Damping in Schrödinger’s equation for macroscopic variables

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Abstract
A simple damping term is proposed, to be added to the Hamiltonian in Schrödinger’s equation. It is shown that this term removes energy without altering the wave-function normalization. It is also demonstrated that the dynamics of the damped wave function are in reasonable agreement with (in one dimension): classical motion in a harmonic potential; tunneling in a cubic potential in a Caldeira-Leggett oscillator bath; and spreading in a flat potential, also in the oscillator-bath model. The use of this new damping term should allow direct simulation in the time domain of several problems, including the dynamic behavior of nanometer scale Josephson junctions.