 Solution

In the $S_z$ basis $\alpha ,\beta$, $S_x=(\hslash /2)\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$. Its normalized eigenstates are $(\alpha +\beta )/\sqrt{2}$ and $(\alpha -\beta )/\sqrt{2}$, with eigenvalues $+\hslash /2$ and $-\hslash /2$. Expanding the initial state in this basis and evolving the two phases gives, with $\Omega =\lambda B$,The sign of $\Omega$ fixes rotation sense; the probability is insensitive to that sign.

In the Stern-Gerlach experiment, a collimated atomic beam passes through a strongly nonuniform magnetic field. The force component $F_z\simeq {\mu }_z{\partial }_z{B}_z$ separates magnetic-moment projections spatially. Silver atoms provide an effective spin-half moment, yielding two resolved components rather than a continuous distribution of deflections. To test the rotation above, use a first analyzer and aperture to prepare $+z$, pass that selected beam through a controlled approximately uniform $x$-directed field, and analyze it with a second $z$-oriented Stern-Gerlach magnet. Varying field strength or interaction time changes the two output populations as the calculated cosine and sine squares. A narrow velocity distribution and controlled field-entry geometry preserve the contrast by reducing variations of interaction time and unwanted spin evolution.