## transition probability for a step function perterbation

1. The problem statement, all variables and given/known data

a particle with mass=m is in a 1-D infinite square well of width a. The particle is initially in ground state. A delta function potential V1=k δ(x-a/2) is turned on at t= -t1 and turned off at t=t1. A measurement is made at t2, where t2>t1. What is the probability that the particle will be found in the third excited state (n=3)?

2. Relevant equations
recall the normalized solution to schodinger's eq for the unperturbed infinite square well:

Psi(x) = sqrt(2/a)sin(npix/a), E(n)= ((h/2pi)(pi)(n))^2/(2ma^2); n=1,2,3...

P(i -> f) = 1/hbar^2 (<psi(final) lHl psi(initial)>)^2(integral of f(t)e^(i(E(final)-E(initial))t/hbar)dt from t(initial) to t(final)

3. The attempt at a solution

the wave function is subject to H(t); H(t) can be factored into a time independent operator, H, and a time dependent piece, f(t), which does not operate on the wave function.
hence H(t) = Hf(t)
It would be really annoying to type out my work so ill try and explain what I am stuck on.

V1 basically acts as an infinite potential barrier at x= a/2.

I initially tried solving shrodingers eq with V(x) = V1.

My intuition tells me that in order to solve this differential equation I need to use a Laplace transformation, however when I try to work it out, I get an answer that I know isnt correct (i.e. psi(x) = 0).

So I scraped that plan and solved schodingers eq for two wave functions, one for (0<x<a/2) and the second for (a/2<x<a).
This worked out ok; I was not sure how to pick the coefficients for each wave function, im assuming they need to be continuos at x=a/2, but also equal to zero at x=0, x=a/2,x=a ?

Furthermore, for the relation H(t) = Hf(t), I am not sure what H or f(t) are.

Since I took took the δ(x-a/2) into account by deriving two wave functions, I made an educated guess and took H=k, and f(t) =1 (-t1<t<t1) and f(t)=0 (t1<t<t2).

I put all this mess into the equation given above for P(i -> f) and simplified, continued simplifying, and eventually decided to request assistance from the internet.

Usually our HW problems aren't this tedious so I think I probably set the problem up completely wrong.

Can anyone share some wisdom to a tired, haggard undergrad?
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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