Mathematical Basis of Bohr-Sommerfeld

In summary, the Bohr-Sommerfeld rule approximates the spectra of integrable systems accurately, especially for high quantum numbers. The rule states that the phase-space of an integrable system is foliated by invariant Lagrangian tori, with some special tori being selected based on the values of the Hamiltonian and other integrals of motion. This approximation has been mathematically proven to be good, at least for high quantum number asymptotics, and has received attention from both mathematicians and physicists. Experimental confirmation has also been found in the millimeter wave spectrum of cyanogen isothiocyanate.
  • #1
Squark
Hello everyone!

The Bohr-Sommerfeld rule allows approximating the spectra of integrable
systems in a quite accurate way, at least for high quantum numbers.
The most general formulation of the rule is as follows (I converted a
less
high-brow formulation into this form, I hope I got it right):

We know the phase-space (X, omega) of an integrable system is foliated
by invariant Lagrangian tori (btw, does it mean the phase-space is a
locally trivial fibration with toric fiber? Or are there exceptional
fibers?)
Lets choose a U(1) bundle L and connection A over the phase space,
such that omega is the curvature of A (together with the Lagrangian
foliation we have for free, these comprise precisely the data for
geometric quantization!) Then, the restriction of A on any of the the
tori
is flat (since the tori are Lagrangian). However, some tori are
special:
the restriction on them is not only flat but trivial (all of the
monodromies
are trivial). These are the tori "selected" by the Bohr-Sommerfeld
rule,
and the values of the Hamiltonian (and the other integrals of motion)
on
them form the predicted quantum spectrum.

The question is, has anyone shown the approximation to be "good", in
some sense, in a mathematically rigorous way? At least for the high
quantum number asymptotics?Squark
 
Physics news on Phys.org
  • #2
Squark wrote:


> We know the phase-space (X, omega) of an integrable system is foliated
> by invariant Lagrangian tori (btw, does it mean the phase-space is a
> locally trivial fibration with toric fiber? Or are there exceptional
> fibers?)


There can be singular fibers, in fact this is the case already in many
simple examples of integrable systems (see below).

> The question is, has anyone shown the approximation to be "good", in
> some sense, in a mathematically rigorous way? At least for the high
> quantum number asymptotics?


Yes this has been shown (e.g. section 5 of the second reference below).

The study of Bohr-Sommerfeld rules in the case of integrable systems
with singularities has received a lot of attention recently from both
mathematicians and physicists, in relation to the idea of "quantum
monodromy" (basically the lattice formed by the quantum states is
usually not a simple periodic one and the quantum numbers are not
globally valid).

Here are three recent references which provide some background:

1. Singular Bohr-Sommerfeld rules for 2D integrable systems
Yves Colin de Verdière and San Vu Ngoc
Annales Scientifiques de l'École Normale Supérieure
Volume 36, Issue 1 , March 2003, Pages 1-55.
doi:10.1016/S0012-9593(03)00002-8
preprint: http://www-fourier.ujf-grenoble.fr/~svungoc/articles/bs2d.pdf [Broken]

2. Symplectic Techniques for Semiclassical Integrable Systems
San Vu Ngoc
2004
preprint:
http://www-fourier.ujf-grenoble.fr/~svungoc/articles/stsis.pdf [Broken]

3. Hamiltonian monodromy as lattice defect
B. I. Zhilinskií
in: Topology in Condensed Matter,
(Springer Series in Solid-State Sciences, Vol. 150), 2006, pp. 165-186.
preprint: http://pca3.univ-littoral.fr/~zhilin/prepub/MLD.psExperimentalists are also starting to study the issue, see

4. Experimental Confirmation of Quantum Monodromy: The Millimeter Wave
Spectrum of Cyanogen Isothiocyanate NCNCS
B. P. Winnewisser at al.
Phys. Rev. Lett. 95, 243002 (2005)
http://link.aps.org/abstract/PRL/v95/e243002Regards,
---
PP
 
Last edited by a moderator:
  • #3
Hello Squark,

Thank you for sharing your understanding of the Bohr-Sommerfeld rule and its mathematical basis. The rule is indeed a powerful tool for approximating the spectra of integrable systems, and it has been extensively studied and validated by mathematicians and physicists alike.

