(Solved): 32. Consider a Hermitian operator \( \hat{Q} \). Answer each question explaining your reasoning in ...

32. Consider a Hermitian operator \( \hat{Q} \). Answer each question explaining your reasoning in detail. a) Explain what is meant by a Hermitian operator. [4] b) Assuming \( \alpha \) is a complex number - under what condition on \( \alpha \) is \( \alpha \hat{Q} \) hermitian? [ 4 ] c) Show that the eigenvalues of a Hermitian operator are real. [4] d) Assume that \( f(x) \) and \( g(x) \) are two eigenfunctions of \( \hat{Q} \) with eigenvalue \( q \). Show that any linear combination of \( f \) and \( g \) is again an eigenfunction of \( \hat{Q} \) with eigenvalue \( q \). [4] e) Consider the relation \[ \frac{d}{d t}\langle Q\rangle=\frac{i}{\hbar}\langle[\hat{H}, \hat{Q}]\rangle, \] where \( Q \) is an observable and \( \hat{H} \) is a Hamiltonian. Evaluate the right hand side of the equation for \( \hat{Q}=\hat{H} \) and explain the meaning of the result.