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:

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)

• Ce^ik2a + De^–ik2a = Fe^ik1a

• ik₂Ce^ik2a – ik₂De^–ik2a = ik₁Fe^ik1a

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.