A ball and an electron at a wall
A ball with less energy than the height of a hill cannot get over it. This is the rule of everyday (classical) physics.
A tiny particle like an electron is different. In quantum physics it behaves like a wave. A wave does not end at a sharp edge when it meets a barrier. This changes the answer.
What happens inside the barrier
Inside the barrier the particle wave gets smaller step by step (it decays). It does not drop to zero straight away. If the barrier is thin, some wave is left when the barrier ends. That wave continues on the other side, smaller but real.
The size of the wave squared tells the chance of finding the particle. So there is a small chance that the particle is found beyond the barrier. This is quantum tunnelling. The particle does not dig a hole or gain extra energy. It is an effect of being a wave.
How the chance depends on the wall
The chance of passing, T, is roughly T ≈ e−2κL. Here L is the width of the barrier, and κ gets larger when the barrier is higher or the particle is heavier.
- Thicker barrier: T falls very fast.
- Higher barrier: T falls.
- Heavier particle: T falls. That is why we never see a cricket ball tunnel through a wall: its chance is almost zero.
Where tunnelling is used and seen
- Scanning tunnelling microscope (STM): a very sharp metal tip is held a tiny distance above a surface. A small voltage makes electrons tunnel across the gap. A change of the gap by the width of a fraction of an atom changes the current a lot. The tip moves to keep the current fixed, and this map of motion shows atoms.
- The Sun: nuclei tunnel past their push-away force so fusion happens at the Sun’s temperature.
- Alpha decay: an alpha particle tunnels out of a nucleus.
- Flash memory and tunnel diodes: electrons tunnel through thin layers.
Try it
In the 3D set the height to 4 and move the width from 0.5 to 3. Write down how many of 100 electrons pass at 0.5, 1.5 and 3. Predict first: will doubling the width halve the number, or reduce it much more?
At home: stand a thin sheet of paper and a thick book. Ask friends which one blocks light. Light also leaks a little through thin materials, a bit like waves do.
Key formulas and definitions
- Transmission chance: T ≈ e^(−2κL)
- κ = √(2m(U − E)) / ħ (m = mass, U = barrier height, E = particle energy)
- Thicker or higher barrier, or heavier particle → smaller T
- Chance of finding the particle ∝ (wave size)²
Worked examples
1. A ball does not have enough energy to climb a hill. Where will it end up?
It comes back. A ball does not tunnel because it is heavy, so the chance is almost zero.
2. For an electron, κL = 1. Estimate the chance of tunnelling.
T ≈ e^(−2) ≈ 0.135, so about 13.5%.
3. The barrier width is doubled so κL goes from 1 to 2. Estimate the new chance.
T ≈ e^(−4) ≈ 0.018, about 1.8%. The chance fell by a factor of about 7.
4. T = 9%. Out of 100 electrons, how many do you expect to pass?
About 9. Which ones is random.
5. In a microscope each extra 0.1 nm of gap cuts the current to one tenth. The gap grows by 0.3 nm. By what factor does the current fall?
10 × 10 × 10 = 1000. The current becomes one thousandth.
6. Why can an electron tunnel but a person cannot walk through a wall?
A person is huge compared with an electron and has an enormous mass. So κ is huge and T is almost exactly zero.
Common mistakes
- Thinking the particle digs a tunnel. Nothing is dug; the wave simply leaks through.
- Thinking the particle gains energy to climb over. Its energy stays the same.
- Thinking all particles tunnel equally. Light, small particles and thin barriers matter most.
- Thinking each electron splits into a big and small part. Each electron is found in one place; only the chance is small.