Introduction to Quantum Mechanics · Problem 4.22

Quantum Mechanics in Three Dimensions — David J. Griffiths

Physics — Quantum Mechanics · angular_momentum_commutators

Problem

(a) Starting with the canonical commutation relations for position and momentum (Equation 4.10), work out the following commutators:

[L_z, x] = i\hbar y, \quad [L_z, y] = -i\hbar x, \quad [L_z, z] = 0

[L_z, p_x] = i\hbar p_y, \quad [L_z, p_y] = -i\hbar p_x, \quad [L_z, p_z] = 0.

(b) Use these results to obtain

[L_z, L_x] = i\hbar L_y

directly from Equation 4.96. (c) Find the commutators

[L_z, r^2]

and

[L_z, p^2]

(where, of course,

r^2 = x^2 + y^2 + z^2

and

p^2 = p_x^2 + p_y^2 + p_z^2

). (d) Show that the Hamiltonian

H = (p^2/2m) + V

commutes with all three components of

\mathbf{L}

, provided that

V

depends only on

r

. (Thus

H

L^2

, and

L_z

are mutually compatible observables.)