Question

Which of the following is a Hermitian operator?

• A. \( \hat{p}_x \)
• B. \( \frac{d}{dx} \)
• C. \( \frac{d^2}{dx^2} \)
• D. \( -\frac{d^2}{dx^2} \)

Hint:

Hermitian operators correspond to observable physical quantities in quantum mechanics.

Answer 1

The Correct Option is D

An operator \( \hat{O} \) is Hermitian if:
\[ \langle \psi | \hat{O} \phi \rangle = \langle \hat{O} \psi | \phi \rangle \] For differential operators:
\[ \left(\frac{d}{dx}\right)^\dagger = -\frac{d}{dx} \] which is not Hermitian.
\[ \left(\frac{d^2}{dx^2}\right)^\dagger = \frac{d^2}{dx^2} \] is not negative definite, so it's not Hermitian.
\[ \left(-\frac{d^2}{dx^2}\right)^\dagger = -\frac{d^2}{dx^2} \] which satisfies the Hermitian condition.