Problem Set (due 10/28/02)

Exercise 31: Solve problems 6.5 and 6.6 of reference 1.

Exercise 32: Prove that the angular momentum operator $L=r\times p$ is hermitian.

Exercise 33: Prove that,

$$\Psi (x+a) = e^{(i/\hbar)ap}\Psi(x),$$

where $p= - i\hbar \partial/\partial x$, and $a$ is a finite displacement.

Exercise 34: Let $\hat{H}$ be the Hamiltonian operator of a system. Denote $\psi_k$ the eigenfunctions of $\hat{H}$ with eigenvalues $E_k$. Prove that $<\psi_n \vert[\hat{Q}, \hat{H}]\vert\psi_k> =0$, for any arbitrary operator $\hat{Q}$, when $n=k$.

Exercise 35: Prove that,

$$[x, H]= i \hbar p /m,$$

where, $H=p^2/(2m) +V(x)$.

Exercise 36: Prove that,

$$L_-Y_l^m = \hbar \sqrt{(l+m)(l-m+1)}Y_l^{m-1},$$

where $L_zY_l^m = m \hbar Y_l^m$, and $L^2 Y_l^m=\hbar^2 l (l+1) Y_l^m$.

Exercise 37: Consider a system described by the Hamiltonian matrix,

$$H=\begin{pmatrix}-E_0 & \Delta \\ \Delta & E_0 \\ \end{pmatrix},$$

where the matrix elements $H_{jk} =<\psi_j\vert\hat{H}\vert\psi_k>$. Consider that the system is initially prepared in the ground state, and is then influenced by the perturbation $W(t)$ defined as follows,

$$W(t)= \begin{pmatrix}0 & e^{-t^2/\tau^2 - i\omega t}\\ e^{-t^2/\tau^2 + i\omega t} &0\\ \end{pmatrix}.$$

Calculate the probability of finding the system in the excited state at time $t >> \tau$.