Semiclassical quantization of non-Hermitian 2-D systems: Classical (Lie transform) Perturbation theory

Asiri Nanayakkara

doi:10.4038/jnsfsr.v37i2.1066

Abstract

Both real and complex semiclassical eigen energies of two dimensional non-Hermitian Hamiltonian systems are obtained by classical (Lie transform) perturbation theory requiring the action variables I 1 and I 2 to satisfy the quantization condition I 1 =( n 1 +(1/2)) ℏ and I 2 =( n 2 +(1/2)) ℏ respectively where n 1 , n 2 are integers. Classical perturbation theory with Lie transform makes classical trajectories, which are non-periodic or non-quasi-periodic, periodic. It was observed that this method produces accurate eigen energies even when classical trajectories are not periodic or quasi-periodic. Eigen energies obtained by classical perturbation theory are compared with the same, determined by Rayleigh-Schroedinger perturbation theory

Highlights

Real non-separable multidimensional Harmonic Oscillator Systems have been widely used in the past for investigating classical chaos and its manifestation in quantum mechanics in Hamiltonian systems[1,2,3,4,5,6,7,8,9,10,11,12]
In the first place they may expose important clues hidden in the correspondence between classical and quantum mechanics, especially when classical motion becomes chaotic or PT symmetry of quantum states are spontaneously broken
We studied two 2D non Hermitian complex

Summary

INTRODUCTION

Real non-separable multidimensional Harmonic Oscillator Systems have been widely used in the past for investigating classical chaos and its manifestation in quantum mechanics in Hamiltonian systems[1,2,3,4,5,6,7,8,9,10,11,12]. Non-Hermitian multidimensional Harmonic oscillator systems do not provide such a practical insight yet, semiclassical investigations of nonHermitian complex Hamiltonian systems are important from both the fundamental and practical point of view. Classical perturbation theory based on Lie transforms has been used to obtain constants of motion of 2-D coupled Harmonic oscillator systems[14,15] and to determine semiclassical eigen energies and linking quantum avoided crossing with classical chaos in real Hermitian systems in the past[16,17]. We use Lie transforms to investigate the classical frequencies and real and complex semiclassical eigen values of 2-D non Hermitian Hamiltonian systems analytically.

H H0 HH1

Classical frequencies

Semiclassical eigen values

Concluding remarks