In one line
Quantum tunnelling is the effect in which a particle has a non-zero probability of passing through a potential barrier even when its energy is less than the barrier height, which is impossible classically.
Explained simply
Classically, a ball rolling towards a hill without enough energy to climb it simply rolls back. A quantum particle is described by a wave, and waves do not stop dead at a barrier: they fade gradually inside it. If the barrier is thin enough, a small part of the wave survives on the far side, so there is a small chance the particle appears there.
How it works
- Barrier: height \(V_0\), width \(a\); particle energy \(E < V_0\).
- Outside the barrier (\(V = 0\)), Schrödinger's equation gives oscillating waves (incident, reflected and transmitted).
- Inside the barrier, \(E - V_0 < 0\), so the equation becomes \(\dfrac{d^2\psi}{dx^2} = \kappa^2\psi\) with \(\kappa = \sqrt{2m(V_0 - E)}/\hbar\). The solution decays exponentially: \(\psi \propto e^{-\kappa x}\).
- Continuity of \(\psi\) and \(d\psi/dx\) at both edges lets a reduced wave continue beyond the barrier.
- Transmission probability is approximately \(T \approx e^{-2\kappa a}\). It falls sharply with a wider barrier, a higher barrier or a heavier particle.
Examples: alpha decay of nuclei, nuclear fusion in the Sun, the tunnel diode, the scanning tunnelling microscope (STM), and charge storage in flash memory.
Key formula
\(T \approx e^{-2\kappa a}\), \(\kappa = \dfrac{\sqrt{2m(V_0 - E)}}{\hbar}\). \(T\): transmission probability (no unit), \(a\): barrier width (m), \(V_0\): barrier height (J), \(E\): particle energy (J), \(m\): mass (kg), \(\hbar = 1.055\times10^{-34}\) J s. This is an approximate (thick-barrier) form.
Quick example
Electron with \(E = 1\) eV meets a barrier \(V_0 = 2\) eV, width 1 nm. \(\kappa = \sqrt{2\times9.11\times10^{-31}\times1.6\times10^{-19}}/1.055\times10^{-34} \approx 5.1\times10^9\) m−1. \(2\kappa a \approx 10.2\), so \(T \approx e^{-10.2} \approx 3.6\times10^{-5}\). About 4 in every 100,000 electrons get through. Double the width and \(T\) drops to about \(10^{-9}\).
Common mistakes
- Saying the particle gains energy to jump over the barrier; its energy stays the same.
- Thinking tunnelling probability is large for any barrier; it falls exponentially with width.
- Saying \(\psi\) is zero inside the barrier; it decays but is not zero (unless the barrier is infinite).
How MU asks it
Asked as "Explain quantum mechanical tunnelling with examples" (3 to 5 marks). Draw the barrier with the incident, decaying and transmitted wave, state the classical versus quantum result, give the \(e^{-2\kappa a}\) dependence, and list two or three applications.