A path integral approach to quantum fluid dynamics: application to double well potential

Abstract

In this work we develop an alternative approach for solution of Quantum Trajectories using the Path Integral method. The state-of-the-art technique in the field is to solve a set of nonlinear, coupled partial differential equations simultaneously. We opt for a fundamentally different route. We first derive a general closed form expression for the Path Integral propagator valid for any general potential as a functional of the corresponding classical path. The method is exact and is applicable in many dimensions as well as multi-particle cases. This, then, is used to compute the Quantum Potential, which, in turn, can generate the Quantum Trajectories. As a model application to illustrate the method, we solve for the double-well potential, both analytically (using a perturbative approach) and numerically (exact solution). Using this we delve into seeking insight into Quantum Tunnelling. The work formally bridges the Path Integral approach with Quantum Fluid Dynamics, an issue of fundamental importance.