Wigner current in multidimensional quantum billiards

TL;DR

This paper derives a simplified expression for the Wigner current of a particle in multidimensional billiards, using a boundary condition method involving convolution, and connects it to an alternative delta-function approach.

Contribution

It introduces a novel convolution-based method to compute the Wigner current in multidimensional billiards, simplifying the boundary condition implementation.

Findings

Derived a simplified surface integral expression for the Wigner current.

Connected the convolution method to an alternative delta-function boundary approach.

Generalized the boundary condition method to multiple dimensions.

Abstract

In the present paper we derive the Wigner current of the particle in a multidimensional billiard -- the compact region of space in which the particle moves freely. The calculation is based on proposed by us previously method of imposing boundary conditions by convolution of the free particle Wigner function with some time independent function, defined by the shape of the billiard. This method allowed to greatly simplify the general expression for the Wigner current, representing its $\mathbf{p}$-component as a surface integral of the product of the shifted particles wave functions. The results are also connected to an alternative approach, which takes into account the boundary conditions by adding the $\propto \delta'(x)$ term to the Hamiltonian. The latter is also generalized to the multidimensional case.