Quantum-classical transition equation with complex trajectories

Abstract

A quantum-classical transition equation in complex space is derived in the framework of the complex quantum Hamilton–Jacobi formalism. The transition equation is obtained by subtracting a complex-valued quantum potential term from the complex-extended time-dependent Schrödinger equation (TDSE). It is shown that the nonlinear transition equation is equivalent to a linear scaled TDSE with a rescaled Planck's constant. Employing the quantum momentum function defined by the gradient of the complex action, we can analyze the quantum-classical transition of physical systems using complex transition trajectories. Complex transition trajectories are presented for the free Gaussian wave packet, the harmonic oscillator, and the Morse potential. This study demonstrates that the transition equation provides a continuous description for the quantum-classical transition of physical systems in complex space.