---
title: "Klein Wave-Packet Tunnelling Interactive Lab | Sandbox Physics"
description: "Why can a higher barrier let a Dirac wave packet through when a lower one reflects it? Because a high enough barrier opens a negative-energy channel inside it."
canonical_url: "https://sandboxphysics.com/labs/klein-wavepacket-tunneling/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/klein-wavepacket-tunneling/"
last_modified: "2026-10-27"
---

Dirac dynamics / A higher barrier lets the wave through

# Klein Wave-Packet Tunnelling

**Why can a higher barrier let a Dirac wave packet through when a lower one reflects it?** Because a high enough barrier opens a negative-energy channel inside it. Raising the barrier lifts both energy branches there. Once the plateau is high enough, the packet's energy lines up with the negative-energy branch, and the packet has a way through. A lower barrier simply reflects it. That is Klein tunneling.

Send a positive-energy wave packet into a smooth barrier. Compare a barrier whose gap blocks the packet with a higher one that opens a channel for negative-energy propagation. Move the observation gate and collect independent position samples.

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

This is a one-dimensional, single-particle Dirac analogue in a given electrostatic potential, not a quantum-field simulation. The barrier has smooth edges. The probability in each region at a finite time is not the long-time reflection or transmission coefficient. Pair creation, disorder, loss and physical trap hardware are not modeled.

FLOW RIGHT FLOW LEFT BARRIER

BEYOND THE GATE $0$

Height: density × 18 · width is an illustrative extrusion · arrows: local current direction · gold: observation gate

$0$

Model → fresh preparations → records

## Make it measurable.

0

500 position measurements and 500 velocity measurements use different fresh preparations. These are ideal simulated outcomes, not a joint measurement or a particle trajectory.

Recorded mean position

$\text{—}$

Recorded mean velocity

$\text{—}$

Recorded gate occupancy

$\text{—}$

Numerical norm error

$\text{—}$

01 / MODEL

### The whole journey

Horizontal: position. Downward: time. Brightness: density. Pale line: model mean. Click to choose a time and gate. Arrow keys move them; Shift takes larger steps.

$xE_0/(\hbar c)$ · $tE_0/\hbar$

02 / OBSERVATION

### One frame. Independent shots.

Cyan: model density. Gold bars: recorded position frequencies. Coral: the gate. Counts do not reconstruct the current.

Move observation gate $0$

03 / READ THE CAUSE

### One wave packet split across three regions.

Cyan: right region. Coral: left region. Grey: barrier neighborhood. They sum to one. Fixed region edges lie two length units outside each nominal barrier edge.

04 / ENERGY BRANCHES

### Why the barrier reopens.

Position horizontally, energy vertically. Shaded: local gap. Gold: energy at the selected central momentum, a reference for a finite-bandwidth packet.

$0$

$xE_0/(\hbar c)$ · $E/E_0$

Last 40 raw outcomes

| Trial | Measurement | Outcome |
| --- | --- | --- |

Model, units & limits

Choose an arbitrary simulator energy scale. Position, momentum and time use the units below; the simulated speed is one. This is a one-dimensional spinor field, not a three-dimensional particle path.

$$
x_{\rm unit}=\frac{\hbar c}{E_0},\quad p_{\rm unit}=\frac{E_0}{c},\quad t_{\rm unit}=\frac{\hbar}{E_0}
$$

$$
i\partial_t\psi=(-i\sigma_x\partial_x+\mu\sigma_z+V)\psi,\quad \mu=mc^2/E_0
$$

$$
\rho=\psi^\dagger\psi,\quad j=\psi^\dagger\sigma_x\psi,\quad \partial_t\rho+\partial_x j=0
$$

$$
V(x)=\frac{V_0}{2}\left[\tanh\frac{x+d/2}{0.4}-\tanh\frac{x-d/2}{0.4}\right]
$$

Unitary Strang splitting, time step at most 0.025. A plateau is locally propagating when the squared energy relative to the potential exceeds the squared mass. A finite packet and two smooth interfaces also introduce reflection, resonances and delay.

1024 points on a periodic interval of length 128; displayed window is cropped. The edge probability and norm error below monitor numerical limits. Position outcomes are grid-cell centers. Sampling errors do not include discretization or experimental noise.

$0$

Smooth diagrams always show the model; they are not reconstructed from the displayed shots. The observation gate is an analysis boundary that does not disturb the wave. Pair creation, detector inefficiency and the trapped-ion apparatus are not modeled.

[Gerritsma et al. · Dirac simulation (2010)](https://arxiv.org/abs/0909.0674) · [Gerritsma et al. · Klein simulation (2011)](https://arxiv.org/abs/1007.3683)

## From this model to the evidence

Can raising the barrier reopen a channel?

