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Quantum Torus Wave Function

Shuang-Bo Yang

Published Sep 20, 2010• 0 citations
OpenAlex

Abstract

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.

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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.
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.
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.
The first application of quantum ideas to a system with a nonuniform potential energy is to the harmonic oscillator: a pendulum, or a mass on a spring. The result is a quantization of energy with equally spaced energy levels. The wave function is localized in the potential well but can do quantum tunneling beyond the turning points into classically forbidden regions. A wave function can be a negative quantity, but the related position probability is always positive. The wave function shows the particle is not in any single place; in a sense it is in many places simultaneously. The fact that the lowest energy quantum state does not vanish is an example of zero-point energy. States of very high energy show a strong analogy to classical oscillator behavior, an example of the Bohr correspondence principle.
We study theoretically two consequences of the mixed classical phase space for three repulsively interacting bosonic particles in a circular trap. First, we show that the energy levels of the corresponding quantum system are well described by a Berry-Robnik distribution. Second, we identify stationary quantum states whose density is enhanced along the stable classical periodic trajectories, and calculate their energies and wave functions using the semiclassical Einstein-Brillouin-Keller (EBK) theory. Our EBK results are in excellent agreement with our full-fledged finite-element numerics. We discuss the impact of discrete symmetries, including bosonic exchange symmetry, on these classically localized states. They are within experimental reach, and occur in the same range of energies as the quantum scar reported in our previous work [Phys. Rev. A 107, 022217 (2023)].