294
9 Semiclassical Theory: Path Integrals
invariant tori. The Bohr-Sommerfeld quantization condition could not be used to
quantize chaotic systems and until recently no method existed to connect classically
chaotic systems with their quantum counterpart.
Semiclassical methods used to treat quantum systems are usually associated with
the Wentzel–Kramers–Brillouin, or WKB, method (or Van Vleck’s generalization
of WKB to systems with several degrees of freedom Van Vleck 1928) for obtaining
semiclassical expressions for the spectrum and eigenfunctions of quantum systems.
However, this method only applies to integrable systems and cannot be used to
quantize chaotic systems.
The first hint that path integrals might afford a means to quantize classically
chaotic systems is due to Selberg (1956) and McKean (1972), who obtained a
path integral formulation for a particle moving on a Riemann surface of negative
curvature, which is a chaotic system. Path integrals had been used earlier to
describe semiclassical integrable systems. In 1951, Morette (1951) obtained Van
Vleck’s formula from the quantum path integral by taking a semiclassical limit,
and Choquard (1955) generalized her results. However, these were both short-time
results because they did not include a means to deal with caustics. Once the problem
of dealing with caustics was worked out, the semiclassical path integral could be
extended to long times (Schulman 1981).
It was Gutzwiller who first successfully applied path integrals to Hamiltonian
systems that are classically chaotic. In 1982, Gutzwiller (1982) showed that a
semiclassical approximation to the Feynman path integral (Feynman and Hibbs
1965) could be used to compute approximate values for the energy eigenvalues
of a quantum mechanical system (the anisotropic Kepler system) whose classical
counterpart is chaotic. Gutzwiller’s work on the anisotropic Kepler system was the
culmination of a long series of papers by him (Gutzwiller 1967, 1970, 1971, 1973,
1977, 1980, 1982) and by Balian and Bloch (1972, 1974) linking the periodic orbits
of a classical system to the spectrum of the corresponding quantum system. In this
chapter, we will derive the semiclassical path integral expressions for the spectral
density of quantum systems whose classical counterparts are either integrable or
chaotic, and we shall illustrate our results for several model systems.
We begin in Sect. 9.2 by obtaining the Green’s function of a Hamiltonian system,
and we shall obtain from it a general expression for the density of states. In Sect. 9.3,
we write the Green’s function in terms of a path integral. The Green’s function can
be used to describe the dynamical evolution of the quantum system from point x 0
and time t 0 to point x and time t. In the path integral formulation of this quantity,
we must integrate over all intermediate paths whether they are physical or not.
A semiclassical expression for the Green’s function is obtained in Sect. 9.4 by
performing a stationary phase approximation. We obtain an expression involving
only those intermediate paths that extremize the action and therefore are physically
realizable, and we do so by neglecting terms of order
√ ¯
h and smaller. This gives us
the semiclassical path integral expression for the Green’s function. In Sect. 10.3.5,
we obtain the semiclassical energy Green’s function and we use it to obtain the
density of states for a particle with one degree of freedom confined to a potential
well. We obtain the expected WKB results.
9 Semiclassical Theory: Path Integrals
invariant tori. The Bohr-Sommerfeld quantization condition could not be used to
quantize chaotic systems and until recently no method existed to connect classically
chaotic systems with their quantum counterpart.
Semiclassical methods used to treat quantum systems are usually associated with
the Wentzel–Kramers–Brillouin, or WKB, method (or Van Vleck’s generalization
of WKB to systems with several degrees of freedom Van Vleck 1928) for obtaining
semiclassical expressions for the spectrum and eigenfunctions of quantum systems.
However, this method only applies to integrable systems and cannot be used to
quantize chaotic systems.
The first hint that path integrals might afford a means to quantize classically
chaotic systems is due to Selberg (1956) and McKean (1972), who obtained a
path integral formulation for a particle moving on a Riemann surface of negative
curvature, which is a chaotic system. Path integrals had been used earlier to
describe semiclassical integrable systems. In 1951, Morette (1951) obtained Van
Vleck’s formula from the quantum path integral by taking a semiclassical limit,
and Choquard (1955) generalized her results. However, these were both short-time
results because they did not include a means to deal with caustics. Once the problem
of dealing with caustics was worked out, the semiclassical path integral could be
extended to long times (Schulman 1981).
It was Gutzwiller who first successfully applied path integrals to Hamiltonian
systems that are classically chaotic. In 1982, Gutzwiller (1982) showed that a
semiclassical approximation to the Feynman path integral (Feynman and Hibbs
1965) could be used to compute approximate values for the energy eigenvalues
of a quantum mechanical system (the anisotropic Kepler system) whose classical
counterpart is chaotic. Gutzwiller’s work on the anisotropic Kepler system was the
culmination of a long series of papers by him (Gutzwiller 1967, 1970, 1971, 1973,
1977, 1980, 1982) and by Balian and Bloch (1972, 1974) linking the periodic orbits
of a classical system to the spectrum of the corresponding quantum system. In this
chapter, we will derive the semiclassical path integral expressions for the spectral
density of quantum systems whose classical counterparts are either integrable or
chaotic, and we shall illustrate our results for several model systems.
We begin in Sect. 9.2 by obtaining the Green’s function of a Hamiltonian system,
and we shall obtain from it a general expression for the density of states. In Sect. 9.3,
we write the Green’s function in terms of a path integral. The Green’s function can
be used to describe the dynamical evolution of the quantum system from point x 0
and time t 0 to point x and time t. In the path integral formulation of this quantity,
we must integrate over all intermediate paths whether they are physical or not.
A semiclassical expression for the Green’s function is obtained in Sect. 9.4 by
performing a stationary phase approximation. We obtain an expression involving
only those intermediate paths that extremize the action and therefore are physically
realizable, and we do so by neglecting terms of order
√ ¯
h and smaller. This gives us
the semiclassical path integral expression for the Green’s function. In Sect. 10.3.5,
we obtain the semiclassical energy Green’s function and we use it to obtain the
density of states for a particle with one degree of freedom confined to a potential
well. We obtain the expected WKB results.