homework and exercises - Questions About Quantum Delta Function Potentials

1. 
   

   I didn't think that it would be possible for a wave function to get through the delta function because there is no "leakage" of the wave function through an infinite potential barrier. I can understand why a particle could get through a non infinite barrier because of this exponential leakage but I don't understand how a particle could get through an infinite delta potential. My thinking right now is that maybe there is some imaginary (as apposed to real) leakage, similar to how a quadratic that doesn't cross the $x$ axis in the real plane may cross the $x$ axis in the imaginary plane. Is this the reason?

   
   



2. 
   

   I tried to solve for the reflection and transmission coefficients by solving for the two solutions outside the delta function and you obviously get 4 different exponentials with imaginary exponents. I know that these two functions have to touch so you equate them at $x=0$ and get $A+B=C+D$. Then I did the finding the change in slope trick and I get another relation between these 4 coefficients. The problem is that this gives me 2 equations and 4 coefficients to solve for, so I'm not sure what to do now (also a k value so 5 variables I guess). For the bound solution the coefficients were much nicer and it was easy to solve, but I'm not sure how to solve this one. My initial thinking is that I can get rid of the D coefficient since I only want the transmitted wave to be right-moving, but that only cuts me down to 3 variables when I need either 2 or another equation.

   
   

Answer

Even with a delta function potential, continuity of the wave function is still required. (Please see comment from ACuriousMind below on this).

The derivative of the wavefunction is obviously not continuous, however. You can find the discontinuity by integration about the delta function from +s to -s, where s is a small parameter.

You then let s go to 0 and check the behaviour of the derivative. You don't state the wavefunction(s) you are using, but here is a plot of a typical one.

You can find the actual calculations here:

Delta Function Wikipedia