Quantum Index Theory: Relations between Quantum Phase and Quantum Number

摘要

This paper extends the index theory originally valid only for classical nonlinear systems to quantum mechanical systems. Based on the Hamilton quantum mechanics, probabilistic behavior described by a wavefunction ψ(x) can be represented equivalently by nonlinear, complex-valued Hamilton equations of motion, from which we can identify fixed points, evaluate their indices and derive quantum index theory. Three quantum indices are discovered for quantum systems. The momentum index n_p counting the net change of the momentum phase is the counterpart of the classical index and is an indication of whether a quantum trajectory is closed. The wavefunction index n_ψ counting the net change of the wavefunction phase, called Berry's geometrical phase, is responsible for the Aharonov-Bohm effect and for the Wilson-Sommerfeld quantization law. The combined index n_ψ' = n_p + n_ψ counts the net phase change of the wavefunction derivative ψ'. Apart from their geometric meanings, it is pointed out here that the three indices, n_p, n_ψ and n_ψ', act as quantum numbers to represent quantization levels, respectively, for the kinetic energy E_k, the quantum potential energy Q and the combined energy E_k + Q.