DIRAC DYNAMICS / A BARRIER REOPENS

Klein Wave-Packet Tunnelling

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

A single-particle one-dimensional Dirac analogue with a prescribed electrostatic potential, not a quantum-field simulation. The barrier has smooth edges. Region probabilities at a finite time are not asymptotic reflection or transmission coefficients. No pair creation, disorder, loss or physical trap hardware.

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

The same wave packet. 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 show the model, never a reconstruction of the displayed shots. The observation gate is a non-invasive analysis boundary. No pair creation, detector inefficiency or trapped-ion apparatus is modeled.

Gerritsma et al. · Dirac simulation (2010) · Gerritsma et al. · Klein simulation (2011)

Start with the essentials

Focus question: Raise the barrier. Open a channel.
One-sentence intuition: The potential shifts both branches. A higher plateau can place the packet energy in the negative branch rather than in the gap.

Core mathematical model

Evolution

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

Dimensionless position, momentum and time use an arbitrary simulator energy scale. The velocity operator is the first Pauli matrix.

Energy branches

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

The plateau gap and negative branch explain the three presets. Interfaces and finite packet bandwidth still affect transmission.

Conserved probability

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

Density and current are calculated from the same spinor. The observation gate does not collapse or absorb the packet.

Common difficulties

What the picture means

Typical misconceptionThe density ribbon traces a real particle, and negative energy demonstrates pair creation.

Better mental modelA single-particle one-dimensional Dirac analogue with a prescribed electrostatic potential, not a quantum-field simulation. The barrier has smooth edges. Region probabilities at a finite time are not asymptotic reflection or transmission coefficients. No pair creation, disorder, loss or physical trap hardware.

Run the experiment

01
Predict first
Choose The gap blocks it and scrub from Prepare to Finish.
What to observe: Most probability returns to the left. Finite-time occupancy is shown throughout.
02
Change the cause
Raise the barrier with the second preset, then remove the mass with the third.
What to observe: The high barrier admits a propagating negative-energy channel; the massless chiral limit passes without reflection.
03
Collect and compare
Choose a frame. Collect 500 position and 500 separate velocity measurements; pin them, change the preparation and restore.
What to observe: The histogram and sample means approach model predictions with sampling uncertainty. Every raw outcome is in the CSV.