What is Quantum Tunneling Anyway?

When Electrons Start To Act Funny…

Quantum mechanics is very different from the classical approach.
Since my Ph.D. involves solving quantum transport equations, I also have to learn one or two things about quantum effects. One such effect is quantum tunneling.

And as I built my first numerical algorithm for the Liouville-von-Neumann equation based on a resonant tunneling diode, I needed to understand this effect early on.

Image by geralt on pixabay.

Again, we can start with classical mechanics and work toward the quantum world.

Imagine a rubber ball at the foot of a hill. Crossing it requires the rubber to have the same amount of kinetic energy (or more) than the potential energy of the hill.

In other words, we need to push the ball hard enough to roll over the hill. If we are unable do so, the rubber ball will simply roll back to our feet. I tried to visualize this in the following picture:

Fig. 1: Classical mechanics: Ball rolling over a hill. Image by author.

Now imagine the same constellation. However, instead of a ball to our feet, we have an electron in front of the hill. And this time, the hill is not a hill but a potential barrier…