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Communications in Theoretical Physics

Unitary Operator and Density Matrix for a Ring of Coupled Oscillator Model

Fan Hong‐Yi, Xu Zhihua, Fan Yue

Published Oct 30, 1998• 2 citations
Publisher

Abstract

A unitary operator which can directly diagonalize the Hamiltonian of a ring of coupled oscillators is found. With use of the technique of integration within an ordered product of operators we see that includes a squeezing transformation. For large it turns out that, except a zero-mode, the ring Hamiltonian has the same spectrum as the one-dimensional monatomic lattice model plus the Born–Von–Karmen boundary condition. The density matrix of this model is calculated by the transformation, which further leads us to derive the heat capacity of the coupled oscillators in the large limit.

DOI
10.1088/0253-6102/30/3/423

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A unitary operator which can directly diagonalize the Hamiltonian of a ring of coupled oscillators is found. With use of the technique of integration within an ordered product of operators we see that includes a squeezing transformation. For large it turns out that, except a zero-mode, the ring Hamiltonian has the same spectrum as the one-dimensional monatomic lattice model plus the Born–Von–Karmen boundary condition. The density matrix of this model is calculated by the transformation, which further leads us to derive the heat capacity of the coupled oscillators in the large limit.
We derive the density matrix ρ(x1, x2; x1', x2', β) for two coupled harmonic oscillators by taking a unitary transformation approach. The corresponding unitary operator U is characteristic of including frequency-jump squeezing transformation. By virtue of the technique of integration within ordered product and the coordinate representation of U we manifestly show that the ground state of the coupled oscillators is a squeezed state. Based on the derived ρ(x1, x2; x1', x2', β) we calculate the average coupling energy.
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In view of a recent paper [F. Michelot, Phys. Rev. A 45, 4271 (1992)] tackling the Hamiltonian of an arbitrary number of harmonically coupled oscillators, we present a coordinate representation of the unitary operator that can diagonalize the Hamiltonian. The normally ordered form of the unitary operator, which manifestly connects two Fock spaces associated with the uncoupled and coupled oscillators, is also derived by virtue of the technique of integration within an ordered product of operators.