Answer to Question #86639 in Atomic and Nuclear Physics for Anand

Question

For a motion of a particle of mass μ in a spherically symmetric potential show that
L²and Lzcommute with the Hamiltonian

Answer 1

The Hamiltonianof a particle of mass μ in a spherically symmetric potential:

$$H = \frac{p^2}{2\mu}+V(r)$$

Angular momentum operators:

$$L_z = -i\hbar \left(x\frac{\partial}{\partial y} - y\frac{\partial}{\partial x}\right), L^2 = L^2_x+L^2_y+L^2_z$$

Angular momentum in spherical coordinates:

$$L_z = -i\hbar \frac{\partial}{\partial \phi}, L^2 = - \hbar^2 \left(\frac{1}{\sin{\theta}}\frac{\partial}{\partial \theta}\left( \sin{\theta}\frac{\partial}{\partial \theta} \right)+\frac{1}{\sin^2{\theta}}\frac{\partial^2}{\partial \phi^2} \right)$$

As the hamiltonian is independent of $\theta, \phi$ coordinates,

$$[L_z, H] =0, \, [L^2, H] =0$$