PHYSICS · Subjective · 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
A state ∣j,m⟩ is rotated by β about the y axis, and J^z is then measured on many identically prepared copies. (a) For j=1, use the columns of d(1)(β) to show that the mean value of J^z after the rotation is mℏcosβ for each of m=1,0,−1. (b) Show that the same formula holds for j=21, m=21, using d(1/2)(β). (c) A classical angular momentum of length L along z, turned by β about y, has z component Lcosβ. Say what the results of (a) and (b) share with this picture and what they do not, and explain why, for j=21, every probability at β=2π equals its value at β=0 although d(1/2)(2π)=−1.