Discussion
Consider an operator \(\hat{A}\) which is not Hermitian. Find the possible values of \(c\) and \(d\) such that the operator \((c\,\hat{A} - d\,\hat{A}^{\dagger})\) is Hermitian.
Answer: A
The operator is Hermitian for \(c = i\) and \(d = i\).
Take the adjoint, using \((c\hat{A})^{\dagger} = c^{*}\hat{A}^{\dagger}\):
\[ \begin{aligned} (i\hat{A} - i\hat{A}^{\dagger})^{\dagger} &= -i\hat{A}^{\dagger} + i\hat{A} \\ &= i\hat{A} - i\hat{A}^{\dagger} \end{aligned} \]
It equals itself, so it is Hermitian. In general the adjoint of \(c\hat{A} - d\hat{A}^{\dagger}\) is \(c^{*}\hat{A}^{\dagger} - d^{*}\hat{A}\), and matching terms needs \(d = -c^{*}\). For c = d = 1 this gives \(\hat{A} - \hat{A}^{\dagger}\), which is anti-Hermitian. For c = i, d = −i it gives \(i(\hat{A} + \hat{A}^{\dagger})\), also anti-Hermitian, and c = −1, d = i fails the condition since \(-c^{*} = 1 \neq i\).
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