Wigner function dynamics with boundaries expressed as convolution
- URL: http://arxiv.org/abs/2304.14928v3
- Date: Fri, 21 Jul 2023 12:26:44 GMT
- Title: Wigner function dynamics with boundaries expressed as convolution
- Authors: S. S. Seidov
- Abstract summary: A method of finding the dynamics of the Wigner function of a particle in an infinite quantum well is developed.
The solution is brought to a form of convolution of the free particle solution with some function, defined by the shape of the well.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- 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.
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