Solving QM Problem: Fermi's Golden Rule & Transitional Probability

[ashah99]

Homework Statement

Consider a free particle of mass m and energy E moving from left to right in onedimension with periodic boundary conditions on length L. This means a planewave (traveling wave) with positive momentum.

Suppose there is perturbation V(x) = w*δ(x-x_0) with x a real number. What is the probability per unit time that the particle is scattered so that it moves from right to left (traveling wave with negative momentum) with energy E after scattering? Compute your answer to lowest nonvanishing order in time-dependent perturbation theory.

Relevant Equations

transition probability per unit of time = (2*pi)/(h_bar)*(|<f|H'|i>|^2)*p(Ef)

Hello all, I would like some guidance on how to approach/solve the following QM problem.

My thinking is that Fermi's Golden Rule would be used to find the transitional probability. I write down that the time-dependent wavefunction for the free particle is [1/sqrt(L)]*exp(i*p*x/h_bar)*exp(-i*E*t/h_bar), where p is the momentum and E is energy. My understanding is that p = 2*pi*n*h_bar/L and E = p^2/2m. But, now when trying to set up to the probability per unit time that the particle is scattered so that it moves from right to left, I am a bit lost. Any ways to solve would be appreciated.

 

[Nate810]

Fermi's Golden Rule is indeed the appropriate method to solve this question. To set it up, you need to calculate the matrix element between the initial and final states of the particle. The initial state is the momentum eigenstate of the particle, given by [1/sqrt(L)]*exp(i*p*x/h_bar). The final state is the same but with a negative sign in the exponential, which corresponds to a momentum pointing in the opposite direction. The matrix element is given by M = <initial state|V|final state> where V is the scattering potential. This is the probability per unit time that the particle is scattered so that it moves from right to left. Once you have calculated the matrix element, you can plug it into the expression for Fermi's Golden Rule, which gives the transition rate:Rate = (2pi/h_bar)*|M|^2. This is the probability per unit time that the particle is scattered so that it moves from right to left. I hope this helps!