Chapter 9
Semiclassical Theory: Path Integrals
Abstract The Wentzel–Kramers–Brillouin, or WKB, method for obtaining semiclassical expressions for the spectrum and eigenfunctions of quantum systems only
applies to integrable systems and cannot be used to quantize chaotic systems.
However, path integrals do provide a means to quantize classically chaotic systems.
Selberg obtained a path integral formulation for a particle moving on a Riemann
surface of negative curvature, which is a chaotic system. Gutzwiller successfully
applied path integrals to Hamiltonian systems that are classically chaotic, and was
able to compute approximate values for the energy levels of the anisotropic Kepler
system, whose classical counterpart is chaotic. Gutzwiller focused on the trace of the
energy Green’s function and showed, by means of a stationary phase approximation,
that the trace could be expressed in terms of a sum over all classical stable and
unstable periodic orbits.
Experimental results on the absorption spectrum of diamagnetic hydrogen clearly
show the effect of underlying classical orbits on quantum dynamics. Semiclassical
path integral calculations using a few periodic orbits and closed orbits are able
to reproduce the essential features of the experimental absorption spectrum data.
The effects of bifurcation of underlying classical orbits also show up clearly in the
experimental data.
Keywords Bohr-Sommerfeld quantization · WKB method · Path integrals ·
Caustics · Green’s function · Semiclassical path integral · Anisotropic Kepler
system · Gutzwiller trace formula · Unstable periodic orbits · Diamagnetic
hydrogen
9.1 Introduction
The “old quantum theory,” which is based on the Bohr-Sommerfeld quantization
condition, provided a means of quantizing a classical mechanical system by quantizing the action variables associated with KAM tori. (For a historical discussion,
see Born 1960.) However, it was recognized by Einstein, as early as 1917 (Einstein
1917), that this method could only be used for systems in which trajectories lie on
© Springer Nature Switzerland AG 2021
L. Reichl, The Transition to Chaos, Fundamental Theories of Physics 200,
https://doi.org/10.1007/978-3-030-63534-3_9
293
Semiclassical Theory: Path Integrals
Abstract The Wentzel–Kramers–Brillouin, or WKB, method for obtaining semiclassical expressions for the spectrum and eigenfunctions of quantum systems only
applies to integrable systems and cannot be used to quantize chaotic systems.
However, path integrals do provide a means to quantize classically chaotic systems.
Selberg obtained a path integral formulation for a particle moving on a Riemann
surface of negative curvature, which is a chaotic system. Gutzwiller successfully
applied path integrals to Hamiltonian systems that are classically chaotic, and was
able to compute approximate values for the energy levels of the anisotropic Kepler
system, whose classical counterpart is chaotic. Gutzwiller focused on the trace of the
energy Green’s function and showed, by means of a stationary phase approximation,
that the trace could be expressed in terms of a sum over all classical stable and
unstable periodic orbits.
Experimental results on the absorption spectrum of diamagnetic hydrogen clearly
show the effect of underlying classical orbits on quantum dynamics. Semiclassical
path integral calculations using a few periodic orbits and closed orbits are able
to reproduce the essential features of the experimental absorption spectrum data.
The effects of bifurcation of underlying classical orbits also show up clearly in the
experimental data.
Keywords Bohr-Sommerfeld quantization · WKB method · Path integrals ·
Caustics · Green’s function · Semiclassical path integral · Anisotropic Kepler
system · Gutzwiller trace formula · Unstable periodic orbits · Diamagnetic
hydrogen
9.1 Introduction
The “old quantum theory,” which is based on the Bohr-Sommerfeld quantization
condition, provided a means of quantizing a classical mechanical system by quantizing the action variables associated with KAM tori. (For a historical discussion,
see Born 1960.) However, it was recognized by Einstein, as early as 1917 (Einstein
1917), that this method could only be used for systems in which trajectories lie on
© Springer Nature Switzerland AG 2021
L. Reichl, The Transition to Chaos, Fundamental Theories of Physics 200,
https://doi.org/10.1007/978-3-030-63534-3_9
293