Electron Tunneling

Physics:	Quantum Mechanics, Electron Tunneling
Grade Range:	Elementary School, Middle School, High School
Format:	Stage

This demonstration is a part of the Quantum Mechanics Show. This demonstration is not typically performed by itself, since it does require background knowledge in the topic to be understood. In the Quantum Mechanics Show, this demonstration is scripted, so although this write-up does not provide a script, one should be aware of the differences that happen between scripted and non-scripted demonstrations.

In order to perform this demonstration, you need two presenters.

Materials

• Box from Schrodinger's Cat demonstration
• Tennis Ball
• Two Small Identical Balls (Ping-pong balls or similar)
• Electron Tunneling Diagrams

Safety Precautions

Please read the General Safety Precautions section of the Demonstration Safety page before performing this demonstration.

Demonstration

1. Before the demonstration begins, be sure that Presenter A and Presenter B already each have one of the identical balls in a pocket. Have Presenter B set up the box while Presenter A introduces the idea of tunneling through a wall.
2. Presenter A only: Ask for a volunteer from the audience, and give the volunteer the tennis ball. Ask the volunteer to try and throw the tennis ball through the wall of the box. Try as they might, the volunteer will fail.
3. Presenter A can either ask for a new volunteer for keep the volunteer on stage. Presenter B will now adjust the box so there is only one panel standing, angled so it looks like a thin wall. Presenter A during this time will explain the change to be a way to think of a very thin barrier.
4. Have the volunteer now stand with Presenter B. Presenter B will show the audience the small ball and give it to the volunteer, explaining that this will represent an electron.
5. Presenter A will now ask the volunteer to throw the small ball at the barrier. Presenter B should whisper to the volunteer to throw it over the barrier. Once it is thrown over, Presenter A will toss it back, explaining how this would have worked.
6. Presenter A will not ask the volunteer to throw the small ball one last time. Presenter B should whisper to the volunteer to fake throwing it. When it is "thrown", Presenter A will pretend to "catch" it, and reveal the ball that they had hidden. Presenter B should quickly and secretly hide the ball the volunteer had.

Why This Works

As is demonstrated at the beginning of this demo, a large object cannot tunnel through a thick barrier unless it has a high amount of energy. However, this changes when we look at atoms and electrons, where an electron can sometimes make it through a barrier when it lacks the energy to do so. Electron Tunneling is the phenomenon which allows electrons to travel through barriers that they otherwise shouldn’t be able to go through. To understand what really happens with electron tunneling, one needs to look at the time-independent Schrödinger equation for particle-wave duality:

Time Independent Schrödinger Equation

d2 / dx2 Ψ(x) = (2m) / (ħ2) M(x) Ψ(x)	M(x): Quantified Energy of the Particle	M(x)= (V(x) - E(x))
If M(x) is constant and positive:	V(x): Potential Energy	E(x): Kinetic energy (x-axis)
(2m)/(ħ2) M(x) Ψ(x) = κ2 Ψ(x)	Where κ2 = (2m)/(ħ2) M	ħ: "h-bar", or the Planck Constant (h)/2pi
If M(x) is constant and negative:		Ψ(x): The Wave Function
(2m)/(ħ2) M(x) Ψ(x) = - k2 Ψ(x)	Where k2 = - (2m)/(ħ2) M	m: Mass of the Particle

Without going too deeply into the math, we can make out what the equations are saying. The first equation is showing the second derivative of the time independent Schrödinger equation. This means that we are looking at the Schrödinger equation without the time factor, so the only factors we will focus on are:

• The mass of the particle, m
• The quantified energy of the particle, M(x)
• The Wave Function, Ψ(x)

We know what the mass of an electron is, so we need to solve for the energy of our particle. If the particle has a positive and constant M value, then we can solve for κ². If the value is negative, then we are solving for k. In either case, we will be solving for the Probability of the particle making it through the barrier.

If a particle has really high energy, it might be able to make it over the barrier without having to tunnel through. This we represent with the ball being thrown over the barrier. Although this is good for our particle, it is not the effect we are looking for with the particle. Instead, we are looking for any situations where the particle doesn't have the energy to go over the barrier, but manages to somehow make it to the other side. This is because the particle, although it doesn’t go over the barrier, still has some chance of going through.

The probability of a particle making it through a barrier can be thought of as a percentage comparison. If the potential energy of a particle is equal to, for example, 70% of the energy barrier’s value, then the particle would have a 70% chance of tunneling through the barrier on the first try. This is because tiny particles can also behave like waves, in what is called Particle-Wave Duality. This means a particle can travel through a barrier by acting like a wave instead of a particle. This allows electrons to get through energy barriers that they should be blocked by, since their wave function can potentially last long enough to make it through rather than the physical particle bouncing off! This is what is demonstrated at the end with the sleight of hand trick; The particle somehow makes it through the barrier without bouncing off, even though it didn't have the energy to go over the barrier.

Additional Information

• Electron Tunneling is the phenomenon that allows for touchscreen technology! A touchscreen detects where the voltage increases across the potential gap, which happens due to an increase in electrons tunneling across.
• This demonstration is a part of the Quantum Mechanics Show