PT-Symmetric Hamiltonian H=p^2-(ix)^N: Welcome to the Complex World

The Hermiticity from conventional quantum mechanics guarantees that the energy spectrum is real. However, if replace this mathematical condition by the physically transparent condition of parity-time reflection symmetry (PT-symmetry), the non-Hermitian Hamiltonian still guarantees that its entire energy spectrum is real if the Hamiltonian has unbroken PT-symmetry. If its PT-symmetry is broken, then two cases can happen - its entire energy spectrum is complex for the first case, or a finite number of real energy levels can still be obtained for the second case. This was “officially” discovered since 1998. After that, the developments in PT-symmetric quantum theory rapidly grew in the last 15 years - with more than 20 international conferences and symposia, and over 2000 research papers about PT-symmetry already published. Furthermore, at least 50 experiments to observe PT-symmetric system were published during the last 10 years. Those experiments told us that it was possible to experimentally measure complex eigenvalue and observe broken and unbroken PT-symmetry. Admittedly, PT-symmetric quantum theory is a young and new field - currently, still not many professors and researchers familiar with this subject. That is why this thesis comes in, and tries to serve a role to introduce this subject to wide audience from students to professors. In this thesis, the energy spectrum from the PT-symmetric Hamiltonian H = p^2 −(ix)^N with x ∈ C, N ∈ R and N ≥ 1 was studied in detail by using numerical and WKB approximation. What the corresponding eigenfunctions look like were also examined in numerical way. Lastly, a few interesting and weird phenomena from PT-symmetric non-relativistic classical mechanics were explored in brief. We hope that this study could not only demystify but also help people appreciate many aspects of PT-symmetry.