# Questions About Quantum Delta Function Potentials [closed]

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.

## closed as unclear what you're asking by CuriousOne, ACuriousMind♦, Gert, user36790, honeste_vivereJul 16 '16 at 19:56

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• You can solve or look up the solution for the rectangular well (or barrier). What happens to the solution when you vary the parameters in such a way that the sequence of barriers approximates a delta function? – CuriousOne Jul 14 '16 at 20:03
• See e.g. Wikipedia. – Qmechanic Jul 14 '16 at 20:09
• I'm not exactly sure what the question here is. What about the standard solution to the $\delta$-function potential is unclear to you? – ACuriousMind Jul 14 '16 at 21:18
• I don't understand why a particle can get through a delta potential. – Shrodinger 2016 Jul 14 '16 at 21:21
• The Dirac potential is just a mathematical device, it's not literally a spike up to infinity. There is nothing in the real world that is infinite, potentials do fade away given enough distance. – user108787 Jul 14 '16 at 21:41