Problem Set (due 10/01/02)

Exercise 11: Consider a distribution of charges $Q_i$, with coordinates $r_i$, interacting with plane polarized radiation. Assume that the system is initially in the eigenstate $\Phi_j$ of the unperturbed charge distribution.

(A) Write the expression of the sinusoidal perturbation in terms of $Q_i, r_i$, and the radiation frequency $\omega$ and amplitude $\epsilon_0$.

(B) Expand the time dependent wave function $\psi$ of the charge distribution in terms of the eigenfunctions $\Phi_k$ of the unperturbed charge distribution.

(C) Find the expansion coefficients, according to first order time dependent perturbation theory.

(D) What physical information is given by the square of the expansion coefficients?

(E) What frequency would be optimum to populate state k? Assume $E_k \geq E_j$.

(F) Which other state could be populated with radiation of the optimum frequency found in term (E)?

(G) When would the transition $j\rightarrow k$ be forbidden?

Exercise 12: A particle in the ground state of a square box of length $\vert a\vert$ is subject to a perturbation $\omega(t) = a x e^{-t^2/\tau}$.

(A) What is the probability that the particle ends up in the first excited state after a long time $t >> \tau$?

(B) How does that probability depend on $\tau$?

Exercise 13:

(a) Compute the minimum energy stationary state for a particle in the square well (See Fig.1) by solving the time independent Schrödinger equation.

(b) What would be the minimum energy absorbed by a particle in the potential well of Fig.1?

(c) What would be the minimum energy of the particle in the potential well of Fig.1?

(d) What would be the minimum energy absorbed by a particle in the potential well shown in Fig.2? Assume that $\lambda$ is a small parameter give the answer to first order in $\lambda$.

Exercise 14: (a) Prove that $\hat{P} = e^{-\hat{H}}$ is a hermitian operator.

(b) Prove that $\hat{P} =$   Cos$(\hat{H})$ is a hermitian operator.