Paper: CSIR-NET June 2011 · Subject: Physical Chemistry · Chapter: Quantum Chemistry · Topic: Quantum Chemistry – General · Marks: 2 · Difficulty: Medium
A constant of motion is defined by the equation:
(a)$[\mathrm{H}, \mathrm{A}]=0$
(b)$\langle[\mathrm{H}, \mathrm{A}]\rangle=0$
(c)$\mathrm{A}=\mathrm{f}(\mathrm{H})$
(d)$\mathrm{A}^{\dagger}=\mathrm{A}$
Answer
Answer: A ✓ checked by 4AB · confidence high
The source book printed B; on checking, A is correct — see the explanation.
Printed B; the definition is the operator commutator, not its expectation value.
Explanation
An observable A (with no explicit time dependence) is a constant of motion when its operator commutes with the Hamiltonian, since d⟨A⟩/dt = (i/ħ)⟨[H, A]⟩ vanishes for every state only if [H, A] = 0. A vanishing expectation value in one state (b) is not sufficient, and A = f(H) or A† = A do not define it. Answer: (a).