Potential step with infinite potential barrier (D.A.B Miller problem 2.8.6)

[roz77]

Homework Statement

We have a potential structure as follows: from z=negative infinity to z=-b, the potential is 0. At z=-b, the potential jumps up to some V_0. Then, at z=0, there is a potential barrier of infinite height. I need to solve for the magnitude of the reflected wave (i.e. the wave propagating to the left). Assume that E is greater than V_0.

Homework Equations

We know that to the left, where the potential is 0, the wavefunction has the from Aexp(ikz) + Bexp(-ikz), where k=sqrt(2mE)/h-bar. To the right, where the potential is equal to V_0, the wavefunction is of the form Csin(fz), where f=-sqrt(2m(E-V_0))/h-bar.

The Attempt at a Solution

I'm really unsure. I know that we have the boundary conditions at z=-b, that the wavefunction has to be continuous and the derivative has to be continuous as well. It get's really messy though, because z=-b, and not 0. The TA also told us to utilize the fact that the probability over the entire region (from negative infinity to 0) is 1. Problem is though, I'm having trouble normalizing everything. Anybody have any ideas?

 

[betel]

Normalizing is not possible in this case. You have a free wave on an infinite domain, so no chance of normalization.

So will have to keep one parameter, B as strength of the incoming wave and solve for the other two. You have two equations so there should be no problem, although the coefficients might be complicated.

 

[betel]

Actually there is a faster way without doing any calculations.

Maybe you misunderstood the TA but the fast ways takes into account a similar argument.

 

[roz77]

So what would the faster way to do it be?

 

[betel]

Well, you have an incoming wave of particles from the left hitting an infinite barrier and getting reflected.
How many particles will return?

 

[roz77]

See, I do get that. Clearly the entire wave eventually is reflected back because of the infinite barrier. However, it asks for the magnitude of the wave propagating to the left, and I took that to mean the wave that is reflected when it hits the step at z=-b. I also have to solve for C, so if the magnitude of the reflected wave is just 1, I don't see how we can possibly solve for C.

 

[betel]

You cannot distinguish between a wave that is reflected ab -b or at 0. They both contribute to the total magnitude which then of course has to be 1.
If you want to split it, you can only measure the intensity of the wave in the domain -b to 0. This will then consist of the incoming and the refelcted wave. To get this coefficient C you have to solve the two equations. This is not difficult. It will be a 5 line calculation (including the two equations you start with).

 

[roz77]

So would the magnitude of the reflected wave just be B+C?

 

[roz77]

Duh. Magnitude is 1. Nevermind.