Correct Statements About Hermitian Operator

TAMIL NADU SET 2024

Which of the following statements related to Hermitian Operator are correct?

(i) The eigenvalues of a Hermitian operator are real.
(ii) Two eigenfunctions of a Hermitian operator that correspond to different eigenvalues are orthogonal.
(iii) The eigenvalues of a Hermitian operator are not real.
(iv) Two eigenfunctions of a Hermitian operator that correspond to different eigenvalues are orthonormal.
(v) If two Hermitian operators commute, their product is also a Hermitian operator.

• (A) (i), (ii), (iii)
• (B) (ii), (iii), (iv)
• (C) (i), (ii), (v)
• (D) (iii), (iv), (v)

Correct Answer: (C) (i), (ii), (v)

Explanation:

• (i) Real Eigenvalues: Eigenvalues represent observables; for Hermitian operators, these must be real.
• (ii) Orthogonality: For $\hat{A}\psi_1 = a_1\psi_1$ and $\hat{A}\psi_2 = a_2\psi_2$, if $a_1 \neq a_2$, then $\langle\psi_1|\psi_2\rangle = 0$.
• (v) Commutation: The product $(\hat{A}\hat{B})^\dagger = \hat{B}^\dagger \hat{A}^\dagger = \hat{B}\hat{A}$. This product is Hermitian only if $\hat{B}\hat{A} = \hat{A}\hat{B}$.