Quantization of Quasiperiodic Orbits in the Bohr-Sommerfeld Model

In summary, the Bohr-Sommerfeld quantization scheme from the early days of quantum mechanics was limited to bound orbits that permitted closed paths. However, it can still be used for quasiperiodic systems if a complete set of action-angle variables is found. This is discussed in Sommerfeld's textbook "Atomic Structure and Spectral Lines." The quantization can be done for any system with a complete set of action-angle variables. The recent paper by Mir and Miranda further explores the quantization of quasiperiodic systems using geometric quantization.
  • #1
Couchyam
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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?
 
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  • #2
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).
 
  • #3
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.)
 
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  • #4
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.
 
  • #5
vanhees71 said:
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?
 
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  • #6
Couchyam said:
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

 
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Related to Quantization of Quasiperiodic Orbits in the Bohr-Sommerfeld Model

1. What is Bohr-Sommerfeld quantization?

Bohr-Sommerfeld quantization is a model of atomic structure proposed by Niels Bohr and Arnold Sommerfeld in the early 20th century. It is based on the idea that electrons in an atom can only occupy certain discrete energy levels, rather than being able to have any amount of energy.

2. How does Bohr-Sommerfeld quantization differ from classical physics?

In classical physics, electrons are thought to move in continuous orbits around the nucleus of an atom. However, in Bohr-Sommerfeld quantization, electrons are only allowed to occupy specific energy levels, which are determined by the angular momentum of the electron.

3. What is the significance of Bohr-Sommerfeld quantization?

Bohr-Sommerfeld quantization was a major breakthrough in understanding the behavior of electrons in atoms. It provided a more accurate model of atomic structure than classical physics, and helped to explain the spectral lines observed in the emission and absorption of light by atoms.

4. How does Bohr-Sommerfeld quantization relate to quantum mechanics?

Bohr-Sommerfeld quantization was one of the first steps towards the development of quantum mechanics. It laid the foundation for the later work of physicists such as Werner Heisenberg and Erwin Schrödinger, who developed the mathematical framework for understanding the behavior of particles on a subatomic level.

5. Is Bohr-Sommerfeld quantization still relevant today?

While Bohr-Sommerfeld quantization has been largely replaced by more advanced theories, it is still relevant today in the sense that it provided the basis for our current understanding of atomic structure. It also serves as a reminder of the progress that has been made in the field of quantum mechanics over the past century.

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