# How Particles Pass Through Potential Barriers That Have Less Energy

When you are working with potential barrier of height V_{0} and width *a* where E > V_{0}, 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

*ik*_{1}(A – B) =*ik*_{2}(C – D)C

*e*^{ik}^{2}+ D^{a}*e*^{–}^{ik}^{2}= F^{a}*e*^{ik}^{1}^{a}*ik*_{2}C*e*^{ik}^{2}–^{a}*ik*_{2}D*e*^{–}^{ik}^{2}=^{a}*ik*_{1}F*e*^{ik}^{1}^{a}

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 *k*_{1} goes to *k*_{2}, 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 > V_{0} probability density,

looks like for the potential barrier in the figure.

That completes the potential barrier when E > V_{0}.