Integration of the Heisenberg equations of motion for quartic potentials.

Abstract

A direct integration of the Heisenberg equations of motion yields expansions of the position and momentum operators x(t) and p(t), each as an infinite series in terms of the initial Weyl-ordered basis set {${\mathit{S}}_{\mathit{m},}$n} formed from x(0) and p(0), with c-number time-dependent coefficients. This method is applied to the problem of tunneling in a symmetric and an asymmetric quartic potential. The expectation values of the position and momentum operators with minimum uncertainty wave packet 〈0\ensuremath{\Vert}x(t)\ensuremath{\Vert}0〉 and 〈0\ensuremath{\Vert}p(t)\ensuremath{\Vert}0〉 can be calculated accurately for a maximum time that is short compared to the period of oscillation of the wave packet. From the result of this calculation, with the help of Prony's method one can determine the level spacings for the low-lying states. In addition, in this formulation the wave packet retains its shape; therefore, one can study the trajectory 〈0\ensuremath{\Vert}x(t)\ensuremath{\Vert}0〉 and 〈0\ensuremath{\Vert}p(t)\ensuremath{\Vert}0〉 as the quantal analogue of the motion of the system in phase space.