sorry your question of HOW can't be answered at your level.

Thanks
Bill

 

[vanhees71]

Read:
https://arxiv.org/abs/1204.0653

Thanks
Bill

Of course Schrödinger didn't derive his equation from the path integral. You cannot derive it to begin with, because it's a basic postulate of QT. Schrödinger's heuristics was to take classical mechanics in the form of the Hamilton-Jacobi partial differential equation (unfortunately usually not taught in the physics course lectures although it's among the most elegant tools to solve even practical problems; Schwarzschild was a master in using it for astronomical and, in his last paper, old Bohr-Sommerfeld quantum mechanics) as the leading-order eikonal approximation of a wave equation describing de Broglie's matter waves. It's worthwhile to read the famous series of original papers, which are also available in English as a collection in book form.

Of course, the eikonal approximation is equivalent to the saddle-point approximation of the path integral. In this sense Bhobba is right in pointing to the above quoted paper.

 

[jeremyfiennes]

Thanks. With a doubt about the origin of the Schroedinger equation, I looked for an open thread on the subject and found this one. Not being a quantum physicist, my interest is not in the maths, which is beyond me. You say "To derive the time-dependent equation Schrodinger ...". My question is simply "To derive the time-dependent Schrodinger equation for what?" What, in his own mind, was Schroedinger doing when deriving his famous equation.

 

[vanhees71]

There was de Broglie's famous PhD thesis, where he formulated the idea of a wave description of particles. I'm not sure what he concretely had in mind his waves are describing, but I think he thought his waves describe particles in analogy of "wave-particle dualism" of electromagnetic waves (or light). One must not forget that in this time there was not yet a consistent theory describing atomic and subatomic phenomena.

Schrödinger went one step further and looked for a equation to describe de Broglies waves quantitatively. His idea was that an electron is in fact a wave phenomenon in the sense of a classical field theory like the electromagnetic field, described mathematically by Maxwell's equations, is a description of light. Then they used by analogy Einstein's "wave-particle dualism" idea concerning electromagnetic waves (photons) to particles like electrons, i.e., the electron was described by Schrödingers field and at the same time had particle properties, depending on how you observed them.

That's quite ununderstandable, and old quantum theory didn't work really well quantitatively. Also the idea of an electron as a "smeared" field like entity is indeed never observed. A single electron leaves a pointlike spot on a screen but doesn't appear as a continuous intensity picture. That's why finally Born came to the conclusion that de Broglies and Schrödinger's waves are "probability amplitudes", i.e., its modulus squared gives the probability distribution to find the electron at a place.

 

[jeremyfiennes]

That's a very nice clear answer on an unclear topic, and has answered my query. Thanks.

 

[bhobba]

In this sense Bhobba is right in pointing to the above quoted paper.

I pointed it to him for section 8 which details how Schrodinger went about it.

The paper of course examines it in light of the background of the sections that went before that includes the path integral etc.

It is indeed sad the Hamilton-Jacobi equation is not taught in typical physics courses these days. All I can suggest to anyone interested in what I consider THE elegant presentation of Classical Mechanics that, while a bit terse in Landau's typical style (he makes the most profound of statements just as sort of off-hand remakes) is Lanadu - Mechanics. It also emphasizes the very important role of symmetry which become more important again in QM and absolutely fundamental in QFT. Strangely though he does it all without even mentioning Noether.

The mathematical requirements for Landau's book isn't that great - just your typical calculus 1,2,3 sequence in the US or advanced calculus here in Aus that good math students do in grade 12 and get university credit. Some have said its not an easy read - I disagree with that - its quite straightforward going - but like fine wine as you learn more and re-read it you get more out of it. It of course is a an ageless classic.

Thanks
Bill