How Particles Pass Through Potential Barriers That Have Less Energy

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.

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