Delta function perturbation switched on and off
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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?
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Solution Summary
We explain how, using first order time dependent perturbation theory, one can compute the transition probabilities. We give a rigorous proof that only transitions to odd n states are possible to all orders in perturbation theory.
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Denote the eigenstates corresponding to the eigenvalue E_n as |n>. The wavefunction if the mass is in state
|n> is then
psi_n(x) = <x|n> = sqrt(2/a) sin(n pi x/a)
We can always express the state of the mass in the form:
|psi(t)> = sum over n from n = 1 to infinity of c_n(t) Exp(-i E_n t/hbar) |n>
The amplitude c_n to first order in perturbation theory is given by:
c_n(t) = 1/(i hbar) Integral from -t_1 to t of <n|V(t)|1> exp(i omega t) dt
where V(t) is the perturbation Kdelta(x-a/2)F(t) and omega = (E_n - E_1)/hbar = hbar pi^2(n^2 -1)/(2ma^2)
and F(t) = 1 if -t_1 < t< t_1
The c_n(t) are constant when t > t_1, as the perturbation is switched off and the system's evolution proceeds from that ...
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