Solving QM Griffiths 2.28: Transmission Coefficient

[genxhis]

The problem asks for the transmission coefficient for a wave packet with energy approximately E passing through a potential function with dirac delta wells of strength alpha at x = -a and x = +a. To solve the problem I split the region into the three obvious intervals [-inf, -a], [-a, a], and [a, +inf]. For the first two regions I expressed the solutions as (A or C) exp(i k x) + (B or D) exp(-i k x) and for the last as E exp(i k x) where k = sqrt(2 m E)/hbar. I then applied the two contraints at the two boundary conditions to reduce the five unknowns to just one. Finally I did some more algebra to find the trasmission coefficient as the square of (E/A). But the entire process was lengthy and tedious. I was wondering if someone could validate this answer:

$$T = \frac{1}{1 + 2 \beta^2[ (1+\beta^2) + (1-\beta^2)\cos 4ka -2\beta \sin4ka ]}$$​

where $$\beta = (m \alpha)/(\hbar k)$$.

 

[dextercioby]

There's no better checking than the one you can do it yourself by making sure your method & calculations were correct.

I think no one around here will do the calculations at this problem,just to agree or disagree with your answer.

Daniel.

 

[genxhis]

your right, sorry. i was hoping someone could recognize this as a special case of a more general problem or simply point out that the answer is unviable. but, I've looked it over more carefully, and i think it is. in any case, recapitulating what i did helps me understand it a little better.