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Journal of the Physical Society of Japan

The Coherent-State Representation of the Wigner Distribution Function

Yutaka Kano

Published Jan 1, 1974• 6 citations
Publisher

Abstract

The relation between the Wigner cistribution function and the P-representation of the density matrix for describing a boson system is investigated. The coherent-state representation of the density matrix is introduced for a Bose particle system. Especially the P-representation of the density matrix describing an ideal boson system in thermal equilibrium is obtained. It can be seen that the distribution function of the complex eigenvalues of the destruction operator is non-negative in this particular case. The coherent-state representation of the Wigner function is then obtained. The coherent-state representation of the components in the decomposition of the Wigner function is also found. The London-Placzek formula can be derived by using one of these components.

DOI
10.1143/jpsj.36.39

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The relation between the Wigner cistribution function and the P-representation of the density matrix for describing a boson system is investigated. The coherent-state representation of the density matrix is introduced for a Bose particle system. Especially the P-representation of the density matrix describing an ideal boson system in thermal equilibrium is obtained. It can be seen that the distribution function of the complex eigenvalues of the destruction operator is non-negative in this particular case. The coherent-state representation of the Wigner function is then obtained. The coherent-state representation of the components in the decomposition of the Wigner function is also found. The London-Placzek formula can be derived by using one of these components.
Quantum statistical mechanics of spin systems is formulated by the use of the representation of atomic coherent state of spin, which is specified by a complex number. After introducing a newly defined distribution function of spin, the Bloch equation for the equilibrium density matrix, the von Neumann equation for the nonequilibrium density matrix and the Heisenberg equation of motion are converted into partial differential equations of the type of separating variables respectively. Formal solutions of them are presented. Any operator which is composed of spins can be converted uniquely into a differential operator. Thus we succeed in formulating equilibrium and nonequilibrium quantum statistical mechanics in the complex variable space, which differs from the previous one in the diagonal representation of density matrix. We further introduce the interaction representation for the perturbational expansion. The linear response formula is written down in the present representation.
The first part is a systematical representation of the Wigner quasiprobability in representations by canonical variables and pairs of complex conjugate variables with particular attention to the reconstruction of the density operator from the Wigner quasiprobability to show their equivalence. All formulae are derived from basic postulates of quantum mechanics. Problems of operator orderings are touched. The second part considers the integration over the amplitude in the Wigner quasiprobability and its specialization for coherent states and for even and odd coherent states and provides phase distributions which are presented in graphics that are new in their kind. It is demonstrated that the Heitler number phase uncertainty relation is incorrect. The problem of the transition from quantum to classical distribution functions by ℏ→0 is discussed with the conjecture that the inertial point of the distributions plays a basic role.
The coherent-state representation of the density operator for an ideal Bose gas in a thermal equilibrium is introduced. It is shown that a pure coherent state can be an eigenstate of the Hamiltonian of the non-interacting boson system when the symmetry breaking term is added to it. However, at the limit of the vanishing “classical field” only zero-energy states can be a coherent state. It is also proved that the pure coherent state is the lowest energy state among all the zero-momentum states. The temperature dependence of the coherent state is then discussed. The diminution of the coherent state can clearly be seen as the temperature rises from zero to the critical temperature.