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Communications in Theoretical Physics

Holstein–Primakoff Boson Realization for the Hydrogen Atom

Bong-Yi Fan, Zhongxi Zhang

Published Oct 30, 1994• 0 citations
Publisher

Abstract

We apply the Holstein–Primakoff transformation to the orbital angular momentum and the Runge–Lenz operators of a hydrogen atom to study its classical Kepler's problem. In this way the classical Kepler's frequency can be directly obtained without introducing Schwinger's boson realization and boson coherent states.
DOI
10.1088/0253-6102/22/3/373

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The orientation in space of a Cartesian coordinate system can be indicated by the two vectorial constants of motion of a classical Keplerian orbit: the angular momentum and the Laplace-Runge-Lenz vector. In quantum mechanics, the states of a hydrogen atom that mimic classical elliptic orbits are the coherent states of the SO(4) rotation group. It is known how to produce these states experimentally. They have minimal dispersions of the two conserved vectors and can be used as direction indicators. We compare the fidelity of this transmission method with that of the idealized optimal method.
A modification of the Holstein-Primakoff transformation is proposed such that creation and annihilation operators for a bosonic field are rewritten as operators of a SU(2)SU(2)SU(2) algebra. Once it is applied to a quantum Hamiltonian, a subsequent application of the prescription by Lieb to obtain the classical limit for spin operators allows one to write efficiently a classical Hamiltonian for the system. This process is illustrated for the MMM-atom Jaynes-Cummings model.