Quantum Physics For Dummies
Book image
Explore Book Buy On Amazon

When you are working with potential barrier of height V0 and width a where E > V0, this means that the particle has enough energy to pass through the potential barrier and end up in the x > a region. This is what the Schrödinger equation looks like in this case:


The solutions for


are the following:


In fact, because there's no leftward traveling wave in the x > a region,


So how do you determine A, B, C, D, and F? You use the continuity conditions, which work out here to be the following:


Okay, from these equations, you get the following:

  • A + B = C + D

  • ik1(A – B) = ik2(C – D)

  • Ceik2a + Deik2a = Feik1a

  • ik2Ceik2aik2Deik2a = ik1Feik1a

So putting all of these equations together, you get this for the coefficient F in terms of A:


Wow. So what's the transmission coefficient, T? Well, T is


And this works out to be


Whew! Note that as k1 goes to k2, T goes to 1, which is what you'd expect.

So how about R, the reflection coefficient? Without going into the algebra, here's what R equals:


You can see what the E > V0 probability density,


looks like for the potential barrier in the figure.


That completes the potential barrier when E > V0.

About This Article

This article is from the book:

About the book author:

Steven Holzner is an award-winning author of technical and science books (like Physics For Dummies and Differential Equations For Dummies). He graduated from MIT and did his PhD in physics at Cornell University, where he was on the teaching faculty for 10 years. He’s also been on the faculty of MIT. Steve also teaches corporate groups around the country.

This article can be found in the category: