Hermitian Operator Condition Derivation

An operator $\hat{O}$ is Hermitian if it is equal to its Hermitian conjugate (adjoint), denoted by $\hat{O}^\dagger$. Mathematically, the condition is:

$ \hat{O}^\dagger = \hat{O} $

Operator Adjoint Calculation

We are given the operator $\hat{O} = c\hat{A} - d\hat{A}^\dagger$. To find when $\hat{O}$ is Hermitian, we first calculate its adjoint $\hat{O}^\dagger$. Using the properties of adjoints, specifically $( \alpha \hat{X} )^\dagger = \alpha^* \hat{X}^\dagger$ and $(\hat{X} - \hat{Y})^\dagger = \hat{X}^\dagger - \hat{Y}^\dagger$, where $\alpha^*$ is the complex conjugate of $\alpha$, we get:

$ \hat{O}^\dagger = (c\hat{A} - d\hat{A}^\dagger)^\dagger $

$ \hat{O}^\dagger = (c\hat{A})^\dagger - (d\hat{A}^\dagger)^\dagger $

$ \hat{O}^\dagger = c^* (\hat{A})^\dagger - d^* (\hat{A}^\dagger)^\dagger $

Since $(\hat{A}^\dagger)^\dagger = \hat{A}$, the expression simplifies to:

$ \hat{O}^\dagger = c^* \hat{A}^\dagger - d^* \hat{A} $

Condition for Hermiticity

For $\hat{O}$ to be Hermitian, we must equate $\hat{O}$ and $\hat{O}^\dagger$:

$ c\hat{A} - d\hat{A}^\dagger = c^* \hat{A}^\dagger - d^* \hat{A} $

Rearranging the terms to group $\hat{A}$ and $\hat{A}^\dagger$:

$ (c + d^*) \hat{A} = (c^* + d) \hat{A}^\dagger $

Assuming $\hat{A}$ and $\hat{A}^\dagger$ are linearly independent (which is true since $\hat{A}$ is not Hermitian), the coefficients must satisfy:

$ c + d^* = 0 \quad \text{and} \quad c^* + d = 0 $

These two conditions are equivalent to:

$ c = -d^* \quad \text{and} \quad d = -c^* $

Testing the Options

Let's check the given options using the conditions $c = -d^*$ and $d = -c^*$. The correct answer is Option A.

• Option A: $c = i, d = i$
  • Check $c = -d^*$: $i = -(i)^* = -(-i) = i$. (True)
  • Check $d = -c^*$: $i = -(i)^* = -(-i) = i$. (True)
  Both conditions hold.
• Option B: $c = 1, d = 1$
  • Check $c = -d^*$: $1 = -(1)^* = -1$. (False)
• Option C: $c = -1, d = i$
  • Check $c = -d^*$: $-1 = -(i)^* = -(-i) = i$. (False)
• Option D: $c = i, d = -i$
  • Check $c = -d^*$: $i = -(-i)^* = -(i) = -i$. (False)

Only Option A satisfies both conditions required for the operator $(c\hat{A} - d\hat{A}^\dagger)$ to be Hermitian when $\hat{A}$ is not Hermitian.