INTEGRATION OF THE HEISENBERG EQUATION OF MOTION FOR QUANTUM TUNNELING

A method of integration of the Heisenberg equation of motion is proposed in which the time evolution of the basis set {T(m,n)(t)} of the Weyl-ordered operators introduced by Bender and Dunne [Phys. Rev. D 40, 3504 (1989)] is obtained from the Taylor expansion and is expressible in terms of the initial {T(m,n)(0)}'s. In the absence of damping forces, the constant values of the energy and the position-momentum commutation relation are used to check the accuracy of the integration. This method is applied to obtain the mean position and velocity of the particle as a function of time as well as the dwell time of the particle inside the barrier. In the example that is considered here, the potential is assumed to be the sum of a harmonic and a cubic term, and the calculation is done with and without dissipative coupling.