Rich-Context
Q7R.1 Suppose we have a box that emits electrons in a
definite but unknown spin state |\psi\rangle. If we send elec-
trons from this box through an SG\(_\theta\)_ device with \(\theta\) set at
120\(^{\circ}\), we find that 3/4 are determined to have \(S_\theta = +\frac{1}{2}\hbar\).
Assuming that the components of |\psi\rangle are real, argue
that there are two distinct \(q\)-vectors for |\psi\rangle consistent
with this result (that is, that differ by more than an over-
all sign). If we send electrons from this box through an
SG\(_x\)_ device and find that 93% are determined to have
\(S_x = +\frac{1}{2}\hbar\), what is |\psi\rangle (up to an overall sign)? (Hint: Let
the components of |\psi\rangle be a and b. Write an expression for
the probability in the first case in terms of a and b, use the
requirement that |\psi\rangle be normalized to eliminate a, and
then solve for b. You should find two solutions, one quite
simple, though the simple solution is not necessarily the
correct one.)
Q7R.2 Suppose we have a box that emits electrons in a
definite but unknown spin state |\psi\rangle. If we send electrons
