Group Representations and Selection Rules

Abstract

One of the main efforts of Chapter 5 was to determine explicitly the normalized wave functions Ψ nlm of a hydrogenic atom; see equation (5.2.12). Here n ∈ {1,2,3,...},l ∈ {0,l,2,...,n −l},m ∈ {−l,−l + 1,...,−1,0,1,2,..., l} are the total, azimuthal, and magnetic quantum numbers, respectively. The triples (n,l,m) = (5,2,−2), (5,4,−2) define, for example, two quantum states with the same energy E (5) (by (5.1.29)) of the hydrogenic atom. One could ask the following question: Is it possible for the atom to proceed directly from the state (5,2,−2) to the state (5,4,−2): is the transition (5,2,−2) ← (5,4,−2) always physically possible? The answer is no, as we shall see in this chapter. Moreover the answer comes by way of some beautiful applications of group theory. This example, of whether a certain transition is allowable or forbidden, illustrates one of many selection rules in quantum mechanics. The answer no just asserted turns out to depend on whether or not a certain integral (depending on the quantum numbers (5,2, −2), (5,4, −2)) vanishes.KeywordsIrreducible RepresentationSelection RuleUnitary RepresentationContinuous RepresentationComplex Vector SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.