Semiclassical calculations of vibrational energy levels for nonseparable systems using the Birkhoff–Gustavson normal form

Abstract

We present a semiclassical method of calculating vibrational energy levels for a system of nonseparable coupled oscillators. For a Hamiltonian written as a power series in which the leading terms are given by a sum of one-dimensional harmonic oscillator Hamiltonians, the method involves transforming the original classical Hamiltonian via a succession of canonical transformations into a normal form which is a power series originally defined by Birkhoff and later generalized by Gustavson. Two cases are distinguished. If the harmonic oscillator frequencies in the unperturbed Hamiltonian are incommensurable, then the normal form is a power series whose terms are products of one-dimensional harmonic oscillator Hamiltonians; if the frequencies in the unperturbed Hamiltonian are commensurable, then additional terms which cannot be written as products of one-dimensional harmonic oscillator Hamiltonians enter into the normal form. Once the normal form is obtained, semiclassical quantization of action variables is straightforward. The incommensurable case yields a formula for the energy spectrum which is a power series in the quantum numbers. The commensurable case is more complicated, and yields a form from which energy levels may be obtained individually by numerical calculation and quantization of a one-dimensional phase integral. Nonseparable two-dimensional examples are treated for each case. The results obtained for both cases show excellent agreement with quantum mechanical calculations. The quantum calculations indicate that all of the energy levels fall into a regular pattern.