I read today about Ehrenfest's theorem on mechanical systems (with adiabatic hamiltonian evolution).(adsbygoogle = window.adsbygoogle || []).push({});

Well in the book, it says that the Ehrenfest's theorem was also used by Bohr and Sommerfield who proposed the quantization of quantities such as:

[itex]\oint p dq [/itex]

According to Bohr-Sommerfield's theory quantities as this acquire discrete values. Their reasoning for such a universal law was that this quantity is adiabatically invariant. For example if we find out that for an Harmonic Oscillator the relation:

[itex] \oint p dq= h (n+\frac{1}{2})[/itex]

[itex] h[/itex] Planck's constant and [itex]n \in N[/itex]

holds, then this relation should be universally valid.

Why? because changing adiabatically the parabolic potential of the Harmonic Oscillator we can achieve any kind of potential, even that of the Hydrogen atom. So for every potential the above relation should hold. As such, the law should be of fundamental and universal validity.

Well, I have two questions...

First of all, I am still unable to understand what is, in fact, in common between the Harmonic Oscillator let's say and the Hydrogen atom. The expression given above looks pretty similar to the energy of the HO, (in fact it's the energy over frequency [itex]\frac{E}{ω}[/itex]). How is that associated with the Hydrogen atom? the H doesn't have such kind of spectrum. If that's true then I think it must hold for EVERY kind of physical potential, because from a smooth function you should be able to create some other smooth one (?)

Also if the above quantity is a constant for every system, it must correspond to some symmetry? what is that symmetry? is it the time translation one? I am not sure...

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# Ehrenfest Theorem QM

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