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First order time dependent perturbation theory problem

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A system is in an eigenstate | psi_i > with energy E_i. The perturbation

V(t) = H'exp(-((alpha)^2)(t^2))

is turned on at t_i = -infinity and left on until t_f = infinity. Here H' is independent of time, and alpha is a constant. Show that at t_f = infinity, the probability that the system has evolved into the eigenstate | psi_f > with energy E_f is

P(i -> f) = (pi)/((hbar^2)(alpha^2)|< psi_f | H' |psi_i >|^2 exp[-((E_f - E_i)^2)/(2(hbar^2)(alpha^2)].

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Solution Preview

If we denote the time dependent wave function of the system by |psi(t)> and the eigenstates with energy E_r by |r>, then we can write:

|psi> = sum over r of c_r (t) exp(-E_r t/hbar) |r>

The coefficients c_r(t) would not depend on time if there were no time dependent perturbation. If the system is initially in the state |i>, then that means that initially c_i = 1 and all other coefficients are zero. If we compute the coefficients in perturbation theory, we find ...

Solution Summary

We consider a time dependent perturbation of the form

V(t) = H'exp[-((alpha)^2)(t^2)]

and compute to first order in perturbation theory the transition probability for an eigenstate with energy E_1 at
t = -infinity to evolve to an eigenstate with energy E_2 at t = infinity.

Similar Posting

Quantum Mechanics: Time dependent perturbation problem

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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