# Two shallow wells moving together

1. Feb 26, 2015

### WalkThePlanck

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

Particle is placed in a field of two shallow one-dimensional potential wells approaching each so that:

$V(x,t)=U(x-\frac{L(t)}{2})+U(x+\frac{L(t)}{2})$

(Reminder: shallow well is a well that can support only one bound state). At t=-∞ the wells are infinitely far apart from each other and the particle is bound by one of them. The distance between the wells L(t) very slowly decreases so that at some instant they fuse into single well V(x)=2U(x) which still remains shallow. What is the probability that the particle will remain bound?

2. The attempt at a solution

I'm not really sure how to start this. I was thinking of using the WKB approximation to find wavefunctions before and after and finding the overlap by treating the merging of the wells as a sudden perturbation, but since the function of the potential isn't specified I'm not sure how to do that.

2. Feb 26, 2015

### HoursofHomework

I am working on this problem too, and am also having issues.

So if we use the WKB approximation, we really just need to find p(x) and p(xo), which will be

$$p(x) = \sqrt{2m} (E-U(x - \frac{L(t)}{2})-U(x+\frac{L(t)}{2}))^{1/2}$$

right? I'm not sure what else to do with it though.