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Quantum Tunnelling

Key idea: Understand quantum tunnelling as a wavefunction effect: finite transmission through a potential barrier even when energy is below barrier height (optional enrichment for A Level Physics).

  • A-level H2 Physics topic extensions
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Learning objectives

  • Explore potential-barrier transmission, tunnelling and X-ray production and spectra beyond the H2 syllabus.
Optional / Enrichment (9478 scope)

Quantum tunnelling is not explicitly listed in the 9478 quantum learning outcomes. Treat this lesson as enrichment that builds on the syllabus idea that particles are described by wavefunctions and probability density.

1. Definitions (Must Know)

A. Potential barrier

A potential barrier is a region where the particle has higher potential energy U than in neighbouring regions.

In a simple 1D model, a barrier has:

  • height U₀,
  • width L.
Matter wave at a finite potential barrierA particle with energy E below rectangular barrier height U zero approaches a barrier of width L. The incident and reflected wave components are on the left; the wavefunction decays exponentially within the classically forbidden barrier, and a smaller transmitted wave continues on the right.Potential-energy profileUxparticle energy E < U₀barrier height U₀width LWavefunction amplitude ψ(x)incident + reflectedexponential decaysmaller transmitted waveclassically forbidden, but ψ ≠ 0
For E below U₀, the wavefunction decays through the width L but remains non-zero, leaving a smaller transmitted wave beyond the barrier.

B. Classically allowed vs forbidden

  • If E > U₀, the particle is classically allowed to cross the barrier.
  • If E < U₀, the barrier region is classically forbidden (the particle “shouldn’t” cross in classical physics).

C. Quantum tunnelling

Quantum tunnelling is when there is still a non-zero probability of finding the particle on the far side of the barrier even when E < U₀.

2. Key Ideas (What Earns Marks)

  • In quantum physics, particles are described by a wavefunction ψ; probabilities come from |ψ|².
  • In a classically forbidden region (E < U), the wavefunction does not become zero immediately; it decays (often exponentially).
  • A thicker or “higher” barrier makes transmission less likely (probability decreases strongly, often exponentially).

Tunnelling probability vs barrier width (illustrative)

An exponential-style decay showing that transmission probability drops rapidly as barrier width increases.

Scroll across the graph to read all labels.

An exponential-style decay showing that transmission probability drops rapidly as barrier width increases.An exponential-style decay showing that transmission probability drops rapidly as barrier width increases.
This is why tunnelling is extremely sensitive to distance: even small increases in barrier width can reduce the current/probability by orders of magnitude.
Open full-size graph
View figure data
Values for Tunnelling probability vs barrier width (illustrative)
Barrier width (L/L0)T ∝ e^(−2κL) (scaled)
01
0.50.368
10.135
1.50.05
20.018
2.50.007
30.0025

3. Detailed Explanations

The figure above combines the barrier profile with the incident, reflected, decaying and transmitted wavefunction regions.

A. Why tunnelling is possible (wavefunction view)

If a particle is described by a wavefunction, it is spread out in space: it does not have a single exact position.

At a barrier, the wavefunction must behave smoothly (it does not “snap” to zero at the boundary). Inside a region where E < U, the wavefunction typically decreases rapidly, but it can remain non-zero throughout the barrier.

If ψ is still non-zero at the far side, then |ψ|² is non-zero there too, so there is a finite probability the particle is detected beyond the barrier.

4. Common Mistakes

  • Saying tunnelling “breaks conservation of energy” (it does not; the particle energy E stays the same in this simple model).
  • Treating tunnelling as a particle “teleporting” (it is a probability effect from the wavefunction).
  • Thinking E < U₀ means “impossible” (it is classically forbidden, not quantum-impossible).

5. Exam Tips

  • If the question asks “what changes the tunnelling probability?”, state the trends clearly:
    • barrier width L increases ⇒ transmission probability decreases,
    • barrier height (or U₀-E) increases ⇒ transmission probability decreases,
    • particle mass increases ⇒ transmission probability decreases.
  • Use precise language: “probability decreases strongly (often exponentially)”.

6. Worked Examples

Modelled example 1

Trend question (no calculation)

Core

Problem

A particle with E < U₀ approaches a finite barrier. State and explain what happens to tunnelling probability when width L increases.
Study the worked solution

  1. Describe the forbidden region

    Method

    The wavefunction decays rather than becoming immediately zero inside the barrier.

    Reason

    E < U₀ makes the region classically forbidden.

    Working

    ψ∼ e^(-κ x)

  2. Increase the width

    Method

    A wider barrier gives the wavefunction a longer decay distance.

    Reason

    Its amplitude at the far boundary becomes smaller.

    Working

    L↑ ⇒ |ψ_far|↓

  3. State the probability trend

    Method

    Transmission probability decreases strongly, often exponentially.

    Reason

    Detection probability depends on squared wavefunction magnitude.

    Working

    T ∝ e^(-2κ L)

Common misconception 2

Comparing two particles

Find and correct the mistake

Learner claim

An electron and proton have the same E and face the same U₀,L. A learner says their tunnelling probabilities are identical because their energies match. Diagnose the claim.

Try this before viewing the solution

More likely to tunnel

View solution step by step

  1. Identify the missing variable

    Method

    Barrier penetration depends on mass as well as energy and barrier properties.

    Reason

    The decay constant increases with particle mass.

    Working

    κ ∝ square root of (m(U₀-E))

  2. Compare masses

    Method

    The electron is much less massive than the proton.

    Reason

    It therefore has a smaller decay constant for the same barrier.

    Working

    mₑ≪ mₚ ⇒ κₑ < κₚ

  3. Conclude

    Method

    The electron is more likely to tunnel.

    Reason

    Its wavefunction penetrates farther and produces a larger T.

    Working

    Tₑ > Tₚ

7. Mind Stretchers

Mind stretcher 1: Why does tunnelling matter for microscopes?Extension

Show Answer

In a scanning tunnelling microscope, electrons tunnel across a tiny gap between a sharp tip and a conducting surface. The tunnelling current depends extremely strongly on the gap size, so measuring current can map surface height at near-atomic resolution.

8. Optional (Enrichment)

A. A common model for transmission probability

In many 1D models, the transmission probability has the form: T ∝ e^(-2κ L) where κ depends on the particle mass and on the energy gap (U₀-E).

B. Video intuition (optional)

Continue with the next resource in this course.

Course and syllabus information
Course
A-level H2 Physics topic extensions
Syllabus scope
Beyond the syllabus
Edition
A-level H2 Physics topic extensions