Quantum dynamics of a one degree-of-freedom Hamiltonian saddle-node bifurcation

TL;DR

This paper investigates how the depth of a potential well influences quantum dynamics in a one-degree-of-freedom Hamiltonian undergoing a saddle-node bifurcation, using spectral and phase space analysis.

Contribution

It provides a detailed analysis of quantum effects near a saddle-node bifurcation by computing eigenvalues, eigenvectors, and phase space distributions for varying potential depths.

Findings

Quantum energy levels vary with potential well depth.

Position uncertainties are affected by bifurcation parameters.

Wigner functions reveal phase space changes during bifurcation.

Abstract

In this paper, we study the quantum dynamics of a one degree-of-freedom (DOF) Hamiltonian that is a normal form for a saddle node bifurcation of equilibrium points in phase space. The Hamiltonian has the form of the sum of kinetic energy and potential energy. The bifurcation parameter is in the potential energy function and its effect on the potential energy is to vary the depth of the potential well. The main focus is to evaluate the effect of the depth of the well on the quantum dynamics. This evaluation is carried out through the computation of energy eigenvalues and eigenvectors of the time-independent Schr\"odinger equations, expectation values and position uncertainties for position coordinate, and Wigner functions.