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arXiv:quant-ph/9804074 · Journal of Physics A Mathematical and General

Dispersion of Klauder's temporally stable coherent states for the hydrogen atom

Paolo Bellomo, Carlos R. Stroud

Published Jun 19, 1998• 13 citations
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Abstract

We study the dispersion of the `temporally stable' coherent states for the hydrogen atom introduced by Klauder. These are states which under temporal evolution by the hydrogen atom Hamiltonian retain their coherence properties. We show that in the hydrogen atom such wavepackets do not move quasiclassically; i.e. they do not follow, with no or little dispersion, the Keplerian orbits of the classical electron. The poor quantum - classical correspondence does not improve in the semiclassical limit.

DOI
10.1088/0305-4470/31/24/001

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Coherent States for the Hydrogen Atom

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The long-standing problem of finding coherent states for the (bound state portion of the) hydrogen atom is positively resolved. The states in question: (i) are normalized and parameterized continuously, (ii) admit a resolution of unity with a positive measure, and (iii) enjoy the property that the temporal evolution of any coherent state by the hydrogen atom Hamiltonian remains a coherent state for all time. Harmonic-oscillator coherent states In modern terms, the distinguished set of states Schrodinger introduced for the harmonic oscillator 1 are now commonly known as coherent states 2 and are given, for all z 2 I C, by jzi j e (za y \\Gammaz a) j0i = e \\Gamma 1 2 jzj 2 1 X n=0 z n p n! jni; (1) where [a; a y ] = 1 as usual. Here jni; n = 0; 1; 2; : : :, denote normalized eigenstates of the number operator N , N jni = njni, which may be identified with the eigenstates of the harmonic oscillator Hamiltonian H 0 = !N = !a y a (¯h = 1). As such it follows that ...

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We study the dispersion of the `temporally stable' coherent states for the hydrogen atom introduced by Klauder. These are states which under temporal evolution by the hydrogen atom Hamiltonian retain their coherence properties. We show that in the hydrogen atom such wavepackets do not move quasiclassically; i.e. they do not follow, with no or little dispersion, the Keplerian orbits of the classical electron. The poor quantum - classical correspondence does not improve in the semiclassical limit.
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 construct wave packets for the hydrogen atom labeled by the classical action-angle variables with the following properties. (i) The time evolution is exactly given by classical evolution of the angle variables. (The angle variable corresponding to the position on the orbit is now noncompact and we do not get exactly the same state after one period; however, the gross features do not change. In particular, the wave packet remains peaked around the labels.) (ii) The resolution of the identity using this overcomplete set involves exactly the classical phase-space measure. (iii) The semiclassical limit is related to Bohr-Sommerfeld quantization. (iv) They are almost minimum uncertainty wave packets in position and momentum.
Klauder's recent generalization of the harmonic oscillator coherent states [J. Phys A 29, L293 (1996)], is applicable only in nondegenerate systems, requiring some additional structure if applied to systems with degeneracies. The author suggests how this structure could be added, and applies the complete method to the hydrogen atom problem. To illustrate how a certain degree of freedom in the construction may be exercised, states are constructed, which are initially localized and evolve semiclassically, and whose long time evolution exhibits ``fractional revivals.''
The long-standing problem of finding coherent states for the (bound state portion of the) hydrogen atom is positively resolved. The states in question: (i) are normalized and parameterized continuously, (ii) admit a resolution of unity with a positive measure, and (iii) enjoy the property that the temporal evolution of any coherent state by the hydrogen atom Hamiltonian remains a coherent state for all time. Harmonic-oscillator coherent states In modern terms, the distinguished set of states Schrodinger introduced for the harmonic oscillator 1 are now commonly known as coherent states 2 and are given, for all z 2 I C, by jzi j e (za y \\Gammaz a) j0i = e \\Gamma 1 2 jzj 2 1 X n=0 z n p n! jni; (1) where [a; a y ] = 1 as usual. Here jni; n = 0; 1; 2; : : :, denote normalized eigenstates of the number operator N , N jni = njni, which may be identified with the eigenstates of the harmonic oscillator Hamiltonian H 0 = !N = !a y a (¯h = 1). As such it follows that ...