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Journal of Mathematical Physics

Symmetry of the Two-Dimensional Hydrogen Atom

Arturo Cisneros, Harold V. McIntosh

Published Feb 1, 1969• 93 citations
Publisher

Abstract

A paradox has arisen from some recent treatments of accidental degeneracy which claim that, for three degrees of freedom, SU(3) should be a universal symmetry group. Such conclusions are in disagreement both with experimentally observed spectra and with the generally accepted solutions of Schrödinger's equation. The discrepancy occurs in the transition between classical and quantum-mechanical formulations of the problems, and illustrates the care necessary in forming quantum-mechanical operators from classical expressions. The hydrogen atom in parabolic coordinates in two dimensions, for which the traditional treatment of Fock, extended by Alliluev, requires the symmetry group O(3), is a case for which the newer methods of Fradkin, Mukunda, Dulock, and others require SU(2). Although these groups are only slightly different, SU(2) fails to be the ``universal'' symmetry group on account of the multiple-valuedness of the parabolic representation. This conclusion extends a result of Han and Stehle: that, for rather similar reasons, SU(2) cannot be the classical symmetry group for the two-dimensional hydrogen atom.

DOI
10.1063/1.1664843

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A paradox has arisen from some recent treatments of accidental degeneracy which claim that, for three degrees of freedom, SU(3) should be a universal symmetry group. Such conclusions are in disagreement both with experimentally observed spectra and with the generally accepted solutions of Schrödinger's equation. The discrepancy occurs in the transition between classical and quantum-mechanical formulations of the problems, and illustrates the care necessary in forming quantum-mechanical operators from classical expressions. The hydrogen atom in parabolic coordinates in two dimensions, for which the traditional treatment of Fock, extended by Alliluev, requires the symmetry group O(3), is a case for which the newer methods of Fradkin, Mukunda, Dulock, and others require SU(2). Although these groups are only slightly different, SU(2) fails to be the ``universal'' symmetry group on account of the multiple-valuedness of the parabolic representation. This conclusion extends a result of Han and Stehle: that, for rather similar reasons, SU(2) cannot be the classical symmetry group for the two-dimensional hydrogen atom.
Recently several authors have proposed a universal symmetry group and demonstrated the classical validity of the concept. Supposedly, an SU(n) symmetry group could be constructed for whatsoever system of n degrees of freedom. The claim is assuredly valid for all classically degenerate systems, but is in contradiction with most of the well-known and widely accepted solutions of Schrödinger's equation. We examine the reasons for this discrepancy on the quantum-mechanical level. The construction of the universal symmetry group requires ladder operators, which in most cases are the ladder operators of Infeld and Hill. Complications which owe their origin entirely to numerical relationships imposed by quantization prevent these operators from forming a von Neumann algebra and, in turn, an SU(n) group of constants of the motion. Two important effects are those imposed by anisotropy, wherein not all quanta have the same size, and by non-Cartesian coordinates, wherein quanta in some dimensions are restricted in size by those in other dimensions. These effects can be seen quite clearly in two-dimensional systems: anisotropy in the Cartesian anisotropic harmonic oscillator, and conflict between dimensions in the polar form of the isotropic oscillator. Further complications arise when the two effects are combined, as in the harmonic oscillator or hydrogen atom with ``excess'' angular momentum. Enough residue of the universal symmetry concept remains that many similarities between the hydrogen atom and harmonic oscillator may be understood, including the fact that some levels of the hydrogen atom, which ordinarily transform according to an orthogonal group, may form irreducible representations of the unitary group.
The Schrödinger equation does not account for the 2n2 degeneracy of the hydrogen atom, which it dismisses as an “accidental” degeneracy. The factor of “2” in the 2n2 degeneracy is well-accounted-for in the relativistic formulation by the two spin states of the electron. The n2 degeneracy is nevertheless not quite an “accident”; it is due to the SO(4), rather than SO(3), symmetry of the hydrogen atom. This result is well known, but is inadequately commented upon in most courses in quantum mechanics and atomic physics, leaving the student wondering about the origins of the n2 degeneracy of the hydrogen atom. A pedagogical analysis of this interesting aspect, which highlights the fundamental principles of quantum mechanics, is presented in this article. While doing so, not only is the n2 degeneracy of the hydrogen atom explained, but its energy spectrum and eigenfunctions are obtained without even using the Schrödinger equation, employing only the fundamental principles of quantum mechanics rather than the Schrödinger equation.
The problem of degeneracy in quantum mechanics is related to the existence of groups of contact transformations under which the Hamiltonian is invariant. The correspondence between transformations in classical and quantum theories is developed. The Fock-Bargmann treatment of the symmetry group of the hydrogenic atom comes under this theory. The symmetry group of the 2-dimensional Kepler problem is found to be the 3-dimensional rotation group; that of the nnn-dimensional isotropic oscillator is isomorphic to the unimodular unitary group in nnn dimensions. The 2-dimensional anisotropic oscillator has the same symmetry as the isotropic oscillator in classical mechanics, but the quantum-mechanical problem presents complications which leave its symmetry group in doubt.
In the previous chapter we saw that for a particle in a one-dimensional potential well, physically acceptable solutions to the one-dimensional Schro¨dinger equation are possible only for particular discrete values of the total energy. Moreover, in the case of an electron in a well of atomic dimensions, the spacings between these energy levels are in qualitative agreement with the separations experimentally observed in atoms; we also interpreted the square of the wave function as a probability distribution for the position of the particle. These ideas led us to phenomena, such as quantum-mechanical tunnelling, that have been observed experimentally. The real world, however, is three-dimensional and, although one-dimensional examples often provide useful insights and analogies, we shall have to extend our theory into three dimensions before we can make quantitative predictions of most experimental results. In the present chapter, therefore, we shall set up the three-dimensional Schro¨dinger equation and obtain its solutions in a number of cases, culminating in a discussion of the hydrogen atom where we shall find that theory and experiment agree to a remarkable degree of accuracy.
Symmetry is a cornerstone of modern physics, offering profound insights into the fundamental laws governing the universe. In quantum mechanics, symmetries lead to conserved quantities and simplify the resolution of complex systems. This paper explores the universal symmetries inherent in three paradigmatic quantum mechanical systems: the one-dimensional quantum harmonic oscillator, the hydrogen atom, and the angular momentum operator. While renowned for their exact solvability, their underlying symmetry structures extend beyond explicit Hamiltonian invariances. We examine both geometric and dynamical (hidden) symmetries, which arise from non-trivial conserved quantities and often explain ``accidental'' degeneracies in energy spectra. For the quantum harmonic oscillator, we delve into the SU(1,1) or SU(2) dynamical symmetry associated with its creation and annihilation operators. The hydrogen atom's unique SO(4) symmetry, stemming from the conservation of the Runge-Lenz vector in addition to angular momentum, is thoroughly investigated. The Lie algebra of angular momentum operators forms the basis of SO(3) or SU(2) symmetry, governing rotational invariance. By employing Lie groups and Lie algebras, we reveal the deep mathematical unity underpinning these seemingly disparate systems. This comparative analysis demonstrates how these universal symmetry principles not only explain solvability and degeneracy patterns but also offer a general framework for understanding quantum phenomena. The paper hypothesizes that the recurring algebraic structures of su(2) and its extensions across these systems signify a deeper, universal blueprint for quantum system behavior, serving as a foundational guide for understanding both exactly solvable and inherently non-solvable quantum phenomena through the lens of symmetry breaking or approximation. This exploration provides a robust foundation for advanced topics in quantum field theory and particle physics.