Through the Wall: Modeling Quantum Tunneling

Project Overview:

In classical physics, if a ball does not have enough energy to roll over a hill, it will always roll back down, no exceptions. In quantum physics, particles can sometimes appear on the other side of an energy barrier even without enough energy to classically cross it, a real effect called quantum tunneling. It sounds like magic, but it is measurable, well understood, and the entire reason technologies like scanning tunneling microscopes and even some parts of nuclear fusion in the sun actually work. In this project, you will build a physical demonstration of the classical picture, then use a short calculation to see just how real tunneling probability behaves, and why you never notice it happening to everyday objects like yourself.

Materials Required:

A marble, small ball, or coin

A ramp or a stack of books to create a slope, a cardboard tube also works well

A small barrier, such as a thick book or block, to act as the energy hill

A phone or calculator

Optional, a computer with internet access to run a short calculation in Google Colab

Part One, the Classical Picture

Set up a ramp leading up to your barrier, positioned so a rolling ball would need a certain amount of speed to get over the top.

Release the ball from a low starting height so it clearly does not have enough energy to clear the barrier. Watch it roll partway up and roll back down every time. Repeat this five or six times to confirm it never randomly ends up on the other side.

Now raise the starting height until the ball finally has enough energy to roll over. This is the classical rule, energy in equals energy out, and there are no exceptions or surprises.

Part Two, Why Quantum Particles Break This Rule

In quantum mechanics, a particle is not a tiny ball with a fixed position, it is described by a wave that represents the probability of finding it in different places. When that wave hits an energy barrier, most of it reflects back, matching the classical picture, but a small piece of the wave leaks through to the other side, even though the particle technically did not have enough energy to go over the top. If you then measure where the particle is, there is a small but real chance you find it on the far side of the barrier, as if it tunneled straight through.

The probability of tunneling depends heavily on three things, the height of the barrier, the width of the barrier, and the mass of the particle. Thicker or taller barriers make tunneling less likely, and heavier particles make it dramatically less likely, which is the actual reason you have never seen a marble tunnel through a wall.

Part Three, See the Numbers For Yourself

Open Google Colab at colab.research.google.com and run this short calculation, which uses a simplified version of the real tunneling probability formula.

import math

def tunneling_probability(mass, barrier_height, barrier_width, particle_energy):

hbar = 1.0546e-34

k = math.sqrt(2 mass (barrier_height - particle_energy)) / hbar

probability = math.exp(-2 k barrier_width)

return probability

electron_mass = 9.11e-31

prob_electron = tunneling_probability(electron_mass, barrier_height=1e-19, barrier_width=1e-9, particle_energy=0.5e-19)

print("Electron tunneling probability:", prob_electron)

marble_mass = 0.01

prob_marble = tunneling_probability(marble_mass, barrier_height=1e-19, barrier_width=1e-9, particle_energy=0.5e-19)

print("Marble tunneling probability:", prob_marble)

Run it and compare the two printed numbers. The electron's tunneling probability will be a small but very real, nonzero number. The marble's tunneling probability will print as essentially zero, so close to zero that it would never happen even if you waited far longer than the age of the universe. This is the actual mathematical reason tunneling matters for electrons and atoms but is completely irrelevant for anything you can hold in your hand.

Congratulations, you just explored the same physics that allows the sun to shine, since tunneling is what lets hydrogen nuclei fuse together in the sun's core despite not technically having enough energy to overcome their natural repulsion.

Fun Fact:

Quantum tunneling is not just theoretical, it is the working principle behind the scanning tunneling microscope, a real instrument that can image individual atoms on a surface by measuring the tiny tunneling current of electrons jumping between the surface and a needle tip hovering just above it, without ever physically touching it.