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.

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 RIGHTFLOW LEFTBARRIER

BEYOND THE GATE

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

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Model → fresh preparations → records

Make it measurable.

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

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Recorded mean velocity

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Recorded gate occupancy

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Numerical norm error

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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

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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.

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$xE_0/(\hbar c)$ · $E/E_0$

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.

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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) · Gerritsma et al. · Klein simulation (2011)

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)

Dirac’s electron: a successful equation with an unfinished story

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 misconceptionThe density ribbon shows the path of a real particle, and negative energy means pairs are being created.

Better mental modelThis 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

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.
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.
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.