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Modern Physics Letters B

WAVE FUNCTION REPRESENTATION FOR THE GENERALIZED EIGENSTATES OF SU(1, 1) NONCOMPACT GENERATORS

Vittorio Penna

Published Dec 20, 1991• 0 citations
Publisher

Abstract

The SU(1, 1) algebra is represented by a two bosonic operator realization. This allows one to obtain a remarkable simplification of the SU(1, 1) noncompact generator eigenvectors by representing the Hilbert space basis in terms of two harmonic oscillator wave functions.
DOI
10.1142/s0217984991002343

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We find that a three-mode boson realization of the SU(1,1) Lie algebra is involved in solving the three coupled harmonic oscillators' problem. The unitary operator U, which is found to be able to transform the Fock space of a three–dimensional isotropic harmonic oscillator into the space in which the Hamiltonian of three coupled oscillators is diagonized, is further decomposed as a quantum–mechanical rotation in three–mode Hilbert space followed by an SU(1,1) squeezing transformation. The coordinate representation of this SU(1,1) unitary operator is obtained.
The harmonic oscillator is a useful model for the interpretation of many physical phenomena. It is used as a first approximation to the vibration of molecules and crystal lattices and for the quantization of electromagnetic fields, among other applications. One can solve the time-independent Schrodinger equation for the harmonic oscillator in many different ways; operator methods give rise to recurrence relations and generating functions for the stationary states. The coordinate representation renders the coordinate operator diagonal. In this particular representation the eigenvectors of the harmonic oscillator are functions of the coordinate that are commonly expressed in terms of Hermite polynomials. The chapter shows that calculation with the eigenfunctions of the harmonic oscillator is remarkably simple in the realm of the Lie algebras. The application of the bosonic algebra in theoretical physics and chemistry is so vast that it is impossible even to give a reasonable summary. An important application of the bosonic algebra is the modeling of simple quantum-mechanical systems.
We have constructed the multimode bosonic realization of the su(1,1) Lie algebra, on the basis of which the su(1,1) generalized coherent state in the multimode Fock space is derived. It is shown that the multimode coherent state is, in fact, the generalized multimode squeezed vacuum state discussed by Lo and Sollie [Phys. Rev. A 47, 733 (1993)], and that the unitary displacement operator can be identified as the generalized multimode squeezing operator. Hence, similar to the single- and two-mode cases, our multimode bosonic realization of the su(1,1) Lie algebra has immediate relevance to the squeezing properties of boson fields.
The one-dimensional finite oscillator model satisfies that under commutator brackets, all operators close into the (dynamical) Lie algebra su(2); therefore the eigenvalues of position, momentum and energy operators are discrete and the eigenvalue of a compact u(l) central operator fixes the dimension of the su(2) representation. In this work we review the one-dimensional finite oscillator model with u(2) as the dynamical algebra where the wave functions, which involve Kravchuk polynomials, are discrete functions of angular momentum theory.