To answer your question, yes, there have been rigorous mathematical studies on the accuracy of the Bohr-Sommerfeld approximation, particularly in the high quantum number asymptotics. In fact, this has been a topic of ongoing research and there have been several developments in recent years.

One approach is to use semiclassical methods, which combine classical and quantum mechanics, to rigorously derive the Bohr-Sommerfeld rule from the underlying classical dynamics of the system. This has been done for a variety of integrable systems, including the hydrogen atom, the Kepler problem, and the harmonic oscillator, among others.

Another approach is to use geometric quantization, as you mentioned, to provide a mathematical framework for the Bohr-Sommerfeld rule. This has been studied extensively by mathematicians, and they have been able to prove the accuracy of the approximation in certain cases.

Overall, while there is still ongoing research in this area, it is safe to say that the Bohr-Sommerfeld rule has been shown to be a valid and accurate approximation of the spectra of integrable systems, particularly in the high quantum number regime.
 

What is the Bohr-Sommerfeld model?

The Bohr-Sommerfeld model is a mathematical model developed by Niels Bohr and Arnold Sommerfeld in the early 20th century to describe the behavior of electrons in an atom. It is based on the principles of quantum mechanics and classical mechanics, and it was the first successful attempt at explaining the spectral lines of atoms.

What is the mathematical basis of the Bohr-Sommerfeld model?

The mathematical basis of the Bohr-Sommerfeld model is a combination of the classical equations of motion, the quantum mechanical principle of quantization, and the concept of stationary states. This model incorporates the principles of both classical mechanics and quantum mechanics to describe the behavior of electrons in an atom.

How does the Bohr-Sommerfeld model explain the spectral lines of atoms?

The Bohr-Sommerfeld model explains the spectral lines of atoms by considering the electrons to be in quantized energy levels. When an electron transitions between energy levels, it emits or absorbs a specific amount of energy in the form of a photon, resulting in a spectral line. These energy levels and transitions are described by the mathematical equations of the Bohr-Sommerfeld model.

What are stationary states in the Bohr-Sommerfeld model?

Stationary states in the Bohr-Sommerfeld model refer to energy levels in which the electron remains in a stable orbit around the nucleus without emitting or absorbing energy. These states are quantized, meaning that the electron can only occupy certain energy levels and cannot exist in between. This concept of stationary states was a key aspect of the Bohr-Sommerfeld model.

How does the Bohr-Sommerfeld model relate to the modern understanding of atoms?

The Bohr-Sommerfeld model was an important step towards the modern understanding of atoms and their behavior. It laid the foundation for later developments in quantum mechanics and helped to explain the behavior of electrons in atoms. However, the model has since been replaced by more accurate and comprehensive models, such as the Schrödinger equation, which provide a more complete understanding of the atomic structure and spectral lines.

Similar threads

  • MATLAB, Maple, Mathematica, LaTeX
Replies
2
Views
2K
  • General Discussion
Replies
4
Views
3K
  • STEM Academic Advising
Replies
5
Views
2K
  • Beyond the Standard Models
Replies
24
Views
4K
  • STEM Academic Advising
Replies
13
Views
2K
  • Beyond the Standard Models
Replies
10
Views
2K
  • MATLAB, Maple, Mathematica, LaTeX
Replies
6
Views
3K
  • MATLAB, Maple, Mathematica, LaTeX
Replies
7
Views
2K
  • MATLAB, Maple, Mathematica, LaTeX
Replies
9
Views
2K
  • MATLAB, Maple, Mathematica, LaTeX
Replies
1
Views
2K
Back
Top