Physics Problems & Solutions: Quantum Mechanics

The components of the orbital angular momentum operator $\vec{L}=(L_x,L_y,L_z )$ satisfy the following commutation relations.

$\\\left[L_x,L_y \right ]=i \hbar L_z \\\left[L_y,L_z \right ]=i \hbar L_x \\\left[L_z,L_x \right ]=i \hbar L_y$

What is the value of the commutator $\left[L_x L_y,L_z \right ]$ ?

A. $2i\hbar L_x L_y$
B. $i\hbar (L_x^2+L_y^2 )$
C. $-i\hbar (L_x^2+L_y^2 )$
D. $i\hbar (L_x^2-L_y^2 )$
E. $-i\hbar (L_x^2-L_y^2 )$

(GR0177 #43)

Solution:

Commutator identities: [A, B] = − [B, A]

and, [B, AC] = A[B, C] + [B, A]C

Therefore,
$\left[\underset{A}{\underbrace{L_x L_y}},\underset{B}{\underbrace{L_z}}\right] = - \left[\underset{B}{\underbrace{L_z}}, \underset{A}{\underbrace{L_x L_y}}\right]$

$\left[\underset{B}{\underbrace{L_z}}, \underset{A}{\underbrace{L_x}}\underset{C}{\underbrace{L_y}}\right]=L_x [L_z,L_y ]+[L_z,L_x ] L_y$

$\begin{aligned} -[L_z,L_x L_y ]&=-L_x [L_z,L_y ]-[L_z,L_x ] L_y \\&=-L_x (-i\hbar L_x)-i\hbar L_y L_y \\&=i\hbar L_x^2-i\hbar L_y^2 \\&=i\hbar (L_x^2-L_y^2 ) \end{aligned}$

Answer: D

Physics Problems & Solutions

Quantum Mechanics - Commutation Relations

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