Some physical problems of pseudo-hermitian hamiltonians

Abstract

In this thesis we have discussed on one of the recent topics in physics, in particular in quantum mechanics, which is "the non-Hermitian Hamiltonians", and the conditions that make the spectra of these Hamiltonians real. Initially, the Hermeticity was a necessary and su cient condition for the reality of the Hamiltonian spectrum. Then a new quantum theory called the "PT -symmetry" started to emerge, this theory was developed in 1998 by Carl Bender and Stefan Boettcher, where they revealed the existence of a class of non-Hermitian Hamiltonians with real spectra. These Hamiltonians are invariant under the transformation of PT -symmetry, where P is the parity operator and T is the time reversal operator. A few years later, another alternative approach was developed in 2002 by Mostafazadeh who works on pseudo-Hermitian Hamiltonians and he showed that every Hamiltonian with a real spectrum is pseudo-Hermitian. We have found from an application on two examples of a pseudo-Hermitian Hamiltonians " shifted harmonic oscillator " and " cubic anharmonic oscillator " that the energy spectrum of these Hamiltonians are real and positive. Keywords: Hermiticity, PT -symmetry, Pseudo-Hermiticity, Quasi-Hermiticity.