Principles of Quantum Mechanics · Problem 10.1.1

Systems with N Degrees of Freedom — R. Shankar

Physics — Quantum Mechanics · direct-product-operators

Problem

Show the following: (1)

[\Omega_1^{(1)}\otimes I^{(2)}, I^{(1)}\otimes\Lambda_2^{(2)}]=0

for any

\Omega_1^{(1)}

and

\Lambda_2^{(2)}

(10.1.17a) (operators of particle 1 commute with those of particle 2). (2)

(\Omega_1^{(1)}\otimes\Gamma_2^{(2)})(\theta_1^{(1)}\otimes\Lambda_2^{(2)})=(\Omega\theta)_1^{(1)}(\Gamma\Lambda)_2^{(2)}

(10.1.17b) (3) If

[\Omega_1^{(1)}, \Lambda_1^{(1)}]=\Gamma_1^{(1)}

then

[\Omega_1^{(1)\otimes(2)}, \Lambda_1^{(1)\otimes(2)}]=\Gamma_1^{(1)}\otimes I^{(2)} \tag{10.1.17c}

and similarly with

1\to 2

. (4)

(\Omega_1^{(1)\otimes(2)}+\Omega_2^{(1)\otimes(2)})^2=(\Omega_1)^{(1)}\otimes I^{(2)}+(I^{(1)}\otimes(\Omega_2)^{(2)}+2\Omega_1^{(1)}\otimes\Omega_2^{(2)}

(10.1.17d)