Date of Award

5-2017

Document Type

Thesis

Degree Name

Master of Science (MS)

College/School

College of Science and Mathematics

Department/Program

Mathematical Sciences

Thesis Sponsor/Dissertation Chair/Project Chair

Ernest Ma

Committee Member

Aihua Li

Committee Member

Bogdan Nita

Abstract

Quantum Mechanics is an axiomatic theory. One of its axioms states that every observable of a physical system is associated with a Hermitian operator allowing the reality of the energy spectrum and a complete set of eigenfunctions. Furthermore, because of the Hermicity imposed on the observable described by the Hamiltonian H, the time evolution of the system is preserved. In recent years, researchers have shown that the Hermicity requirement may be relaxed by a weaker condition described by the combined actions of P and T symmetries on the operator. Under this new regimen some non-Hermitian Hamiltonians have real spectra such as the complex extension of the harmonic oscillator. However, since the inner product of the PT symmetric Hamiltonian is not always positive definite a new inner product is defined with a new symmetry described by a C operator in order to construct a viable quantum mechanics theory. In this thesis a succinct literature review of the PT symmetric non-Hermitian Hamiltonian theory is presented. The Hamiltonian H = p2 + x2(ix)E is discussed under PT transformation and its eigenvalues are calculated under numerical, asymptotic and semiclassical approximations. A two level Hamiltonian is introduced to showcase the main features of the PT symmetric theory. Moreover, the CPT inner product is constructed. The objective of this thesis is to present to the reader a clear introduction of Quantum Mechanics for Non-Hermitian Hamiltonians with PT Symmetries where the principal concepts are explained step by step in order for the material to be accessible to the new reader.

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