Principles of Quantum Mechanics · Problem 12.5.2

Rotational Invariance and Angular Momentum — R. Shankar

Physics — Quantum Mechanics · angular-momentum-matrices

Problem

(1) Verify that the

2 \times 2

matrices

J_x^{(1/2)}

J_y^{(1/2)}

, and

J_z^{(1/2)}

obey the commutation rule

[J_x^{(1/2)}, J_y^{(1/2)}] = i\hbar J_z^{(1/2)}

. (2) Do the same for the

3 \times 3

matrices

J_i^{(1)}

. (3) Construct the

4 \times 4

matrices and verify that

[J_x^{(3/2)}, J_y^{(3/2)}] = i\hbar J_z^{(3/2)}