The operator is hermitian if

(5 points) The operator is hermitian if A = At The operator is anti-hermitian if A = -At (a) Demonstrate that the expectation value of the anti-hermitian operator is imag- inary. (b) Show that the eigenvalue of the anti-hermitian operator is imaginary. (c) Show that the eigenvectors of an anti-hermitian operator corresponding to dif- ferent eigenvalues are orthogonal. (d) Consider a commutator C = [A, B] . If A and B are hermitian is C hermitian or anti-hermitian ? . If A and B are anti-hermitian is their commutator hermitian or anti-hermitian or neither ? (e) Show that any operator can be decomposed into the sum of hermitian and anti- hermitian operators