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.
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.
The Weyl–Wigner formulation of quantum confined systems poses several interesting problems. The energy stargenvalue equation as well as the dynamical equation do not display the expected solutions. In this paper, we review some previous results on the subject and add some new contributions. We reformulate the confined energy eigenvalue equation by adding to the Hamiltonian a new (distributional) boundary potential. The new Hamiltonian is proved to be globally defined and self-adjoint. Moreover, it yields the correct Weyl–Wigner formulation of the confined system.
The Weyl-Wigner formulation of quantum confined systems poses several interesting problems. The energy stargenvalue equation, as well as the dynamical equation does not display the expected solutions. In this paper we review some previous results in the subject and add some new contributions. We reformulate the confined energy eigenvalue equation by adding to the Hamiltonian a new (distributional) boundary potential. The new Hamiltonian is proved to be globally defined and self-adjoint. Moreover, it yields the correct Weyl-Wigner formulation of the confined system.
Quantum mechanics in phase space (or deformation quantization) appears to fail as an autonomous quantum method when infinite potential walls are present. The stationary physical Wigner functions do not satisfy the normal eigen equations, the *-eigen equations, unless an ad hoc boundary potential is added [Dias-Prata]. Alternatively, they satisfy a different, higher-order, ``*-eigen-* equation'', locally, i.e. away from the walls [Kryukov-Walton]. Here we show that this substitute equation can be written in a very simple form, even in the presence of an additional, arbitrary, but regular potential. The more general applicability of the -eigen- equation is then demonstrated. First, using an idea from [Fairlie-Manogue], we extend it to a dynamical equation describing time evolution. We then show that also for general contact interactions, the -eigen- equation is satisfied locally. Specifically, we treat the most general possible (Robin) boundary conditions at an infinite wall, general one-dimensional point interactions, and a finite potential jump. Finally, we examine a smooth potential, that has simple but different expressions for x positive and negative. We find that the -eigen- equation is again satisfied locally. It seems, therefore, that the -eigen- equation is generally relevant to the matching of Wigner functions; it can be solved piece-wise and its solutions then matched.
We examine the deformation quantization of a single particle moving in one dimension (i) in the presence of an infinite potential wall, (ii) confined by an infinite square well, and (iii) bound by a delta function potential energy. In deformation quantization, considered as an autonomous formulation of quantum mechanics, the Wigner function of stationary states must be found by solving the so-called ∗-genvalue (``stargenvalue'') equation for the Hamiltonian. For the cases considered here, this pseudo-differential equation is difficult to solve directly, without an ad hoc modification of the potential. Here we treat the infinite wall as the limit of a solvable exponential potential. Before the limit is taken, the corresponding ∗-genvalue equation involves the Wigner function at momenta translated by imaginary amounts. We show that it can be converted to a partial differential equation, however, with a well-defined limit. We demonstrate that the Wigner functions calculated from the standard Schr\\"odinger wave functions satisfy the resulting new equation. Finally, we show how our results may be adapted to allow for the presence of another, non-singular part in the potential.
Abstract We consider a non-relativistic particle in a one-dimensional box with all possible quantum boundary conditions that make the kinetic-energy operator self-adjoint. We determine the Wigner functions of the corresponding eigenfunctions and analyze in detail their classical limit, governed by their behavior in the high-energy regime. We show that the quantum boundary conditions split into two classes: all local and regular boundary conditions collapse to the same classical boundary condition, while a dependence on singular non-local boundary conditions persists in the classical limit.
In deformation quantization (a.k.a. the WignerWeylMoyal formulation of quantum mechanics), we consider a single quantum particle moving freely in one dimension, except for the presence of one infinite potential wall. Dias and Prata pointed out that, surprisingly, its stationary-state Wigner function does not obey the naive equation of motion, i.e., the naive stargenvalue (*-genvalue) equation. We review our recent work on this problem that treats the infinite wall as the limit of a Liouville potential. Also included are some new results: (i) we show explicitly that the Wigner-Weyl transform of the usual density matrix is the physical solution, (ii) we prove that an effective-mass treatment of the problem is equivalent to the Liouville one, and (iii) we point out that self-adjointness of the operator Hamiltonian requires a boundary potential, but one apparently different from that proposed by Dias and Prata. PACS Nos.: 03.65.w, 03.65.Ca, 03.65.Ge