GATE 2026 PH – Question 13

Correct answer: (A) $c=i$ and $d=i$

Explanation

An operator $\hat O$ is Hermitian if $\hat O^\dagger=\hat O$. For the combination $\hat O=c\hat A-d\hat A^\dagger$,
$$\hat O^\dagger=c^*\hat A^\dagger-d^*\hat A .$$

**Hermiticity requires** $\hat O^\dagger=\hat O$:
$$c^*\hat A^\dagger-d^*\hat A=c\hat A-d\hat A^\dagger .$$

Comparing the coefficients of $\hat A$ and of $\hat A^\dagger$ (A is not Hermitian, so $\hat A$ and $\hat A^\dagger$ are independent):
- coefficient of $\hat A$: $-d^*=c$,
- coefficient of $\hat A^\dagger$: $c^*=-d$ (the same condition).

So $c=-d^*$.

**Test the options:**
- $c=i,\ d=i$: $-d^*=-(-i)=i=c$ ✓.
- $c=1,\ d=1$: $-d^*=-1\neq1$ ✗.
- $c=-1,\ d=i$: $-d^*=i\neq-1$ ✗.
- $c=i,\ d=-i$: $-d^*=-(i)=-i\neq i$ ✗.

Answer **$c=i$ and $d=i$** (A), that is, the operator $i(\hat A-\hat A^\dagger)$.