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Journal of Physics A Mathematical and Theoretical

The solution of the Schrödinger equation for complex Hamiltonian systems in two dimensions

Fakir Chand, Ram Mehar Singh, Narender Kumar, Shubham Mishra

Published Aug 1, 2007• 16 citations
Publisher

Abstract

We investigate the ground state solutions of the Schrodinger equation for complex (non-Hermitian) Hamiltonian systems in two dimensions within the framework of an extended complex phase-space approach. The eigenvalues and eigenfunctions of some two-dimensional complex potentials are found.
DOI
10.1088/1751-8113/40/33/015

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Abstract In this study, we introduce a two-dimensional complex harmonic oscillator potential with space and time reflection symmetries. The corresponding time-independent Schrödinger equation yields real eigenvalues with complex eigenfunctions. We also construct the coherent state of the system by using a superposition of 12 eigenfunctions. Using the complex correspondence principle for the probability density we investigate the possible modifications in the probability densities due to the non-Hermitian aspect of the Hamiltonian.
Two types of non-Hermitian systems are considered. One of them is both non-Hermitian and non-Linear and an iterative process is used to obtain excited state solutions; the ground state may be solved exactly. The model has been used in many physical systems and the method of calculation uses a simple Hilbert space with a generalised inner product. The second type has a complex term in the Hamiltonian and is a well studied problem in the infinite interval. Here a finite interval is considered and a complete set of eigenfunctions for this interval is used.The relationship between the finite interval states and the infinite interval states is discussed.