Parity sends a direction to

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Symmetry, angular momentum and spin · Symmetry, angular momentum and spin

The Structure of Quantum Mechanics, Ch Angular momentum from the rotation group

Problem

Parity sends a direction n^\hat n to −n^-\hat n, and Section 13.4 showed that Ylm(−n^)=(−1)lYlm(n^)Y_l^m(-\hat n)=(-1)^lY_l^m(\hat n) for every mm in the multiplet. The function cos⁡θ=z/r\cos\theta=z/r changes sign under n^→−n^\hat n\to-\hat n. ((a)) By relabelling the integration variable from n^\hat n to −n^-\hat n, show that I=∫dΩ  Yl′m′∗(n^) cos⁡θ Ylm(n^)I=\int d\Omega\;Y_{l'}^{m'*}(\hat n)\,\cos\theta\,Y_l^m(\hat n) vanishes whenever l+l′l+l' is even, without evaluating any integral. ((b)) Hence show that in any state built only from the 2l+12l+1 states of one multiplet, the mean value of cos⁡θ\cos\theta is zero.

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