---
title: "Can a Particle Cross an Impossible Wall? Interactive Lab | Sandbox Physics"
description: "Why can a particle sometimes be detected beyond a barrier it classically cannot cross?"
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last_modified: "2026-10-27"
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Quantum Lab 02 · forbidden region and transmission

# Quantum Tunneling Chamber

**Why can a particle sometimes be detected beyond a barrier it classically cannot cross?** Inside the barrier the wave's amplitude decays but never quite reaches zero. A barrier of finite width leaves a little amplitude on the far side. Over many repeated trials, the detector beyond it clicks with a small, steady probability.

Send a wave packet at a rectangular barrier that is higher than the particle's energy. Change the barrier height, the width and the particle energy to see the amplitude enter the classically forbidden region and decay inside it. Repeat the measurement many times and a stable transmission probability appears.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

The glowing wave surface shows probability amplitude, not the path of a particle. Each detector flash is the outcome of one measurement. Tunneling does not mean the particle briefly “borrows” energy. Instead, the quantum state does not drop to zero inside the barrier.

Physics tutorial

## Quantum tunneling: how impossible becomes improbable

**Background** A classical particle cannot cross a barrier higher than its energy. A quantum state obeys a wave equation, and its wavefunction must be continuous, so the amplitude reaches into the barrier and decays exponentially.

**Why it matters** Tunneling is a real, measurable effect. Fusion in stars, the scanning tunneling microscope and many electronic devices depend on it.

Start with the essentials

- **Focus question**: If the particle lacks enough energy to climb the barrier, why can a detector beyond it still click?
- **One-sentence intuition**: The amplitude inside the barrier is not zero. If the barrier has finite width, some amplitude is left on the far side, so a few of many repeated trials are detected beyond it.

### Core mathematical model

#### Time-dependent Schrödinger equation

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

The potential changes the spatial evolution of the wavefunction, but a finite barrier does not abruptly cut it to zero.

#### Decay in the forbidden region

$$
\psi(x)\propto e^{-\kappa x},\qquad \kappa=\frac{\sqrt{2m(V_0-E)}}{\hbar}
$$

When $E<V_0$, amplitude decays exponentially inside the barrier instead of propagating as an oscillation.

#### Transmission through a rectangular barrier

$$
T=\left[1+\frac{V_0^2\sinh^2(\kappa a)}{4E(V_0-E)}\right]^{-1}
$$

At fixed energy and barrier height, increasing the width $a$ makes the transmission drop quickly.

### Common difficulties

#### The particle does not borrow energy

**Typical misconception** Energy–time uncertainty lets the particle briefly borrow enough energy to cross.

**Better mental model** In steady-state scattering from a fixed barrier, the incoming and transmitted parts of the wave have the same energy. Tunneling happens because the quantum state is spread out in space, and energy is conserved throughout.

#### The glowing wave is not the particle’s path

**Typical misconception** The luminous waveform traces the particle’s actual route through the wall.

**Better mental model** The waveform shows the probability amplitude. Each detector flash is one measurement at one place. Nothing in the experiment measures a continuous path inside the barrier.

### Run the experiment

1. 01

   #### Build the statistics first

   Choose “Thin barrier,” launch a stream, and watch both detectors.

   **What to observe:** You cannot predict any single result, but the fraction detected beyond the wall gets closer and closer to the predicted transmission.

2. 02

   #### Make the wall thicker

   Hold energy and barrier height fixed while increasing $a$.

   **What to observe:** The amplitude decays more inside the thicker wall, and transmission events become exponentially rarer.

3. 03

   #### Approach the barrier top

   Raise $E$ while keeping $E<V_0$.

   **What to observe:** The decay constant shrinks, the wave reaches deeper into the barrier, and detections beyond the wall become much more likely.

Physical conclusion

In tunneling, the particle does not break through an impenetrable wall. The barrier is finite, the amplitude beyond it is not zero, and that gives a measurable chance of transmission.

## Related experiments and articles

Probes and measurement · Reading path

### Diffraction, phase and surface structure

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **STM: a current contour is more than a height map**](https://sandboxphysics.com/stories/probe-evidence/stm-current-and-surface-inference/)

Condensed matter · Reading path

### Materials that control current and light

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Why a tunnel diode carries less current at a higher voltage**](https://sandboxphysics.com/stories/condensed-matter/esaki-tunnel-diode/)
2. [Lab **Transistor Bias and Output Curves** Transistor output curves](https://sandboxphysics.com/labs/transistor-output-characteristics/)

Condensed matter · Reading path

### Cold liquids and superconducting order

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **How tunneling current revealed the superconducting gap**](https://sandboxphysics.com/stories/condensed-matter/giaever-gap-spectroscopy/)
2. [Lab **BEC–BCS Crossover** Pairing across the crossover](https://sandboxphysics.com/labs/bec-bcs-crossover/)

Quantum & fields · Starter path

## Start from a single measurement

Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)