Ehrenfest Theorem in Quantum Mechanics: Bohr-Sommerfield's Theory

[ChrisVer]

I read today about Ehrenfest's theorem on mechanical systems (with adiabatic hamiltonian evolution).
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:
$\oint p dq$
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:
$\oint p dq= h (n+\frac{1}{2})$
$h$ Planck's constant and $n \in N$

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 $\frac{E}{ω}$). 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...

 

[Matterwave]

This is the quantization of action-angle variables (Born-Sommerfield quantization). It's a pretty archaic way of doing things. Action-angle variables arise from the Hamilton-Jacobi mechanics, which is yet another reformulation of classical mechanics (somewhat distinct from Hamiltonian and Lagrangian, but taking a lot of ques from Hamiltonian mechanics). I'm not super familiar with this, because this is like the 4th or 5th formulation of classical mechanics you learn, and by this point, it seems that the others are sufficient. Are you interested in this just out of curiosity?

 

[ChrisVer]

curiosity, because reading this up just shocked me... (imagining that all kind of different problems in QM could have such a symmetry connecting them and making them "equivalent" under it)... hadn't ever thought over that... although I know that Sommerfield's models are pretty archaic (his model belongs to semiclassical qm), if there's such a symmetry I don't know how it can be disproven with present results ^^...

 

[atyy]

Bohr-Sommerfeld quantization is only a semi-classical method. If you have a classical integrable system, you can always find "action-angle variables" - ie. you can make the system look like an oscillator, and the phase space is just a bunch of tori (I think this is why Bohr-Sommerfeld quantization is nice, in some way). However, this is not true for non-integrable systems. Nowadays, the most common way to quantize is via non-commutation of canonically conjugate observables. Of course, this is also a guess, since it is the quantum system that gives rise to the classical, not the other way round.

http://homepages.vub.ac.be/~gaspard/G.quantum.Encycl.05.pdf
http://chaosbook.dk/chapters/traceSemicl.pdf (very interesting stuff on trying to extend semi-classical quantization to non-integrable systems, not to do real quantum physics, but in the hope that this may provide insight into the statistics of energy levels)

 

[ChrisVer]

I agree with that too Post4. But my main point is on symmetries rather than the theory itself. Symmetries can't be lost from going to classical to quantum world, can they?
For example, the Hydrogen atom has the symmetry of Runge-Lenz vector even in the QM formalism (although it was at first a result of the semi-classical approach)

edit: just saw the links, I'll have a look at them

 

[atyy]

I can't answer your specific question about the Runge-Lenz vector, but in quantum field theory, symmetries can be lost going from classical to quantum. The chiral anomaly http://www.scholarpedia.org/article/Axial_anomaly is a famous example.

 

[Matterwave]

One more point regarding your first post. Remember that action angle variables are in units of action (or angular momentum). As such, the quantization of the e.g. Bohr atom is done by quantizing the angular momentum. Since you have circular symmetry, you know that the p's are constant, and so

$$\oint pdq=2\pi p*r=2\pi L$$

You quantize this L and posit $2\pi L=nh$, which is equivalent to the usual $L=n\hbar$. But for any system you have to first find the action angle variables.

Now, regarding the discussion of symmetries being lost going from classical to quantum mechanics. Our current way of "quantizing" involves looking at the poisson brackets from classical mechanics and positing cannonical commutation relations from them. This, as atyy said, is really just a guess since it's the quantum version which should be the "fundamental" theory, and not the classical version. One easy way to see this is that you have to pick the right cannonical positions and momenta in your quantizing. In classical mechanics, cannonical transformations are allowed, in quantum mechanics, this is apparently not so (don't ask me about the details, I'm not familiar with geometric quantization), one should use the Cartesian x and p in the cannonical commutation relations.

 

[PhilDSP]

It might be worth looking at the Einstein–Brillouin–Keller action quantization procedure also

(For example): http://scitation.aip.org/content/aapt/journal/ajp/72/12/10.1119/1.1768554

"A simple semiclassical means for obtaining correct quantum mechanics results without invoking the full mathematics of the Schroedinger equation is provided by the Einstein–Brillouin–Keller EBK action quantization. [1–3] Although this approach is a well-known tool in contemporary atomic and molecular theory, its usefulness and pedagogical value has been largely overlooked in elementary textbooks. Unfortunately, textbooks tend to perpetuate and then disparage the flawed and archaic Bohr–Sommerfeld–Wilson quantization, [4–6] which as was recognized shortly after its introduction [1] fails to properly account for caustic phase jumps at the classical turning points.

The EBK quantization approach [1–3] involves a path integral over the phase space of each coordinate qi and its conjugate momentum pi..."