Pramod Kumar, Teacher at www.chegg.com, Delhi University, Numerade Educator

33:09

Introduction to Quantum Mechanics

(a) Starting with the canonical commutation relations for position and momentum (Equation 4.10), work out the following commutators:
$\left[L_{z}, x\right]=i \hbar y, \quad\left[L_{z}, y\right]=-i \hbar x, \quad\left[L_{z}, z\right]=0$,
$\left[L_{z}, p_{x}\right]=i \hbar p_{y}, \quad\left[L_{z}, p_{y}\right]=-i \hbar p_{x}, \quad\left[L_{z}, p_{z}\right]=0$
(b) Use these results to obtain $\left[L_{z}, L_{x}\right]=i \hbar L_{y}$ directly from Equation $4.96$.
(c) Evaluate the commutators $\left[L_{z}, r^{2}\right]$ and $\left[L_{z}, p^{2}\right]$ (where, of course, $r^{2}=$ $x^{2}+y^{2}+z^{2}$ and $\left.p^{2}=p_{x}^{2}+p_{y}^{2}+p_{z}^{2}\right)$
(d) Show that the Hamiltonian $H=\left(p^{2} / 2 m\right)+V$ 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.)

Chapter 4: Quantum Mechanics In Three Dimensions

Section 3: Angular Momentum