Quantum Mechanical Angular Distributions and Group Representations on Banach Spaces
In large classes of scattering experiments in atomic, molecular, and nuclear physics the angular distribution of a product of the reaction can be described by simple trigonometric formulae. For example, the angular distribution of a particle ejected from an unpolarized target due to the absorption of a photon via a dipole process takes the phenomenological form A + B cos2 θ. The microscopic parameters describing the detailed structure of the target determine the values of the macroscopic parameters A and B but do not change the phenomenological form. We show that the phenomenological form of an angular distribution may be simply derived from physically induced representations of the rotation group on the space of states and on the space of observables which are represented, respectively, by the trace class operators T(H) and the bounded operators L(H) on a Hilbert space H. Specifically we show that with appropriate topologies the representations on T(H) and L(H) are completely reducible and that Schur’s lemma applies. These results lead to the appropriate phenomenological form for the angular distribution and also provide straightforward procedures for evaluating the macroscopic parameters in terms of microscopic parameters and geometrical factors.
KeywordsAngular Distribution Pure State Unitary Representation Density Operator Observable Space
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