- **Try**: Compare the preset where the gap blocks the packet with the one where the barrier is raised. At the same final time, collect position samples, then move the observation gate.
- **Read**: Compare the fraction of samples to the right of the gate with the model’s occupancy at that finite time. The gate is only an analysis threshold; moving it keeps the samples already collected.
- **Scope**: Here a smooth, finite barrier shifts the one-dimensional Dirac branches. The cited trapped-ion experiment used a linear potential, and the Lab does not reproduce its data or hardware. Occupancy at a finite time differs from the asymptotic transmission coefficient. This is a single-particle calculation, so it has no pair creation.

$$
E_\pm(p)=V\pm\sqrt{p^2+\mu^2}
$$

Model reference: [Gerritsma et al. · Klein simulation (2011)](https://arxiv.org/abs/1007.3683)

[Dirac’s electron: a successful equation with an unfinished story](https://sandboxphysics.com/stories/quantum-evidence/dirac-electron-spin-relativity/)

Physics tutorial

## Klein Wave-Packet Tunnelling

**Background** Send a positive-energy wave packet into a smooth barrier. Compare a barrier whose gap blocks the packet with a higher one that opens a channel for negative-energy propagation. Move the observation gate and collect independent position samples.

**Why it matters** Find out why raising an electrostatic barrier can reopen a channel for the packet to propagate.

Start with the essentials

- **Focus question**: Why can a higher barrier let a Dirac wave packet through when a lower one reflects it?
- **One-sentence intuition**: The potential shifts both branches together. On a higher plateau, the packet's energy can fall in the negative branch instead of in the gap.

### Core mathematical model

#### The Dirac equation in one dimension

$$
i\partial_t\psi=(p\sigma_x+\mu\sigma_z+V)\psi
$$

Position, momentum and time are dimensionless, measured against an arbitrary energy scale of the simulator. The velocity operator is the first Pauli matrix.

#### The two energy branches

$$
E_\pm(p)=V\pm\sqrt{p^2+\mu^2}
$$

The gap and the negative branch on the plateau explain the three presets. The interfaces and the packet's finite bandwidth still affect how much is transmitted.

#### Probability is conserved

$$
\partial_t\rho+\partial_xj=0,\quad j=2\operatorname{Re}(\psi_1^*\psi_2)
$$

The density and the current are both computed from the same spinor. The observation gate does not collapse or absorb the packet.

### Common difficulties

#### What the picture means

**Typical misconception** The density ribbon shows the path of a real particle, and negative energy means pairs are being created.

**Better mental model** This is a one-dimensional, single-particle Dirac analogue in a given electrostatic potential, not a quantum-field simulation. The barrier has smooth edges. The probability in each region at a finite time is not the long-time reflection or transmission coefficient. Pair creation, disorder, loss and physical trap hardware are not modeled.

### Run the experiment

1. 01

   #### Predict first

   Choose The gap blocks it and scrub from Prepare to Finish.

   **What to observe:** Most of the probability comes back to the left. The region probabilities shown are always those at the current, finite time.

2. 02

   #### Change the conditions

   Raise the barrier with the second preset, then remove the mass with the third.

   **What to observe:** The high barrier opens a negative-energy channel that the packet can propagate through. In the massless chiral limit, the packet passes with no reflection.

3. 03

   #### Collect and compare

   Choose a frame. Collect 500 position measurements and a separate 500 velocity measurements, pin them, then change the preparation and restore it.

   **What to observe:** The histogram and the sample means approach the model predictions, within the sampling uncertainty. Every raw outcome is in the CSV.

Physical conclusion

This is a one-dimensional, single-particle Dirac analogue in a given electrostatic potential, not a quantum-field simulation. The barrier has smooth edges. The probability in each region at a finite time is not the long-time reflection or transmission coefficient. Pair creation, disorder, loss and physical trap hardware are not modeled.

## Related experiments and articles

Quantum evidence · Reading path

### Laws and the formation of quantum mechanics

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

1. [Article **Dirac’s electron: a successful equation with an unfinished story**](https://sandboxphysics.com/stories/quantum-evidence/dirac-electron-spin-relativity/)
2. [Lab **Dirac Trembling Motion** Watch a free Dirac wave packet tremble](https://sandboxphysics.com/labs/dirac-trembling-motion/)
3. [Lab **Hydrogen Orbitals & Spectra** Compare the nonrelativistic hydrogen spectrum](https://sandboxphysics.com/labs/hydrogen-orbitals-spectra/)
4. [Lab **Coupled Spin Spectroscopy** Compare spin spectra from an effective model](https://sandboxphysics.com/labs/coupled-spin-spectroscopy/)

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)