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Physical Review A

Symmetry of a Hydrogen Atom in a Weak Magnetic Field

Sadaaki Yanagawa

Published Jul 1, 1973• 7 citations
Publisher

Abstract

The degeneracy problem of a hydrogen atom in a uniform weak magnetic field $H$ is considered from the standpoint of the invariance group that the system obeys. Evidently the degeneracy space has a constant $z$ component of the angular momentum. Under such a classification, the dynamical invariance of the problem is shown to be an isomorph to the Lie group of the linear transformation of the straight line.

DOI
10.1103/physreva.8.46

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The degeneracy problem of a hydrogen atom in a uniform weak magnetic field HHH is considered from the standpoint of the invariance group that the system obeys. Evidently the degeneracy space has a constant zzz component of the angular momentum. Under such a classification, the dynamical invariance of the problem is shown to be an isomorph to the Lie group of the linear transformation of the straight line.
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.
As is known, the hydrogen atom energy levels depend on the principal quantum number n only and are independent of the azimuthal quantum number l. One can say (by using the common but not very suitable term) that there is degeneracy (i.e., the multiplicity of the level) with respect to the azimuthal quantum number. On the other hand, there is a general rule according to which the multiplicity of the Schro¨dinger equation eigenvalues relates to the invariance of the equation under a definite group of transformations. So, for example, the invariance under a common rotation group (spherical symmetry) leads to the energy level independence on the magnetic quantum number m. Therefore, it should be expected that the energy level independence of the azimuthal quantum number is explained by the existence of a certain transformation group that is more general than the three-dimensional rotation group. Up to now, this group of transformations for the Schro¨dinger equation has not been found. In the present work, we shall show that this group is equivalent to the rotation group in the four-dimensional Euclidean space.
The three dimensional rotation group, SO(3), is a symmetry group of the normal hydrogen atom. Each reducible representation of this group can be associated with a degenerate energy level. If this atom is placed in an external magnetic field, the interaction between the orbital magnetic moment with this field will lead to a symmetry breaking where the symmetry group of the atom is a new group distinct from the SO(3) group. This phenomenon describes the normal Zeeman effect, where a degenerate energy level splits into several new energy levels. It is explicitly shown that each of the new energy levels can be associated with an irreducible representation of the new symmetry group.
The classical problem of planar cyclotron motion of a charged particle in a uniform magnetic field possesses symmetries which account for the ``accidental'' degeneracies of the analogous nonrelativistic Schrödinger equation, as found by Johnson and Lippman. The essentially quadratic nature of the Hamiltonian is not changed by considering the particle moving in a harmonic oscillator potential, a ``Zeeman effect'' for the harmonic oscillator. The transitions to the limiting cases of a weak magnetic field (pure harmonic oscillator) or a strong field (pure cyclotron motion) involve the contraction of the corresponding symmetry groups, yielding Larmor precession of the oscillator orbits in the first case, and the drift of the cyclotron orbit in the second. The constants of the motion generate the unitary unimodular group SU2 in all cases except for pure cyclotron motion, in which case one obtains the commutation rules of creation and annihilation operators. Only for certain ratios of magnetic field strength to the oscillator frequency does one obtain bounded closed orbits, and presumably only in these cases do degeneracies exist quantum-mechanically. A transition to a rotating coordinate system reduces the problem to that of a plane harmonic oscillator; however, the time dependencies of the transformation must be allowed for interpreting the constants thereby arising. Moreover, the velocity-dependent forces introduce gauge transformations which also affect the interpretation of the symmetries. There are two kinds of symmetries—inner symmetries involving the canonical coordinates and governing the shape of the orbits, and outer symmetries involving the mechanical coordinates and governing the location of the orbits.