Wigner Functions with Boundaries

We consider a general Wigner function for a particle confined to a finite interval and subject to Dirichlet boundary conditions. We derive the boundary corrections to the ”star-genvalue” equation and to the time evolution equation. These corrections can be cast in the form of a boundary potential contributing to the total Hamiltonian. Moreover, in this context, it is the boundary potential that is responsible for the discretization of the energy levels as is illustrated by the example of the free particle in the infinite potential well. Finally, using the Weyl map we show that the boundary potentials can be obtained directly from standard operator quantum mechanics.