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Journal of Physics A Mathematical and General

Classical limit of the harmonic oscillator Wigner functions in the Bargmann representation

Nicolò Ripamonti

Published Aug 21, 1996• 16 citations
Publisher

Abstract

Writing the Wigner functions of any pair of harmonic oscillator eigenstates in the Bargmann representation, a direct and detailed proof is given of their convergence (in the sense of distributions) to at the classical limit , , , m-n fixed, .
DOI
10.1088/0305-4470/29/16/031

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The Wigner function is a quantum analogue of the classical joined distribution of position and momentum. As such is should be a good tool to study quantum-classical correspondence. In this paper, the classical limit of the Wigner function is shown using the quantum harmonic oscillator as an example. The Wigner function is found exactly for all states. The semi-classical wavefunctions for highly excited states are used as the approach to the classical limit. Therefore, one can found the classical limit of the Wigner function for highly excited states and shown that it gives the classical microcanonical ensemble.
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For a quantum observable AℏA_\hbarAℏ​ depending on a parameter ℏ\hbarℏ we define the notion ``AℏA_\hbarAℏ​ converges in the classical limit''. The limit is a function on phase space. Convergence is in norm in the sense that Aℏ→0A_\hbar\to0Aℏ​→0 is equivalent with ∥Aℏ∥→0\Vert A_\hbar\Vert\to0∥Aℏ​∥→0. The ℏ\hbarℏ-wise product of convergent observables converges to the product of the limiting phase space functions. ℏ−1\hbar^{-1}ℏ−1 times the commutator of suitable observables converges to the Poisson bracket of the limits. For a large class of convergent Hamiltonians the ℏ\hbarℏ-wise action of the corresponding dynamics converges to the classical Hamiltonian dynamics. The connections with earlier approaches, based on the WKB method, or on Wigner distribution functions, or on the limits of coherent states are reviewed.
By using the localized character of canonical coherent states, we give a straightforward derivation of the Bargmann integral representation of the Wigner function ( W ). A non-integral representation is presented in terms of a quadratic form W ∝ V † F V , where F is a self-adjoint matrix whose entries are tabulated functions and V is a vector depending, in a simple recursive way, on the derivatives of the Bargmann function. Such a representation may be of use in numerical computations. We discuss a relation involving the geometry of the Wigner function and the spatial uncertainty of the coherent state basis we use to represent it.