A beam of spin 1/2 atoms goes through a series of Stem-Gerlach-type measurements as follows:

(a) The first measurement accepts s_z = h̄/2 atoms and rejects s_z = -h̄2 atoms.

(b) The second measurement accepts s_n = h̄/2 atoms and rejects s_n = -h̄/2 atoms, where s_n is the eigenvalue of the operator S · n̂, with ft making an angle β in the xz-plane with respect to the z-axis.

(c) The third measurement accepts S_z = -h̄/2 atoms and rejects s_z = h̄/2 atoms. What is the intensity of the final s_z = -h̄/2 beam when the s_z = h̄/2 beam surviving the first measurement is normalized to unity? How must we orient the second measuring apparatus if we are to maximize the intensity of the final s_z = -h̄/2 beam?