Phy5645/Plane Rotator

Problem 1:

Solution:

a) A can be determined from the normalization condition:

$1=\int _{-\pi }^{\pi }d\phi |\psi (\phi )|^{2}=A^{2}\int _{-\pi }^{\pi }d\phi sin^{4}\psi =A^{2}3\pi /4$

Then, we could get $A={\frac {2}{\sqrt {3\pi }}}$

b) The probability to measure the angular momentum to be $\hbar m$ is

<math> P_m = |<\psi_m|\psi>|^2 = |\int_{-\pi}^{\pi}d\phi \frac {e^{-im\phi}}{\sqrt {2\pi}} \psi(\phi)|^2 = \frac {2}{3} \delta_m,0 + \frac {1}{6}(\delta_m,2+\delta_m,-2)