Principles of Quantum Mechanics · Problem 12.5.10

Rotational Invariance and Angular Momentum — R. Shankar

Physics — Quantum Mechanics · spherical-harmonics-differential-equation

Problem

Write the differential equation corresponding to

L^2|\alpha\beta\rangle = a|\alpha\beta\rangle

in the coordinate basis, using the

L^2

operator given in Eq. (12.5.36). We already know

\beta = m\hbar

from the analysis of

-i\hbar(\partial/\partial\phi)

. So assume that the simultaneous eigenfunctions have the form

\psi_{am}(\theta, \phi) = P_a^m(\theta)\, e^{im\phi}

and show that

P_a^m

satisfies the equation

\left(\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\sin\theta\frac{\partial}{\partial\theta} + \frac{a}{\hbar^2} - \frac{m^2}{\sin^2\theta}\right)P_a^m(\theta) = 0