Wigner functions with boundaries

Abstract

We consider the general Wigner function for a particle confined to a finite interval and subject to Dirichlet boundary conditions. We derive the boundary corrections to the "stargenvalue" equation and to the time evolution equation. These corrections can be cast in the form of a boundary potential contributing to the total Hamiltonian which together with a subsidiary boundary condition is responsible for the discretization of the energy levels. We show that a completely analogous formulation (in terms of boundary potentials) is also possible in standard operator quantum mechanics and that the Wigner and the operator formulations are also in one-to-one correspondence in the confined case. In particular, we extend Baker's converse construction to bounded systems. Finally, we elaborate on the applications of the formalism to the subject of Wigner trajectories, namely in the context of collision processes and quantum systems displaying chaotic behavior in the classical limit.

Deformation Quantization of Confined Systems

“…The translation of this equation to the Weyl-Wigner formulation yields a new stargenvalue equation displaying the correct solutions. This problem has been studied before [10]- [13]. Here we reformulate it on simpler grounds and add some new contributions.…”

“…We can solve the new eigenvalue equation (13), both analytically and numerically (with a suitable smooth regularization of the Dirac delta) [10], for Dirichlet boundary conditions and show that all its solutions are of the form (6).…”

“…Quite unexpectedly the deformation approach displays several problems in the case of confined systems [10]- [13]. The dynamical and the eigenvalue equations do not yield the correct solutions.…”

Deformation Quantization and Wigner Functions

“…Consider a particle of mass m in the infinite potential well [15], confined to the interval a ≤ x ≤ b. For simplicity we shall assume Dirichlet boundary conditions.…”

One-dimensional Schrödinger operators with singular potentials: A Schwartz distributional formulation

“…These objects yield exactly solvable models [2,4,5,6,8,11,12,26,38,45] and have been widely used in applications in quantum mechanics (e.g. in models of low-energy scattering [3,13,14,35] and quantum systems with boundaries [22,23,24,27,32]), condensed matter physics [10,17,25] and, more recently, on the approximation of thin quantum waveguides by quantum graphs [1,15,16,20].…”