Approximation of a Phase Space Operator for the Numerical Solution of the Wigner Equation

The application of the Wigner-Veyl transform for the analysis of quantum transport effects leads to several difficulties and to criticism. One of the difficulties is the numerical discretization of the Wigner transport equations, after the Wigner-Veyl transform is applied onto the statistical density matrix as a part of the quantum Liouville equation. The conventional discretization methods are based on a finite-difference approximation of the resulting Wigner transport equation applying upwind difference schemes or hybrid difference schemes. The results obtained do not match with the statistical density matrix based on the solution of the Schrödinger equation. The contribution addresses the difficulties when the conventional discretization approaches are applied. The difficulties arising from this approach are discussed, and the prerequisites for an improved approach are defined. On the basis of this discussion and the conclusions obtained, a new approach is proposed. This approach is based on an evaluation of the characteristics of the Wigner transport equation in the eigenvalue domain. The new approach is discussed and is applied onto a standard resonant tunneling diode structure. The numerical results are compared with the results from the application of the conventional techniques.