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arXiv:2304.14928 · Journal of Physics A Mathematical and Theoretical

Wigner function dynamics with boundaries expressed as convolution

S. S. Seidov

Published Jul 12, 2023• 2 citations
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Abstract

Abstract In the present paper a method of finding the dynamics of the Wigner function of a particle in an infinite quantum well is developed. Starting with the problem of a reflection from an impenetrable wall, the obtained solution is then generalized to the case of a particle confined in an infinite well in arbitrary dimensions. It is known, that boundary value problems in the phase space formulation of the quantum mechanics are surprisingly tricky. The complications arise from nonlocality of the expression involved in calculation of the Wigner function. Several ways of treating such problems were proposed. They are rather complicated and even exotic, involving, for example, corrections to the kinetic energy proportional to the derivatives of the Dirac delta–function. The presented in the manuscript approach is simpler both from analytical point of view and regarding numerical calculation. The solution is brought to a form of convolution of the free particle solution with some function, defined by the shape of the well. This procedure requires calculation of an integral, which can be done by developed analytical and numerical methods.

DOI
10.1088/1751-8121/ace6e5

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Abstract In the present paper a method of finding the dynamics of the Wigner function of a particle in an infinite quantum well is developed. Starting with the problem of a reflection from an impenetrable wall, the obtained solution is then generalized to the case of a particle confined in an infinite well in arbitrary dimensions. It is known, that boundary value problems in the phase space formulation of the quantum mechanics are surprisingly tricky. The complications arise from nonlocality of the expression involved in calculation of the Wigner function. Several ways of treating such problems were proposed. They are rather complicated and even exotic, involving, for example, corrections to the kinetic energy proportional to the derivatives of the Dirac delta–function. The presented in the manuscript approach is simpler both from analytical point of view and regarding numerical calculation. The solution is brought to a form of convolution of the free particle solution with some function, defined by the shape of the well. This procedure requires calculation of an integral, which can be done by developed analytical and numerical methods.
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 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 ∝ δ ′ ( x ) term to the Hamiltonian. The latter is also generalized to the multidimensional case.
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 explore the semi-classical structure of the Wigner functions Ψ(q,p) representing bound energy eigenstates | Ψ 〉 for systems with fdegrees of freedom. If the classical motion is integrable, the classical limit of Ψ is a delta function on the f-dimensional torus to which classical trajectories corresponding to |Ψ〉 are confined in the 2f-dimensional phase space. In the semi-classical limit of Ψ (ℏ small but not zero) the delta function softens to a peak of order Ψ−f and the torus develops fringes of a characteristic ‘Airy’ form. Away from the torus,Ψ can have semi-classical singularities that are not delta functions; these are discussed (in full detail when f = 1) using Thom's theory of catastrophes. Brief consideration is given to problems raised when is calculated in a representation based on operators derived from angle coordinates and their conjugate momenta. When Ψ the classical motion is non-integrable, the phase space is not filled with tori and existing semi-classical methods fail. We conjecture that (a) For a given value of non-integrability parameter ⋲,the system passes through three semi-classical régimes as ℏ diminishes. (b) For states |Ψ〉 associated with regions in phase space filled with irregular trajectories, Ψ will be a random function confined near that region of the ‘energy shell’ explored by these trajectories (this region has more thanks dimensions). (c) For ⋲ ≠ 0, ℏ blurs the infinitely fine classical path structure, in contrast to the integrable case ⋲ = 0, where ℏ imposes oscillatory quantum detail on a smooth classical path structure.
The space-fractional Schrödinger equation for a local potential is difficult to solve because the fractional Riesz operator in it is nonlocal. In fractional quantum mechanics, the infinite wall is a basic problem and has not been solved yet. In this paper, we consider a free particle in an infinite wall region. Making use of the Lévy path integral method, we derive the Lévy path integral amplitude of the particle and take advantage of it to get the solution of the fractional Schrödinger equation for an infinite wall. Then, the infinite wall potential with a delta-function perturbation is also studied. According to Fox’s H-function, we get an equation of bound state energies of a free particle moving in an infinite wall region perturbed by the δ-function for two situations, E < 0 and E > 0. We give a transcendental equation that determines energy levels when E < 0 and prove that there is no bound states when E > 0. The asymptotic behaviors of the center of the delta function approaching to the origin and the infinite are also shown. The results of this paper include special cases in standard quantum mechanics.
In deformation quantization (a.k.a. the Wigner–Weyl–Moyal 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
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.