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Physical Review A

Simple construction of the unitary operator to transform from individual particles to Jacobi coordinates

Fan Hong‐Yi, Jack Vanderlinde

Published Feb 1, 1991• 4 citations
Publisher

Abstract

The unitary operator for transforming individual particles' coordinates to Jacobi coordinates is constructed in terms of a simple mapping of the classical transformation to quantum Hilbert space in Dirac's coordinate representation. The corresponding normally ordered form of the unitary operator is easily obtained by integration within ordered products.

DOI
10.1103/physreva.43.1604

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The unitary operator for transforming individual particles' coordinates to Jacobi coordinates is constructed in terms of a simple mapping of the classical transformation to quantum Hilbert space in Dirac's coordinate representation. The corresponding normally ordered form of the unitary operator is easily obtained by integration within ordered products.
Two types of canonical transformations in two-mode classical phase space are mapped into the quantum mechanical Hilbert space to produce some new normally ordered unitary operators. These operators are evaluated in the coordinate (momentum) representations using the "integration within ordered product" technique, and the mapping is manifestly apparent in the derivation. New generalized coherent states are constructed in terms of these operators, and the uncertainty relations for these states are analysed.
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.
The formalism of mapping a classical canonical transformation to quantum unitary operators for a boson system is generalized to fermion systems. In terms of the ``integration within an ordered product'' technique for fermion systems, the transition of a classical transformation in Grassmann number space to a quantum image is manifestly evaluated in the fermion coherent-state representation. The normally ordered unitary operators that generate the Bogolyubov-Valatin transformation and the SU(2) rotation are derived in this way.
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.