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).

About this simulation

This simulation solves the 1D time-independent Schrödinger equation exactly using the transfer matrix method (TMM): the potential is split into piecewise-constant segments — free space and barrier regions — and each segment contributes a 2×2 complex matrix that propagates the wavefunction amplitude across it. Multiplying these matrices for the whole structure gives the exact reflection and transmission amplitudes, from which the transmission probability T(E) and reflection R(E) = 1 − T(E) are computed with no approximation. Three potential shapes are available: a single square barrier, a double barrier (resonant tunnelling), and a step potential.

🔬 What it shows

The left canvas plots the stationary-state probability density |ψ(x)|² as it penetrates and decays inside the barrier and re-emerges on the far side with reduced amplitude; the right canvas plots the exact transmission coefficient T(E) across the energy range 0 to 2V₀. For the single square barrier, T follows the closed-form result T = [1 + V₀²sinh²(κd)/(4E(V₀−E))]⁻¹, with κ = √(2m(V₀−E))/ℏ the decay constant inside the barrier — this is computed directly by the transfer-matrix code, not read off a formula.

🎮 How to use

Drag Energy E/V₀ to move the particle's energy relative to the barrier height — below 1.0 the particle is classically forbidden yet still tunnels; above 1.0 it is classically allowed yet still partially reflects. Adjust Barrier height V₀ and Barrier width to see T fall roughly exponentially with both. Switch to Double Barrier to reveal the Gap width slider and watch resonant tunnelling: T rises to exactly 1.0 at discrete energies where a standing wave forms in the well between the two barriers, the mechanism behind the resonant tunnelling diode.

💡 Did you know?

The info strip also reports the WKB approximation T ≈ e^(−2κd) alongside the exact transfer-matrix value, so you can see directly how close (or far) the simple exponential estimate comes to the true quantum-mechanical result as barrier width and energy change.

Frequently asked questions

What numerical method does this simulation use?

It uses the transfer matrix method (TMM) applied to the time-independent 1D Schrödinger equation. The potential is broken into piecewise-constant segments; each segment contributes a propagation matrix and each boundary contributes a matching matrix built from the local wave numbers. Multiplying all of these together gives an exact 2×2 matrix relating the incoming and outgoing wave amplitudes, from which the transmission coefficient T and reflection coefficient R = 1 − T follow with no approximation.

How is quantum tunnelling possible if the particle's energy is below the barrier?

Inside a classically forbidden region (E < V₀) the wavefunction does not oscillate but decays exponentially, with decay constant κ = √(2m(V₀−E))/ℏ. If the barrier is thin enough, the wavefunction has not decayed to zero by the far edge, so it re-emerges there as a propagating wave with reduced but non-zero amplitude — giving a transmission probability T > 0 even though a classical particle with the same energy would simply bounce back.

What is resonant tunnelling and why does T reach exactly 1?

In Double Barrier mode, the region between the two barriers acts as a quantum well. At certain particle energies the wave reflected back and forth inside the well interferes constructively, forming a standing wave — a quasi-bound state. At those resonant energies the transfer matrix gives T = 1, meaning the particle passes through both barriers with perfect transmission despite each barrier individually blocking most of the wave. This effect is the operating principle of the resonant tunnelling diode.

Why does the transmission curve T(E) look different for the step potential mode?

The step potential has only one interface instead of a barrier of finite width, so there is no exponential decay-and-recovery region — the wave either propagates (E > V₀, with partial quantum reflection at the step) or decays into the forbidden region and cannot re-emerge (E < V₀, where T = 0 exactly). This produces a curve with T = 0 for all E below V₀ and a smooth rise toward 1 above it, unlike the square and double barrier modes where T oscillates or recovers after the classically forbidden region.

What real physical systems rely on quantum tunnelling like this?

The same square-barrier and double-barrier mathematics modelled here describes alpha decay, where a nucleus's alpha particle tunnels through the Coulomb barrier; the scanning tunnelling microscope, where electrons tunnel across a vacuum gap between a sharp tip and a surface; resonant tunnelling diodes, which use a double-barrier quantum well exactly like the one in this simulation's Double Barrier mode; and flash memory, where electrons tunnel through a thin oxide layer to store or erase a bit.