Transport Theory and Statistical Physics

The Wigner transform and the Wigner-poisson system

Paul F. Zweifel

Published Aug 1, 1993• 22 citations

Abstract

The Wigner method in (quantum) statistical mechanics was introduced in 19321 as an alternative to the standard formulation2 which involves computation of eigenvalues and eigenvectors (and traces) of linear operators in Hilbert spaces. The Wigner approach uses only integrations over phase space –R6N in the case of an N-particle system–of ordinary functions f:R6n → C.

DOI: 10.1080/00411459308203824

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Transport Theory and Statistical Physics

The Wigner transform and the Wigner-poisson system

Paul F. Zweifel

Published Aug 1, 1993• 22 citations

Publisher

Abstract

DOI: 10.1080/00411459308203824

Related papers

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Physics of Particles and NucleiMay 1, 20211 citation

Abstract Two methods for fast Fourier transforms are used in a quantum context. The first method is for systems with dimension of the Hilbert space

D=d^n

D = d n with d an odd integer, and is inspired by the Cooley-Tukey formalism. The ‘large Fourier transform’ is expressed as a sequence of n ‘small Fourier transforms’ (together with some other transforms) in quantum systems with d-dimensional Hilbert space. Limitations of the method are discussed. In some special cases, the n Fourier transforms can be performed in parallel. The second method is for systems with dimension of the Hilbert space

D=d_0...d_{n-1}

D = d 0 . . . d n - 1 with

d_0,...,d_{n-1}

d 0 , . . . , d n - 1 odd integers coprime to each other. It is inspired by the Good formalism, which in turn is based on the Chinese reminder theorem. In this case also the ‘large Fourier transform’ is expressed as a sequence of n ‘small Fourier transforms’ (that involve some constants related to the number theory that describes the formalism). The ‘small Fourier transforms’ can be performed in a classical computer or in a quantum computer (in which case we have the additional well known advantages of quantum Fourier transform circuits). In the case that the small Fourier transforms are performed with a classical computer, complexity arguments for both methods show the reduction in computational time from

\mathcal{O}(D^2)

O ( D 2 ) to

\mathcal{O}(D\log D)

O ( D log D ) . The second method is also used for the fast calculation of Wigner and Weyl functions, in quantum systems with large finite dimension of the Hilbert space.