Time reversal symmetry and the geometric phase
Degree GrantorUniversity of Canterbury
Degree NameDoctor of Philosophy
This thesis examines the quantum-mechanical geometric phase with a view toward time reversal symmetry considerations. The idea of time reversal in quantum mechanics is investigated, disagreements and inconsistencies in the literature are examined, and the action of the time reversal operator is extended to time-dependent Hamiltonians. With this background, and using a definition of time reversal symmetry based on the evolution operator, I demonstrate that the existence of a non-zero geometric phase can in all cases be attributed to a breakdown of time reversal symmetry in some form. This result holds for both adiabatic and nonadiabatic evolutions, and for arbitrary dimensional parameter spaces. I explore the role of the geometric phase in a two-level Kramers system described by a parameter-dependent Hamiltonian such that the two levels can become degenerate for some value of the parameters, and discuss, from a mathematical point of view, the monopole geometric potential that results. I then extend this analysis by considering a pair of Kramers doublets, each doublet degenerate due to time reversal symmetry, where the parameters can be chosen so that each of the pair of doublets becomes degenerate with the other. I find the explicit functional forms for the two resulting nonabelian geometric gauge potentials and show that they can be identified exactly with the only two gauge-inequivalent SU(2) monopole potentials of Yang. Furthermore, following a conformal transformation these potentials can be mapped to those of the SU(2) instanton/anti-instanton pair. Finally I examine the relevance of the geometric phase to the molecular physics of time-odd systems. Time-odd coupling in molecular physics is a much under-studied area, with many potentially interesting results. Specifically I study time-odd coupling in Jahn-Teller systems under the Born-Oppenheimer approximation, where the electronic position states are coupled to the lattice momentum rather than the usual time-even lattice position. As an example I solve the E ⊗ (b₁ ⊕ b₂ ⊕ a₂) Jahn-Teller system exactly, showing that once again monopole-like geometric potentials arise, and comment on how this affects the angular momentum of the lattice subsystem.