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Journal of Mathematical Physics

A new unitary operator for a pair of coupled oscillators

Hong-Yi Fan

Published Jun 1, 1989• 7 citations
Publisher

Abstract

Using the IWOP (integration within ordered product) technique, a new unitary operator is found that is useful in diagonizing the Hamiltonian of a pair of coupled harmonic oscillators. The coordinate representation of the unitary operator is presented, and is well applied to obtain the wave function of energy eigenstate of the coupled oscillator.

DOI
10.1063/1.528303

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Using the IWOP (integration within ordered product) technique, a new unitary operator is found that is useful in diagonizing the Hamiltonian of a pair of coupled harmonic oscillators. The coordinate representation of the unitary operator is presented, and is well applied to obtain the wave function of energy eigenstate of the coupled oscillator.
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.
A two-mode integral form projection unitary operator is introduced. Using the technique of integration within an ordered product （IWOP） of operators, the transformation property of the integral form projection operator is studied, and its normal ordered expression is derived. Then the Hamiltonian of two-mode quantum harmonic oscillator system with coordinate-momentum coupling is diagonalized by virtue of the integral form projection operator, there by the eigenenergy and eigenfunction of the system are exactly solved. Lastly, a special case is discussed.
A double-mode integral form projection unitary operator is introduced. Using the technique of integration within an ordered product (IWOP) of operators, the transformation character of the integral form projection operator is studied. The Hamiltonian of double-mode quantum harmonic oscillator system with kinetic coupling is diagonalized by virtue of the integral form projection operator, and then the eigenenergy and eigenfunction of the system are accurately solved. Finally the particular case is discussed.