Solve the following:
Delta function perturbation switched on and off
Q: A particle with mass m is in a one-dimensional infinite square-well potential of width a, so V(x)=0 for 0 <= x <= a, and there are infinite potential barriers at x=0 and x=a. Recall that the normalized solutions to the Schrodinger equation are
psi_n(x) = sqrt(2/a)sin[(n pi x)/a]
with energies
E_n = (hbar^2 (pi^2 n^2)/(2m a^2)
where n = 1,2,3,...
The particle is initially in the ground state. A delta-function perturbation
H_1 = K(delta(x-a/2))
(where K is a constant) is turned on at time t=-t_1, and turned off at t=t_1. A measurement is made at some later time t_2, where t_2 > t_1.
a) What is the probability that the particle will be found to be in the excited state n=3?
b) There are some excited states n in which the particle will never be found, no matter what values are chosen for t_1 and t_2. Which excited states are these?