# Quantum Tunneling

> A quantum packet hitting a barrier higher than its energy still partly gets through: transmission falls off exponentially with barrier width.

- Category: Physics › Quantum Mechanics
- Level: advanced
- Tags: quantum, tunneling, barrier, transmission, wave-packet
- Updated: 2026-11-01
- URL: https://gdcs.coody.app/physics/quantum-mechanics/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](https://gdcs.coody.app/physics/quantum-mechanics/schrodinger-2d) solver: compare your measured `T` against the formula. It is unrelated to [physics-engine tunneling](https://gdcs.coody.app/systems/collision/tunneling), where fast objects skip through thin colliders between steps.

## Key formulas

- **Time-dependent Schrödinger equation (1D):** $i\hbar\,\frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2 \psi}{\partial x^2} + V(x)\,\psi$
- **Decay constant inside the barrier:** $\kappa = \frac{\sqrt{2m(V - E)}}{\hbar}$
- **Exact transmission, E < V:** $T = \left[1 + \frac{V^2 \sinh^2(\kappa a)}{4E(V-E)}\right]^{-1}$
- **Opaque-barrier limit:** $T \approx 16\,\frac{E}{V}\left(1 - \frac{E}{V}\right) e^{-2\kappa a}\quad (\kappa a \gg 1)$
- **Above the barrier, E > V:** $T = \left[1 + \frac{V^2 \sin^2(k a)}{4E(E-V)}\right]^{-1},\quad k = \frac{\sqrt{2m(E-V)}}{\hbar}$ — 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

Rectangular-barrier transmission (ħ = m = 1), safe at E = V:

```ts
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.

## Interactive visual

The cyan curve is the packet's probability density riding on the dashed yellow energy line, and the red block is the barrier (height V, width a). Lower the energy below the barrier top or widen the barrier and the transmitted bump shrinks; the readout shows the exact T and R.

## Prompt

```text
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 Box](https://gdcs.coody.app/physics/quantum-mechanics/particle-in-box.md) — A particle trapped between hard walls can only hold standing waves, so its energy is quantized as n² and superpositions slosh at beat frequencies.
- [2D Schrödinger Wave Packet](https://gdcs.coody.app/physics/quantum-mechanics/schrodinger-2d.md) — A Gaussian wave packet in 2D drifts, spreads and splits at a potential bump; brightness shows |ψ|² and colour shows the quantum phase.
- [Tunneling & Continuous Collision](https://gdcs.coody.app/systems/collision/tunneling.md) — A fast object can be on one side of a thin wall at one tick and past it at the next. Swept tests (CCD) check the whole path between positions.

## Further reading

- [OpenStax University Physics Vol. 3 — Quantum Tunneling of Particles through Potential Barriers](https://openstax.org/books/university-physics-volume-3/pages/7-6-the-quantum-tunneling-of-particles-through-potential-barriers)

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Source: Game Development Cheat Sheet (https://gdcs.coody.app)