Quantum Tunneling

What it is

Classically, a ball with energy E cannot cross a hill of height V > E. A quantum particle is a wave, and inside the barrier its wavefunction does not stop dead. It decays exponentially as e^(−κx). If the barrier is thin enough, some amplitude is left on the far side and continues as a transmitted wave.

For a rectangular barrier the transmission probability T has an exact closed form. Below the top (E < V) it is dominated by e^(−2κa), so doubling the width squares the suppression. Above the top (E > V) T is still not 1: the packet partly reflects off the step and shows resonances where whole half-waves fit inside the barrier.

Tunneling explains alpha decay, scanning tunneling microscopes, flash memory and tunnel diodes. In code it is a good stress test for a Schrödinger solver: compare your measured T against the formula. It is unrelated to physics-engine tunneling, where fast objects skip through thin colliders between steps.

Key formulas

Time-dependent Schrödinger equation (1D)

$i\hslash \frac{\partial \psi }{\partial t}=-\frac{{\hslash }^2}{2m}\frac{{\partial }^2\psi }{\partial x^2}+V(x)\psi$

Decay constant inside the barrier

$\kappa =\frac{\sqrt{2m(V-E)}}{\hslash }$

Exact transmission, E < V

$T={\left[1+\frac{V^2{\sinh }^2(\kappa a)}{4E(V-E)}\right]}^{-1}$

Opaque-barrier limit

$T\approx 16\frac{E}{V}\left(1-\frac{E}{V}\right)e^{-2\kappa a}(\kappa a\gg 1)$

Above the barrier, E > V

$T={\left[1+\frac{V^2{\sin }^2(ka)}{4E(E-V)}\right]}^{-1},k=\frac{\sqrt{2m(E-V)}}{\hslash }$

T = 1 exactly when ka = nπ (transmission resonance).

Rules of thumb

• In natural units (ħ = m = 1) κ = √(2(V − E)); κa ≳ 3 already means T below about 1 %.
• Handle E = V separately: the limit is T = 1 / (1 + mVa²/(2ħ²)), and the general formula divides by zero there.
• Electron, 1 eV below the top of a 1 nm barrier: κ ≈ 5.1 nm⁻¹, T ~ 10⁻⁴. At 0.5 nm it rises to ~10⁻².
• A packet's group velocity is ħk/m = √(2E/m). Faster packets carry more energy and tunnel more.

Implementation

export function transmission(E: number, V: number, a: number): number {
  if (E <= 0) return 0
  if (Math.abs(E - V) < 1e-9) return 1 / (1 + (V * a * a) / 2)
  if (E < V) {
    const kappa = Math.sqrt(2 * (V - E))
    const s = Math.sinh(kappa * a)
    return 1 / (1 + (V * V * s * s) / (4 * E * (V - E)))
  }
  const k = Math.sqrt(2 * (E - V))
  const s = Math.sin(k * a)
  return 1 / (1 + (V * V * s * s) / (4 * E * (E - V)))
}

// Reflection is whatever is not transmitted (probability is conserved).
export const reflection = (E: number, V: number, a: number) =>
  1 - transmission(E, V, a)

Common mistakes

• Assuming everything passes when E > V. Above-barrier reflection is real and oscillates with width.
• Using sinh(κa) for large κa without care: it overflows near κa ≈ 710. Switch to the exponential approximation first.
• Scaling the packet's height with its energy. |ψ|² integrates to 1 whatever the energy.
• Calling a barrier's T a 'speed' or a delay. It is a probability, measured over many particles.

Prompt

In TypeScript, implement transmission(E, V, a) for a 1D rectangular potential barrier in natural units (ħ = m = 1), covering E < V (sinh form), E > V (sin form) and the E = V limit, returning 0 for E ≤ 0. Then build a Three.js visual: a translucent barrier box of height V and width a, a dashed line at height E, and a Gaussian |ψ|² curve drawn on that line that travels at speed √(2E) and splits into a transmitted part (scaled by T) and a reflected mirror image (scaled by 1−T) at the barrier's left face, updating a preallocated buffer each frame. Unit-test that T + R = 1, that T decreases monotonically with a when E < V, that T = 1 at ka = π for E > V, and continuity across E = V within 1e-4.

Related

• Particle in a Boxwalls force whole half-waves, so energy comes in steps
• 2D Schrödinger Wave Packeta cloud of probability that spreads, scatters and interferes
• Tunneling & Continuous Collisionfast objects skip over thin walls between two ticks

Further reading

• OpenStax University Physics Vol. 3 — Quantum Tunneling of Particles through Potential Barriers