The C(t)PT-inner Product for Time-dependent non-Hermitian Quantum Systems

Abstract

In this thesis, we have studied the analytical solutions ofthe Schrödinger equationfor a class of explicit time dependent non-Hermitian systems. In the first chapter we introduced the basic concepts used in quantum theory for non-Hermitian systems, such PT-symmetry, PT and CPT-inner-products and pseudo-Hermiticity. The second chapter is dedicated to the Lewis-Riesenfeld invariant method for solving the Schrödinger equation forexplicitly time dependent Hermitian and non-Hermitian Hamiltonians. In the last chapter, we have used a unitary transformation F(t) that reduces the non-Hermitian Hamiltonian H(t) to a time-independent PT-symmetric one, and thus the analytical solution of the Schrödinger equation of the initial system is easily obtained. Then, we defined a new C(t)PT-inner product and showed that the evolution preserves it, where C(t)=F⁺(t)CF(t). Moreover, we proved that the expectation value of the time-dependent non-Hermitian Hamiltonian H(t) is real in the C(t)PT-normed states since the transformation F(t) is unitary and [P,F(t)]=0. As an illustration, we have study a class of quantum time-dependent mass oscillators with a complex linear driving force. The expectation value of the Hamiltonian, the uncertainty relation and the probability density have also been calculated. The results of this chapter constitute the main results of this thesis.