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Hydrogenic atoms
Book chapter

Hydrogenic atoms

Robert Gilmore
Lie Groups, Physics, and Geometry, pp 221-258
17 Jan 2008
DOI: https://doi.org/10.1017/CBO9780511791390.015

Abstract

Many physical systems exhibit symmetry. When a symmetry exists it is possible to use group theory to simplify both the treatment and the understanding of the problem. Central two-body forces, such as the gravitational and Coulomb interactions, give rise to systems exhibiting spherical symmetry (two particles) or broken spherical symmetry (planetary systems). In this chapter we see how spherical symmetry has been used to probe the details of the hydrogen atom. We find a hierarchy of symmetries and symmetry groups. At the most obvious level is the geometric symmetry group, SO(3), which describes invariance under rotations. At a less obvious level is the dynamical symmetry group, SO(4), which accounts for the degeneracy of the levels in the hydrogen atom with the same principal quantum number. At an even higher level are the spectrum generating groups, SO(4, 1) and SO(4, 2), which do not maintain energy degeneracy at all, but rather map any bound (scattering) state of the hydrogen atom into linear combinations of all bound (scattering) states. We begin with a description of the fundamental principles underlying the application of group theory to the study of physical systems. These are the principle of relativity (Galileo) and the principle of equivalence (Einstein).IntroductionApplications of group theory in physics start with two very important principles. These are Galileo's principle of relativity (of observers) and Einstein's principle of equivalence (of states).

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Mathematics, Applied
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