Mathematical Basis of Bohr-Sommerfeld

In summary, the Bohr-Sommerfeld rule is a useful tool for approximating the spectra of integrable systems, especially for high quantum numbers. It involves using the phase-space (X, omega) and invariant Lagrangian tori to select certain values of the Hamiltonian and other integrals of motion, which form the predicted quantum spectrum. This method has been rigorously shown to be effective, even for systems with singularities, and has been studied by both mathematicians and physicists. Experimental confirmation of the rule has also been demonstrated in certain cases.
  • #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
 
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  • #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

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

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,

The mathematical basis of the Bohr-Sommerfeld rule lies in the principles of geometric quantization and symplectic geometry. Geometric quantization is a mathematical framework that aims to construct quantum systems from classical systems by using symplectic geometry.

In the Bohr-Sommerfeld rule, the phase-space is foliated by invariant Lagrangian tori, which means that each torus is a closed, smooth, and embedded submanifold of the phase-space that is preserved by the dynamics of the system. This foliation is a consequence of the integrability of the system, which means that there exist enough conserved quantities (integrals of motion) to fully specify the motion of the system.

Now, let's consider the U(1) bundle L and connection A over the phase-space. The curvature of A is the symplectic form omega, which is a fundamental quantity in symplectic geometry. The fact that the restriction of A on the tori is flat (i.e. the curvature is zero) reflects the fact that the tori are Lagrangian, i.e. they are preserved by the symplectic form. This is a crucial property for the Bohr-Sommerfeld rule to work.

The special tori that are selected by the Bohr-Sommerfeld rule are those for which the restriction of A is not only flat but trivial (all of the monodromies are trivial). This means that the holonomy (a measure of the non-triviality of a connection) along these tori is trivial, and this is what allows us to obtain the predicted quantum spectrum.

In terms of mathematical rigor, there have been studies that show the accuracy of the Bohr-Sommerfeld rule for high quantum numbers. One example is the work of Gutzwiller in the 1970s, where he showed that for high quantum numbers, the semiclassical approximation using the Bohr-Sommerfeld rule was in good agreement with the exact quantum results. However, it should be noted that the rule is an approximation and may not hold for all systems.

I hope this helps answer your question.
 

Related to Mathematical Basis of Bohr-Sommerfeld

1. What is the Bohr-Sommerfeld model?

The Bohr-Sommerfeld model, also known as the old quantum theory, is a mathematical model developed by Niels Bohr and Arnold Sommerfeld in the early 1900s to explain the behavior of electrons in an atom. It was the first successful attempt at providing a mathematical basis for understanding the structure of atoms.

2. What are the key features of the Bohr-Sommerfeld model?

The Bohr-Sommerfeld model is based on the following key features:

  • Electrons orbit the nucleus in circular orbits at fixed energy levels.
  • Electrons can only exist in certain discrete energy levels and cannot exist in between.
  • Electrons emit or absorb energy in the form of photons when they move between energy levels.
  • The angular momentum of an electron is quantized, meaning it can only have certain specific values.
  • The model also incorporates classical mechanics and the quantization of angular momentum from the new quantum theory.

3. How did the Bohr-Sommerfeld model contribute to our understanding of atoms?

The Bohr-Sommerfeld model was a significant step in understanding the structure of atoms. It provided the first successful mathematical description of electrons in an atom and explained the existence of discrete energy levels. It also paved the way for future developments in quantum mechanics and our understanding of the behavior of subatomic particles.

4. What were the limitations of the Bohr-Sommerfeld model?

The Bohr-Sommerfeld model was eventually replaced by the more accurate quantum mechanical model, as it could not fully explain the behavior of electrons in more complex atoms. It also did not account for the wave-like nature of particles and could not explain phenomena such as electron spin. Additionally, the model could not explain the fine details of atomic spectra.

5. How is the Bohr-Sommerfeld model still relevant today?

Although the Bohr-Sommerfeld model has been replaced by more advanced theories, it is still relevant in understanding the basic principles of atomic structure and in teaching introductory quantum mechanics. Some of its concepts, such as quantization of energy and angular momentum, are still used in modern quantum theory. The model also played a crucial role in the development of the more accurate quantum mechanical model.

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