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

Date created: 
Stokes wedge

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.

Document type: 
This thesis may be printed or downloaded for non-commercial research and scholarly purposes. Copyright remains with the author.
Senior supervisor: 
Andrei Frolov
Science: Department of Physics
Thesis type: 
(Thesis) M.Sc.