Quantum Tunnelling

Schrödinger equation · Transfer matrix · T(E) transmission curve · Resonant tunnelling

T = 1.00  |  R + T = 1.00  |  κ = 0.00 nm⁻¹ (decay rate inside barrier)  |  WKB: T ≈ e−2κd

⚛️ Quantum Tunnelling

In classical mechanics a particle with energy E < V₀ cannot cross a potential barrier of height V₀ — it simply bounces back. In quantum mechanics the wavefunction penetrates exponentially into the barrier and re-emerges on the other side with a finite transmitted amplitude. This is quantum tunnelling, one of the most radical departures from classical intuition.

🔬 Physics

The time-independent 1D Schrödinger equation −(ℏ²/2m)ψ″ + V(x)ψ = Eψ is solved exactly using the transfer matrix method. Each region (free space or barrier) contributes a 2×2 matrix; multiplying them gives the exact reflection amplitude r and transmission amplitude t. T = |t|² satisfies R + T = 1 (current conservation). For a single square barrier of height V₀ and width d, the exact result is T = [1 + V₀² sinh²(κd) / (4E(V₀−E))]⁻¹ where κ = √(2m(V₀−E))/ℏ.

🎮 How to Use

Use the Energy E/V₀ slider to move the particle energy relative to the barrier. Below 1.0 the particle tunnels; above 1.0 it transmits classically (but still shows quantum reflection). Switch to Double Barrier to see resonant tunnelling: T = 1 at discrete energies where standing waves form inside the well — the basis of the resonant tunnelling diode.

💡 Applications

Quantum tunnelling underlies: alpha decay (nucleus tunnels out), tunnel diodes, scanning tunnelling microscopy (STM), fusion in the Sun's core (proton tunnelling through Coulomb barrier), enzyme catalysis (H transfer), and flash memory (electrons tunnel through gate oxide to store bits).