Showing papers on "Coherent states in mathematical physics published in 1991"

The quantum Heisenberg group H(1)q

Abstract: The structure of the quantum Heisenberg group is studied in the two different frameworks of the Lie algebra deformations and of the quantum matrix pseudogroups. The R‐matrix connecting the two approaches, together with its classical limit r, are explicitly calculated by using the contraction technique and the problems connected with the limiting procedure discussed. Some unusual properties of the quantum enveloping Heisenberg algebra are shown.

Quantum coherent operators: A generalization of coherent states

Abstract: We introduce a technique to compare different, but related, quantum systems, thereby generalizing the way that coherent states are used to compare quantum systems to classical systems in semiclassical analysis. We then use this technique to estimate the dependence of the free energy of the quantum Heisenberg model on the spin value, and to estimate the relation between the ferromagnetic and antiferromagnetic free energies.

Square integrability of group representations on homogeneous spaces. II : Coherent and quasi-coherent states. The case of the Poincaré group

Abstract: The concept of a reproducing triple, developed in the first paper of the series (I), is utilized to give a general definition of a square integrable representation of a group. This definition is applicable to homogeneous spaces of the group, and generalizes earlier attempts at obtaining such notions. Among others, it leads naturally to a notion of equivalence among families of coherent states, and also to the concept of quasi-coherent states, or weighted coherent states. The general considerations are applied to the specific example of the Wigner representation of the Poincare group P+up (1, 1) in one space and one time dimensions. A whole class of equivalent families of coherent states is derived, each of which corresponds to a continuous frame, in the sense of I.

An example of a quantum exponential process

Abstract: In [3],R. L. Hudson andK. R. Parthasarathy showed that the Fock space based on the Heisenberg—Weyl algebra hosts Brownian motion and Poisson processes. In this paper we construct a quantum exponential process acting on the Fock space based on the finite-difference algebra ofP. J. Feinsilver ([2]).

A basis‐independent approach to quantum optics

Abstract: States of the radiation field can be defined without reference to a given basis of normal modes. In this way optical systems for which there is no natural set of orthogonal modes can be handled quantum mechanically in a more rigorous manner. In particular, coherent states and associated phase‐space methods can easily be generalized to this basis‐independent representation.

Quantization, coherent states and diffeomorphism groups

Abstract: We suggest extending the method of coherent-states quantization, which applies to homogeneous spaces for locally compact groups, to the case of infinite-dimensional groups such as diffeomorphism groups. Such a framework could unify a number of different approaches to quantum theory. We review some relevant results and examples, and take a first step in this program by demonstrating that the unitary representation of Diff(IR) obtained from the “single-particle” coadjoint orbit is in fact square-integrable.

Coherent states of hydrogen atom

Abstract: From Kustannheimo-Stiefel transformation, the hygrogen problem in quantum mechanics can be transformed to that of a four dimensional isotropic harmonic oscillator with constrain. The coherent states are defined, for which the expectation values of the position and the momentum are shown to give the classical Kepler orbits. The uncertainty relations for the coherent states are discussed as well.

Product formulas for Q-representatives

Abstract: To obtain a closed and consistent mathematical description of quantum systems in terms of group-related 'phase space' functions one needs counterparts for all the operations usually performed with operators. For functions related to coher ent states, so-called Q-representatives, the relation corresponding to the product of two operators is given for compact Lie groups, especially for SU(2), and for the non-compact Weyl-Heisenberg group. Equations (5-8) show that these formulas, as well as the resulting commutator relation, are especially suited to discuss quantum corrections to classical results. In addition they should simplify the calculation of expectation values for coherent states as they relate the Q-representatives of more complicated operators to those of simpler ones.

On Complete Orthonormal Sets of Coherent and of Squeezed States

Abstract: A complete set of harmonic oscillator orthogonal coherent states is discussed. The properties of the states are studied, in particular towards applications as a basis for nonstationary problems. To make such a basis even more flexible, an orthonormal set of squeezed states is introduced. These states share most of the properties that make the familiar coherent states useful in applications. They are also extremal states of the uncertainty product. Their particular advantage, beyond the obvious one of orthogonality, is in applications to the dynamics of excited states.