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Wigner distribution function for the time-dependent quadratic-Hamiltonian quantum system using the Lewis-Riesenfeld invariant operator. (English) Zbl 1084.81050

The author of this very interesting paper studies quantum properties of the time-dependent quadratic Hamiltonian system (TDQHS for short). The Lewis-Riesenfeld (LR) invariant operator method is applied here. This method can be seen in the works of H. R. Lewis, Jr. (1967) and H. R. Lewis, Jr., W. B. Riesenfeld (1969). The Wigner distribution function in the phase space applied to the general TDQHS has bein proposed by E. Wigner (1932). The Hamiltonian of the most general TDQHS has the form \[ \hat H (\hat q,\hat p, t)=A(t)\hat p^2+B(t)(\hat q \hat p+\hat p\hat q)+ C(t)\hat q^2+ D(t)\hat q+E(t)\hat p+F(t), \] where \(A,B,C,D,E,F\) are time dependent differentiable functions and \(A(t)\neq 0\). The functions \(D,E\) are related to the driving force of the system and \(F\) is related to the reference point of potential energy. The author introduce LR invariant operator in the form \[ \hat I=\hbar \Omega \biggl(\hat a^{† }\hat a+{1\over 2} \biggr), \] where \(\hat a\) is the annihilation operator and \(\hat a^{† }\) is its adjoint, creation operator. They satisfy the relation \([\hat a,\hat a^{† }]=1\), where \([\cdot ,\cdot ]\) is the commutator. The Wigner distribution function of the system under consideration in Fock state, coherent state, squeezed state and thermal state is shown. The author applies the theory to the one-dimensional motion of a Brownian particle and to the driven oscillator with strongly pulsating mass.

MSC:

81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81R30 Coherent states
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