An operator A is defined as $\mathrm{A}=-\mathrm{d} / \mathrm{dx}+\mathrm{x}$. Which one of the following statements is true?
(a)A is a Hermitian operator.
(b)$\mathrm{A}^{\dagger}$ is an antihermitian operator.
(c)Both $\mathrm{AA}^{\dagger}$ and $\mathrm{A}^{\dagger} \mathrm{A}$ are Hermitian.
(d)$\mathrm{AA}^{\dagger}$ is Hermitian, but $\mathrm{A}^{\dagger} \mathrm{A}$ is antihermitian
Answer
Answer: C ✓ checked by 4AB · confidence high
The source book printed no answer; this one was worked out and checked — see the explanation.
Explanation
A = d/dx + x has A† = −d/dx + x, so A is neither Hermitian nor anti-Hermitian; but A†A and AA† are always Hermitian (they are the raising/lowering-operator products of the harmonic oscillator).