M.C. Gutzwiller, "Classical quantization of a Hamiltonian with ergodic behavior"
Physical Review Letters 45 no.3 (1980) 150-153
Abstract
"Conservative Hamiltonian systems with two degrees of freedom are discussed
where a typical trajectory fills the whole surface of constant energy. The
trace of the quantum mechanical Green's function is approximated by a sum over
classical periodic orbits. This leads directly to Selberg's
trace formula for the motion of a particle on a surface of constant negative
curvature, and, when applied to the anisotropic Kepler problem, yields excellent
results for all the energy levels."
Excerpt p.150-151:
"The relations between classical and quantal mechanics are of central
importance to the understanding of physics. Classical mechanics is believed
to be a limit of quantum mechanics when Planck's constant is small. It is
natural to use the classical motion as a starting point in order to find an
approximate solution for Schrödinger's equation. The idea was implicit
in the "old quantum mechanics" of Bohr and Sommerfeld, and was made explicit
in the WKB approximation. The latter can be applied to problems with
several degrees of freedom provided the separation of variables can be carried
out. There are two basic difficultites. First, conserved quantities beyond
the endergy have to be found which would permit the separation of variables,
even if they are not analytic functions of momentum and position. Second,
if there are not any, the whole basis for the quantization conditions of Bohr
and Sommerfeld vanishes.
The first point was clearly understood by Poincaré and became a frequent
topic of investigation by both mathematicians and astronomers in the 1950's. The
second point was first discussed by Einstein in 1917 and has received very little
attention. The emphasis seems mostly on how to handle the occurrence of
nonanalytic constants of motion. Einstein gave the clue when he pointed to
the existence of invariant tori on the surface of constant energy in phase space,
and he showed the way to the proper treatment of the quantization conditions. But
he then goes on to ask how to proceed when there are no invariant tori, not
even nearby.
This Letter has two purposes. First, a large class of examples without
invariant tori will be shown where a direct connection exists between the
energy levels and classical orbits. The relevant
formula was given by Selberg in 1954 and coincides with a formula which
the author proposed some ten years ago [3] for Hamiltonian systems in general.
Second, an entirely different example without invariant tori, the anisotropic
Kepler problem, will be treated. The same formula gives very close agreement
for the energy levels between the solutions of Schrödinger's equation
and a method which is based entirely on classical mechanics. Only conservative
Hamiltonian systems with two degrees of freedom will be discussed where the
typical trajectory (with the exception of a set of measure zero) comes arbitrarily
close to every point on the surface of constant energy in phase space. The
exceptional set is formed by the closed orbits of which there are many. They
are all linearly unstable, i.e., a trajectory which starts nearby in
phase space drifts away exponentially fast."
[3] M.C. Gutzwiller, "Periodic orbits and classical quantization conditions",
J. Math. Phys. 12 (1971) 343-358