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Acta Physica Sinica

The characteristics of a new two mode composite coordinate representation and its application

Shi-Min Xu, Jiang Ji-Jian, Hongqi Li, Xing-Lei Xu

Published Jan 1, 2008• 8 citations
Publisher

Abstract

In order to solve dynamic problems of double non-identical quantum harmonic oscillators with interaction potential and movement coupling, the new two mode composite coordinate representation |η1，η2〉 is proposed by the technique of integration within an ordered product of operators. The two mode squeezing operator U is constructed, and its characteristics are analyzed. Moreover, its application to double non-identical quantum harmonic oscillators with interaction potential and movement coupling is presented for solving some the dynamic problems. Approach of solving some the dynamic problems involving complicated quantum harmonic oscillators is proposed.

DOI
10.7498/aps.57.7430

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In order to solve dynamic problems of double non-identical quantum harmonic oscillators with interaction potential and movement coupling, the new two mode composite coordinate representation |η1，η2〉 is proposed by the technique of integration within an ordered product of operators. The two mode squeezing operator U is constructed, and its characteristics are analyzed. Moreover, its application to double non-identical quantum harmonic oscillators with interaction potential and movement coupling is presented for solving some the dynamic problems. Approach of solving some the dynamic problems involving complicated quantum harmonic oscillators is proposed.
A unitary transformation matrix, n linear-composite coordinate operators and n linear-composite momentum operators are constructed for an n-particle system, and the complete and orthonormal common eigenvectors of the multi-mode linear composite momentum operators are examined by virtue of the technique of integration within an ordered product of operators. The multi-mode linear composite momentum representation is proposed, and its application to a general two-mode forced quantum harmonic oscillator system with kinetic coupling is presented for solving some dynamic problems.
We construct four linear composite operators for a two-particle system and give common eigenvectors of those operators. The technique of integration within an ordered product (IWOP) of operators is employed to prove that those common eigenvectors are complete and orthonormal. Therefore, a new two-mode intermediate momentum-coordinate representation which involves quantum entanglement for a two-particle system is proposed and applied to some two-body dynamic problems. Moreover, the pure-state density matrix |ξ 1 , ξ 2 〉 C,D C,D 〈ξ 1 , ξ 2 | is a Radon transform of Wigner operator.
By virtue of the technique of integration within an ordered product of operators, we derive the normal ordering expansion of a one- and two-mode combination squeezing operator for two harmonic oscillators with coordinate-momentum coupling. It turns out that this squeezing operator just diagonalizes the Hamiltonian H = p 1 2 /2 m 1 + m 1 ω 1 2 x 1 2 /2+ p 2 2 /2 m 2 + m 2 ω 2 2 x 2 2 /2−λ x 2 p 1 , so its ground state is a one- and two-mode combination squeezed state. Quantum fluctuation in the ground state is calculated.
Hamiltonian of two modes coupled harmonic oscillators is general quadric form of generalized coordinates and momenta.After coordinate transformation,it can be expressed as standard quadric form under new bases.Because canonical commutation relation is satisfied among new coordinates,creation operator and annihilation operator are introduced.Thus,coupling between modes is eliminated and Hamiltonian can be simplified as two mode independent harmonic oscillators,problem is solved.This method can be generalized in solving non-identical n modes coupled harmonic oscillators.