Quantization of Quasiperiodic Orbits in the Bohr-Sommerfeld Model

[Couchyam]

TL;DR Summary

Can the Bohr-Sommerfeld approach be adapted to handle quasi-periodicity? (Or would that just be another special case of the WKB approximation?)

Recall that in the semi-classical Bohr-Sommerfeld quantization scheme from the early days of quantum mechanics, bound orbits were quantized according to the value of the action integral around a single loop of a closed path. Clearly this only makes sense if the orbits in question permit closed paths, however, which is not always the case (consider for example a central potential of the form $1/r^{1+\epsilon}$, $\epsilon \neq 0$, in which orbits are generically (in $\epsilon$ as well as initial conditions) quasi-periodic in $\theta(t)$ and $r(t)$. Quasiperiodicity presumably would have been a well-understood (dare I say pedestrian) concept in Bohr's time, and so it must have been considered to some extent. Did Bohr and/or Sommerfeld write anything about this? If not, how might their approach be expanded on to incorporate quasiperiodicity?

 

[vanhees71]

Quasiperiodic systems are no problem, i.e., you can use the Bohr-Sommerfeld quantization condition whenever you find a complete set of action-angle variables for the system. The best resource on old quantum theory is of course Sommerfeld's famous textbook "Atomic Structure and Spectral Lines". There it's in paragraph 6 of Chpt. II (3rd edition of the English translation of 1934).

 

[Couchyam]

Thanks for your answer. I must admit that I'm unfamiliar with Sommerfeld's textbook, but I'll give it a look. The first example of a quasiperiodic system that comes to mind is a particle moving on a two-dimensional torus, in which energy and momentum are conserved. The system is integrable, and has two sets of action-angle variables that allow a dense set of closed orbits (in position space), corresponding I think to the ordinary quantum spectrum. The situation for closed orbits of arbitrary central potential wells is more subtle but seems to be resolved similarly (although I could imagine cases where tunneling effects are essential, as when the radial potential energy is rough.) Meanwhile, however, a two-dimensional harmonic oscillator with incommensurate frequencies in the $x$ and $y$ directions is integrable, again with two sets of action-angle coordinates, but the set of closed orbits appears to be much smaller than the corresponding quantum spectrum. This phenomenon might be somewhat contrived in single particle systems (one must wonder where the incommensurate frequencies come from) but could be more typical when there are multiple particles. Would there be a way of approaching systems like this within the Bohr-Sommerfeld formalism? (One might simply posit that each mode propagates and/or is quantized independently of the others, but this comes across as a bit ad hoc; one justification might be if, in the limit of rational approximations to incommensurate frequencies, the spectrum of closed orbits approaches the complete set of ordinary incommensurate frequencies.)

 

[vanhees71]

I'm not sure, where the problem with the 2D (or any dimension too) harmonic oscillator with arbitrary eigenfrequencies should be. You just have a Bohr-Sommerfeld quantization condition for each action-angle-variable pair, i.e., the quantization can be done for any system for which you have a complete set of action-angle variables.

 

[Couchyam]

I'm not sure, where the problem with the 2D (or any dimension too) harmonic oscillator with arbitrary eigenfrequencies should be. You just have a Bohr-Sommerfeld quantization condition for each action-angle-variable pair, i.e., the quantization can be done for any system for which you have a complete set of action-angle variables.

It's a bit hard to explain exactly, but the idea might be whether the closed orbit criterion is fundamental to the Bohr-Sommerfeld model, or just a "happy accident" in the case of the hydrogen atom. In situations where the central potential has precessing orbits, should the angular quantization condition $\int p_\theta d\theta = nh$ be applied over a single loop, $0\leq \theta < 2\pi$, or over however many loops are required to complete a given closed orbit?

 

[A. Neumaier]

It's a bit hard to explain exactly, but the idea might be whether the closed orbit criterion is fundamental to the Bohr-Sommerfeld model, or just a "happy accident" in the case of the hydrogen atom. In situations where the central potential has precessing orbits, should the angular quantization condition $\int p_\theta d\theta = nh$ be applied over a single loop, $0\leq \theta < 2\pi$, or over however many loops are required to complete a given closed orbit?

A lot has happened since Bohr and Sommerfeld. The quasiperiodic case corresponds to what are called
Bohr-Sommerfeld tori. These can be classified and quantized - not the individual loops. A very recent paper on the topic (from where you can work backwards) is

• Mir, P., & Miranda, E. (2021). Geometric quantization via cotangent models. Analysis and Mathematical Physics, 11(3), 1-35. https://arxiv.org/pdf/2102.02699.pdf