Show that, for general operators A and B that have a commutation |, B| = C, "/ relation of the form one has (B) = f(B+c1) for an arbitrary function f. Hint: You can Taylor-expand the function. (b) (1 point) Apply this result to the harmonic-oscillator ladder operators and at and the number operator = and show that e = e, e-n at en e-^t. e -Xn = (c) (2 points) Show that for a Hermitian operator, the eigenvectors corresponding to different eigenvalues are orthogonal. 1