Heisenberg’s equations of motion for dissipative tunneling

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Abstract
The quantum-mechanical problem of tunneling of a particle coupled to a heat bath that consists of a large number of harmonic oscillators is studied in this paper. Here the equation of motion for the particle which is a nonlinear integro-differential equation is found by eliminating the degrees of freedom of the oscillators. The nonlinear operator equation may be solved by noting that the position operator at the time t can be expanded as a power series in terms of the initial position and momentum with time-dependent c-number coefficients. These coefficients also satisfy nonlinear integro-differential equations. For the problem of dissipative tunneling of a particle in a double-well potential, the first few terms of expansion have been calculated numerically. The expansion parameter in this formulation is a small dimensionless parameter which is inversely proportional to the product of the width and the square root of the height of the barrier.