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Chinese Physics B

New representation of the multimode phase shifting operator and its application

Shuai Wang, Jiang Ji-Jian, Shimin Xu, Hongqi Li

Published Jan 1, 2010• 7 citations
Publisher

Abstract

Based on the rotation transformation in phase space and the technique of integration within an ordered product of operators, the coherent state representation of the multimode phase shifting operator and one of its new applications in quantum mechanics are given. It is proved that the coherent state is a natural language for describing the phase shifting operator or multimode phase shifting operator. The multimode phase shifting operator is also a useful tool to solve the dynamic problems of the multimode coordinate–momentum coupled harmonic oscillators. The exact energy spectra and eigenstates of such multimode coupled harmonic oscillators can be easily obtained by using the multimode phase shifting operator.

DOI
10.1088/1674-1056/19/1/014208

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Based on the rotation transformation in phase space and the technique of integration within an ordered product of operators, the coherent state representation of the multimode phase shifting operator and one of its new applications in quantum mechanics are given. It is proved that the coherent state is a natural language for describing the phase shifting operator or multimode phase shifting operator. The multimode phase shifting operator is also a useful tool to solve the dynamic problems of the multimode coordinate–momentum coupled harmonic oscillators. The exact energy spectra and eigenstates of such multimode coupled harmonic oscillators can be easily obtained by using the multimode phase shifting operator.
Based on the rotation transformation in the phase space and the technique of integration within an ordered product of operators,the coherent state representations of the rotation operator in quantum phase space,the Fourier transform operator and the parity operator are derived. Then the three-mode rotation operator is introduced.By the three-mode rotation operator,the Hamilton of the three-dimensional coordinate-momentum coupled harmonic oscillators is simply decoupled. Then its exact energy spectrums and eigenstates are obtained.
By the properties of the coherent state and the technique of performing integrations within ordered products as in Ref. [1], we derive some new useful operator ordering identities. Their applications, especially in studying the Boglyubov transformations both in one and two dimensional cases, are presented. The explicit non-unitary operators which turn the coordinate or momentum eigen-state to the corresponding coherent state are also given. Moreover, by using the Schwinger angular momentum theory and the technique mentioned above we find the normal product form of the rotation operator, with which the coefficient of the rotation D-matrix can be easily obtained.
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.
Classical linear transformations in coordinate-momentum phase space are mapped to unitary quantum-mechanical operators in Hilbert space to produce a generalization of squeeze operators. The classical to quantum transition is manifestly apparent in the derivation. The unitary operators are evaluated in the coherent-state representation using the ``integration within ordered products'' technique in the one- and two-mode cases. Application of these operators results in a new generalization of coherent states.