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Physical Review A

Coherent states for the Kepler motion. II

Tadashi Toyoda, 澄子 若山

Published Aug 16, 2001• 11 citations
Publisher

Abstract

The coherent states for the quantum Kepler motion proposed in our previous work [Phys. Rev. A 59, 1021 (1999)], which is based on the dynamical $\mathrm{SU}(2)\ensuremath{\bigotimes}\mathrm{SU}(2)$ symmetry and the Duru-Kleinert auxiliary time, are improved by making use of the theory of the conserved-charge coherent states formulated by Bhaumik et al. and by Skagertsam. The expectation values for the angular momentum and the Runge-Lenz-Pauli vector with respect to the improved Kepler coherent states are also discussed.

DOI
10.1103/physreva.64.032110

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The coherent states for the quantum Kepler motion proposed in our previous work [Phys. Rev. A 59, 1021 (1999)], which is based on the dynamical SU(2)\ensuremath⨂SU(2)\mathrm{SU}(2)\ensuremath{\bigotimes}\mathrm{SU}(2)SU(2)\ensuremath⨂SU(2) symmetry and the Duru-Kleinert auxiliary time, are improved by making use of the theory of the conserved-charge coherent states formulated by Bhaumik et al. and by Skagertsam. The expectation values for the angular momentum and the Runge-Lenz-Pauli vector with respect to the improved Kepler coherent states are also discussed.
Majumdar and Sharatchandra recently proposed a set of ``coherent states'' for the hydrogen atom [Phys. Rev. A 56, R3322 (1997)]. These states satisfy some of the typical properties of coherent states, such as, for example, continuity in the parameters. The time evolution of these states is given by the classical evolution of the angle variables, and the expectation values of the quantum observables behave quasiclassically. However, those authors also claimed that, although one does not obtain exactly the same state after a Kepler period, the gross features of their wave packets do not change. We show that this is not correct, and that these wave packets do not remain localized on the classical variables, but spread rapidly over the Keplerian orbit. The localization properties of these states do not improve in the limit of large quantum numbers.
We consider relativistic coherent states for a spin-0 charged particle that satisfy the next additional requirements: (i) the expected values of the standard coordinate and momentum operators are uniquely related to the real and imaginary parts of the coherent state parameter $\ensuremath{\alpha};$ (ii) these states contain only one charge component. Three cases are considered: free particle, relativistic rotator, and particle in a constant homogeneous magnetic field. For the rotational motion of the two latter cases, such a description leads to the appearance of the so-called nonlinear coherent states.
The coherent states for a particle on a sphere are introduced. These states are labelled by points of the classical phase space, i.e. the position on the sphere and the angular momentum of a particle. As with the coherent states for a particle on a circle discussed in Kowalski et al ( 1996 J. Phys. A: Math. Gen. 29 4149 ), we deal with a deformation of the classical phase space related to quantum fluctuations. The expectation values of the position and the angular momentum in the coherent states are regarded as the best possible approximation of the classical phase space. The correctness of the introduced coherent states is illustrated by an example of the rotator.
We construct the system of generalized coherent states for the quantum Kepler problem corresponds to the homogeneous domain SU(2,2)/S(U(2)×U(2))SU(2,2)/S(U(2)\times U(2))SU(2,2)/S(U(2)×U(2)). We show that the SU(2,2)-equivariant momentum map for this domain yields the momentum map for the classical Kepler problem via appropriate limiting passage. We also show that under this passage the average values of quantum observables in this system of coherent states pass into the functions on classical phase space and −i/h-i/h−i/h times commutator pass into the Poisson bracket.