Problem 3.4 – Griffith’s Intro to QM

Problem 3.4

(a) Show that the sum of two hermitian operators is hermitian.
(b) Suppose $\hat{Q}$ is hermitian, and $\alpha$ is a complex number. Under what condition (on $\alpha$ ) is $\alpha \hat{Q}$ hermitian?
(c) When is the product of two hermitian operators hermitian?
(d) Show that the position operator $(\hat{x})$ and the Hamiltonian operator $(\hat{H} = -(\hbar^2 /2m)d^2/dx^2 +V(x))$ are hermitian.