Question

Consider the function 𝜓(𝜃) of the angular variable 𝜃, restricted to the interval −𝜋 ≤ 𝜃 ≤ 𝜋. If the wave functions satisfy the condition 𝜓(𝜋) = 𝜓(−𝜋), show that the operator 𝐿 = ℏ / 𝑖. 𝑑 / 𝑑𝜃 has a real expectation value.

Answer 1

The equation for eigenfunctions is given by

$$L\psi=\lambda\psi$$

$$\frac{\hbar}{i}\frac{d}{d\theta}\psi=\lambda\psi$$

$$\psi(\theta)=Ce^{i/\hbar\lambda\theta}$$

$$\psi(\pi)=Ce^{i/\hbar\lambda \pi},\quad \psi(-\pi)=Ce^{-i/\hbar\lambda \pi}$$

$$e^{i/\hbar\lambda \pi}=e^{-i/\hbar\lambda \pi}$$

$$e^{2i/\hbar\lambda \pi}=1$$

$$2/\hbar\lambda \pi=2\pi n$$

The expectation values of operator

$$\lambda=\hbar n=\rm real$$