Quantum Physics For Dummies, Revised Edition
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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:

image0.png

The solutions for

image1.png

are the following:

image2.png

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

image3.png

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

image4.png

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:

image5.png

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

image6.png

And this works out to be

image7.png

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:

image8.png

You can see what the E > V0 probability density,

image9.png

looks like for the potential barrier in the figure.

image10.jpg

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.

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