In Sec. II we present some of the well-known facts about
the angular momentum eigenvalue problem in quantum mechanics
that will be needed later. Section III contains the
discussion of the probabilities to measure the eigenvalues of
jˆ
u[ejˆ in eigenstates of jˆ z , where e is a unit vector that
makes an angle u with ez . After presenting the general
method, we first give the explicit formulas up to j53
2. For
larger values of j the probabilities are calculated using a
simple recursion relation. Our method is conceptually much
simpler than the approach first proposed by Wigner,5 which
we present for comparison in the appendix. In Sec. IV we
discuss the semiclassical limit j@1 for measuring jˆ x. We
compare the exact results for px ( j)(mx ,mz) with the continuous
classical probability distribution that corresponds to Fig.
1. For j@1, the probabilities px ( j)(mx ,mz) for j2umzu!j can
be expressed as uw j2umzu(mx)u2, where the wn are harmonic
oscillator eigenfunctions. In this way we bring together two
of the most important topics in quantum mechanics. As the
harmonic oscillator is usually treated before angular momentum
in quantum mechanics courses, our results offer the possibility
to use methods already known to students. We do not
know of any earlier discussion of the quasiclassical limit.