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Physical Review A

Addendum to “Direct trajectory method for semiclassical wave functions”

Shuangbo Yang, Michael E. Kellman

Published Feb 5, 2002• 11 citations
Publisher

Abstract

For two kinetically coupled Morse oscillators, we present details of the method of calculation in Ref. 1 of the semiclassical wave function for an Einstein-Brillouin-Keller quantizing torus, using a single trajectory and its stability matrix. This is the ``standard'' method, contrasted in Ref. 1 with our new ``direct'' method, which does not require the calculation of the stability matrix.

DOI
10.1103/physreva.65.034103

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For two kinetically coupled Morse oscillators, we present details of the method of calculation in Ref. 1 of the semiclassical wave function for an Einstein-Brillouin-Keller quantizing torus, using a single trajectory and its stability matrix. This is the standard'' method, contrasted in Ref. 1 with our new direct'' method, which does not require the calculation of the stability matrix.
This paper reports a method to build a semiclassical wave function corresponding to an invariant torus satisfying Einstein-Brillouin-Keller quantization conditions. Instead of calculating the stability matrix of the trajectory at each step, as in the standard method of Keller [Ann. Phys. (N.Y.) 4, 180 (1958)] or the modification of Maslov and Fedoriuk [Semiclassical Approximations in Quantum Mechanics (Reidel, Dordrecht, 1981)], we use the actual density of the trajectory, calculated by running the trajectory and counting passages through cells in coordinate space. The method is tested for a system of coupled Morse oscillators, and found to be comparable in accuracy to the standard method. It may be more useful than the standard method for testing ideas for semiclassical quantization in the chaotic regime.
A method for Einstein-Brillouin-Keller (EBK) semiclassical wave functions is applied to the case where the quantizing torus is close to a chain of resonance islands, and standard EBK wave functions are distorted by the resonance islands. The method tested here is a modification of a recent ``direct trajectory'' implementation of the standard EBK method. The phase factor of the semiclassical wave function is obtained exactly as by Yang and Kellman [Phys. Rev. A 62, 022105 (2000); 65, 034103 (2002)], by running a single trajectory on the quantizing torus. However, rather than obtaining the amplitude factor from this trajectory, the method here uses an ensemble of trajectories uniformly distributed in a region enclosing an area \ensuremath{\Elzxh} around the torus. The agreement between the resulting semiclassical wave function and the quantum wave function is excellent.
For two kinetically coupled identical Morse oscillator system, this paper introduced the concept of quantum torus wave function. Quantum torus wave function comes from the decomposition of a quantum wave function of the system. It reflects the symmetry of the quantum wave function. Physically, it corresponds to a quantizing torus or a quantizing cantorus (when it exists) in phase space. The number of nodes in x and y coordinates in the contour plot, being the same as the quantum number in the Einstein-Brillouin-Keller (EBK) quantization conditions, is the natural assignment of the given energy level. When quantizing torus or cantorus is destroyed by weak resonances in phase space, using quantum torus wave function to assign energy level is easier and reasonable. This paper also shows the application of quantum torus wave function in the study of dynamical tunneling.