When a particle doesn't have as much energy as the potential of a barrier, you can use the Schrödinger equation to find the probability that the particle will tunnel through the barrier's potential. You can also find the reflection and transmission coefficients, R and T, as well as calculate the transmission coefficient using the *Wentzel-Kramers-Brillouin* (WKB) approximation.

Here's how it works: When a particle doesn't have as much energy as the potential of the barrier, you're facing the situation shown in the following figure.

_{0}.

In this case, the Schrödinger equation looks like this:

All this means that the solutions for

are the following:

This situation is similar to the case where E > V_{0}, except for the region

The wave function oscillates in the regions where it has positive energy, *x* < 0 and *x* > *a*, but is a decaying exponential in the region

You can see what the probability density,

looks like in the following figure.

## How to find the reflection and transmission coefficients

How about the reflection and transmission coefficients, R and T? Here's what they equal:

As you may expect, you use the continuity conditions to determine A, B, and F:

A fair bit of algebra and trig is involved in solving for R and T; here's what R and T turn out to be:

Despite the equation's complexity, it's amazing that the expression for T can be nonzero. Classically, particles can't enter the forbidden zone

because E < V_{0}, where V_{0} is the potential in that region; they just don't have enough energy to make it into that area.

## How particles tunnel through regions

Quantum mechanically, the phenomenon where particles can get through regions that they're classically forbidden to enter is called *tunneling*. Tunneling is possible because in quantum mechanics, particles show wave properties.

Tunneling is one of the most exciting results of quantum physics — it means that particles can actually get through classically forbidden regions because of the spread in their wave functions. This is, of course, a microscopic effect — don't try to walk through any closed doors — but it's a significant one. Among other effects, tunneling makes transistors and integrated circuits possible.

You can calculate the transmission coefficient, which tells you the probability that a particle gets through, given a certain incident intensity, when tunneling is involved. Doing so is relatively easy in the above example because the barrier that the particle has to get through is a square barrier. But in general, calculating the transmission coefficient isn't so easy. Read on.

## How to find the transmission coefficient with the WKB approximation

The way you generally calculate the transmission coefficient is to break up the potential you're working with into a succession of square barriers and to sum them. That's called the *Wentzel-Kramers-Brillouin* (WKB) approximation — treating a general potential, V(*x*), as a sum of square potential barriers.

The result of the WKB approximation is that the transmission coefficient for an arbitrary potential, V(*x*), for a particle of mass m and energy E is given by this expression (that is, as long as V(*x*) is a smooth, slowly varying function):

In this equation

So now you can amaze your friends by calculating the probability that a particle will tunnel through an arbitrary potential. It's the stuff science fiction is made of — well, on the microscopic scale, anyway.