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Physical Review A

Connection between the hydrogen atom and the four-dimensional oscillator

Augustine C. Chen, M. Kibler

Published Jun 1, 1985• 34 citations
Publisher

Abstract

The connection between the three-dimensional hydrogen atom and a four-dimensional harmonic oscillator (or equivalently a coupled pair of two-dimensional harmonic oscillators) subjected to a constraint condition is further explored. In particular, the role the constraint condition plays in determining the phase relationship between the pair of two-dimensional oscillators is examined. Furthermore, the connection is discussed in a group-theoretical context involving the Lie algebras of SO(4), SO(3,1), E(3), SO(4,2), and Sp(8,R).

DOI
10.1103/physreva.31.3960

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The connection between the three-dimensional hydrogen atom and a four-dimensional harmonic oscillator (or equivalently a coupled pair of two-dimensional harmonic oscillators) subjected to a constraint condition is further explored. In particular, the role the constraint condition plays in determining the phase relationship between the pair of two-dimensional oscillators is examined. Furthermore, the connection is discussed in a group-theoretical context involving the Lie algebras of SO(4), SO(3,1), E(3), SO(4,2), and Sp(8,R).
The angular momentum states may be realized by the wave functions of two dimensional harmonic oscillators in polar coordinates. Using the generating function of these wave functions, Schwinger's generating function of Clebsh-Gordan in Fock space and Gauss integral, we determine the generating function of the couplings function of two angular momentums in coordinates representations. We deduce from the last function the connection of the wave function of the hydrogen atom and the four dimensional harmonic oscillator. We also find the expression of the relation between the spherical and parabolic wave functions of hydrogen atom. In addition, we find a new bilinear generating function involving the products of Laguerre polynomials.
Quantization of the conformal Kepler problem is defined and studied in order that the quantized system, which will be referred to as a conformal hydrogen atom, may associate the harmonic oscillator with the hydrogen atom. The conformal hydrogen atom shares with the harmonic oscillator the eigenspaces of negative energies. The four-dimensional conformal hydrogen atom reduces to the three-dimensional ordinary hydrogen atom. The symmetry group SO(4) of the hydrogen atom is brought out from the symmetry subgroup of the harmonic oscillator. The conformal hydrogen atom gives an example of those quantum systems of which the configuration spaces are curved Riemannian spaces with nonconstant scalar curvatures and of which the Hamiltonian operators depend on the scalar curvatures.
Until this point, we have been solving problems that involve just two states or are reducible to two-state problems. While we could examine a plethora of complex quantum phenomena, it is not sufficient to describe the behavior of electrons in atoms or how quantum particles interact with each other. That all changes now as we work to set-up the quantum mechanics of the atom. There is much for us to do before we get to the hydrogen atom. We start, in this chapter, with the simple harmonic oscillator in one dimension. We show how to determine the canonical commutation relation between position and momentum from the fact that atomic spectra have sharp lines and then move into describing free particles and free particles on a circle. We introduce two of our operator identities, the Hadamard lemma and the Baker-Campbell-Hausdorff formula and we revisit exponential disentangling, now for the symplectic group Sp(2). Then we show how to solve the harmonic oscillator energy eigenstates using operator algebra, discuss Heisenberg uncertainty, determine harmonic oscillator wavefunctions, and examine coherent and squeezed states. We end with a discussion of applications of the simple harmonic oscillator.