Transition probability for a step function perterbation

1. JFuld

23
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

2. vela

12,561
Staff Emeritus
You probably don't need to solve the Schrodinger equation with the perturbation (unless the problem asks you to). In time-dependent perturbation theory, you work with the eigenstates of the unperturbed Hamiltonian.

In case you do want to solve for the eigenstates when the perturbation is turned on, the boundary conditions are (1) ψL(0)=ψR(a)=0; (2) ψL(a/2)=ψR(a/2); and (3) ψ'R(a/2)-ψ'L(a/2) = some constant, where the two wave functions are the solutions for the left and right halves of the well. You can figure out what the constant is by integrating the Schrodinger equation from x=a/2-ε to x=a/2+ε. The wave function remains continuous at x=a/2, but the delta function causes a discontinuity in the derivative.
Close. f(t) is basically right, though it would be easier to say f(1)=1 for -t1<t<t1 and f(t)=0 elsewhere. The time-independent part needs to include the delta function, so you should have H=k δ(x-a/2). That makes evaluating the matrix elements trivial.