Double Scattering Homework: Solving for Transmission Amplitude

[carllacan]

Homework Statement

We have a double step potential described by
-V0 for |x| < a/2
0 otherwise.

That is, three regions separated at -a/2 and a/2 with 0, -V0 and 0

We have a particle coming from the left with E > 0. Find the transmission amplitude S(E)

Homework Equations

The Attempt at a Solution

To begin with I have "moved" everything by a/2 so the first step is at 0 and the second at a, to simplify things.

Now I see two possible ways to do it. The first one is to

1) Find a solution for each of the three regions, which will involve defining 6 constants.
2) Setting the constant that corresponds to a particle coming from the right to 0, reducing it to 5 constants .
3) Use the normalization and continuity conditions on the wavefunction and its derivative at 0 and a will yield 5 equations, with which in principle I can solve for the constants.
4) Identify the solution corresponding to the particle being in the third region (transmitted) and find the transmission probability.

The second one is

1) Treat it as two scattering processes for a particle of energy E, one with a step from 0 to -V0 at x=0 and other with a step from -V0 to 0 at x=a.
2) Find the transmission probabilities of both processes.
3) The total trnasmission probability is the product of both probabilities.

Now, questions about this:

a) The transmission probability is the integral over ALL space of the wavefunction for the third region. Is that right?
b) Is the transmission "amplitude" just the (positive) root square ot the transmission probability?
c) Is the second method even possible or do I have to consider the infinite reflections and transmissions that the first transmitted wave will suffer inside the well? I guess is the latter, since later on I'm asked to find something called the ressonance factor.
d) In the case I have to consider the reflections between the steps can you give me a hint on where to start? Does my first method already account for this or do I have to postulate an infinite linear combination of solutions in the second region?
e) In the second method (assuming it is right) can I change the values of the second step so that it is at x=0 and from 0 to +V0?

And finally, can you recommend me any book where this problem is treated?

Thank you for four time.

 

[PhysicsGente]

It is a free particle right? So write down your wave equation for each of the three regions and think about what happens at the boundary. Also, you don't need to do any integrals to find the transmission probability and amplitude, look at any intro to QM book for details on rectangular barrier problems (I'm thinking Griffiths).

 

[carllacan]

Yeah, that's what I did, write the equation for three particles (the one in the middle with V0 more energy than the others) and use the continuity equations at 0 and a for find the constants. Thanks for your reference, I hadn't realized that Griffiths had a solution for this exact problem, I'm glad to see my solution was almost the same as his :-)

However I still have a couple of doubts, out of curiosity:

first: in this scenario the wave transmitted at the first step will travel to the right, encounter the second step, and produce two new waves: one that will be transmitted farther to the right and one that will be reflected back to the left; the second one will encounter again the first step and produce yet another couple of waves, a transmitted one which will go to the left and a reflected one that will go to the right, repeating the process (potentially ad eternum). How does this formulation accounts for this infinite reclections?

My guess is that it does so via the interdependency of the constants for each wave. That is, the amplitude of the "first" reflected wave accounts for the sum of all the waves that come out of the second region, and the amplitude of the "final" wave that moves to the right acconts for all the waves that get out of the second region.

And second: would using the second method I mentioned make any sense?

Thank you for your time.