Semiclassical states on Lie algebras
Abstract
The effective technique for analyzing representationindependent features of quantum systems based on the semiclassical approximation (developed elsewhere) has been successfully used in the context of the canonical (Weyl) algebra of the basic quantum observables. Here, we perform the important step of extending this effective technique to the quantization of a more general class of finitedimensional Lie algebras. The case of a Lie algebra with a single central element (the Casimir element) is treated in detail by considering semiclassical states on the corresponding universal enveloping algebra. Restriction to an irreducible representation is performed by “effectively” fixing the Casimir condition, following the methods previously used for constrained quantum systems. We explicitly determine the conditions under which this restriction can be consistently performed alongside the semiclassical truncation.
 Authors:
 King’s College, 133 North River Street, Kingston, Pennsylvania 18702 (United States)
 Publication Date:
 OSTI Identifier:
 22403124
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Journal of Mathematical Physics; Journal Volume: 56; Journal Issue: 3; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; IRREDUCIBLE REPRESENTATIONS; LIE GROUPS; QUANTIZATION; SEMICLASSICAL APPROXIMATION
Citation Formats
Tsobanjan, Artur, Email: artur.tsobanjan@gmail.com. Semiclassical states on Lie algebras. United States: N. p., 2015.
Web. doi:10.1063/1.4914010.
Tsobanjan, Artur, Email: artur.tsobanjan@gmail.com. Semiclassical states on Lie algebras. United States. doi:10.1063/1.4914010.
Tsobanjan, Artur, Email: artur.tsobanjan@gmail.com. 2015.
"Semiclassical states on Lie algebras". United States.
doi:10.1063/1.4914010.
@article{osti_22403124,
title = {Semiclassical states on Lie algebras},
author = {Tsobanjan, Artur, Email: artur.tsobanjan@gmail.com},
abstractNote = {The effective technique for analyzing representationindependent features of quantum systems based on the semiclassical approximation (developed elsewhere) has been successfully used in the context of the canonical (Weyl) algebra of the basic quantum observables. Here, we perform the important step of extending this effective technique to the quantization of a more general class of finitedimensional Lie algebras. The case of a Lie algebra with a single central element (the Casimir element) is treated in detail by considering semiclassical states on the corresponding universal enveloping algebra. Restriction to an irreducible representation is performed by “effectively” fixing the Casimir condition, following the methods previously used for constrained quantum systems. We explicitly determine the conditions under which this restriction can be consistently performed alongside the semiclassical truncation.},
doi = {10.1063/1.4914010},
journal = {Journal of Mathematical Physics},
number = 3,
volume = 56,
place = {United States},
year = 2015,
month = 3
}

Using the transformations from paper I, we show that the Schroedinger equations for (1) systems described by quadratic Hamiltonians, (2) systems with timevarying mass, and (3) timedependent oscillators all have isomorphic Lie spacetime symmetry algebras. The generators of the symmetry algebras are obtained explicitly for each case and sets of numberoperator states are constructed. The algebras and the states are used to compute displacementoperator coherent and squeezed states. Some properties of the coherent and squeezed states are calculated. The classical motion of these states is demonstrated. (c) 2000 American Institute of Physics.

Displacementoperator squeezed states. I. Timedependent systems having isomorphic symmetry algebras
In this paper we use the Lie algebra of spacetime symmetries to construct states which are solutions to the timedependent Schr{umlt o}dinger equation for systems with potentials V(x,{tau})=g{sup (2)}({tau})x{sup 2}+g{sup (1)}({tau})x+g{sup (0)}({tau}). We describe a set of numberoperator eigenstates states, {l_brace}{Psi}{sub n}(x,{tau}){r_brace}, that form a complete set of states but which, however, are usually not energy eigenstates. From the extremal state, {Psi}{sub 0}, and a displacement squeeze operator derived using the Lie symmetries, we construct squeezed states and compute expectation values for position and momentum as a function of time, {tau}. We prove a general expression for the uncertainty relationmore » 
Displacementoperator squeezed states. II. Examples of timedependent systems having isomorphic symmetry algebras
In this paper, results from the previous paper (I) are applied to calculations of squeezed states for such wellknown systems as the harmonic oscillator, free particle, linear potential, oscillator with a uniform driving force, and repulsive oscillator. For each example, expressions for the expectation values of position and momentum are derived in terms of the initial position and momentum, as well as in the ({alpha},z) and in the (z,{alpha})representations described in I. The dependence of the squeezedstate uncertainty products on the time and on the squeezing parameters is determined for each system. {copyright} {ital 1997 American Institute of Physics.}