DIRAC DYNAMICS / TWO ENERGY BRANCHES

Dirac Trembling Motion

Launch a free Dirac wave packet, watch its probability current reverse, then prepare only positive energies. Scrub the full evolution and sample position and velocity on separate fresh preparations.

A dimensionless one-dimensional quantum simulator. The two spinor components are not the two energy branches. Position density and current come from the same evolved state. No real electron trajectory, relativistic field quantization or pair creation is simulated.

FLOW RIGHTFLOW LEFT

MEAN VELOCITY

0

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

0

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

\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

A trembling center. A changing velocity.

Cyan: model mean position. Coral: model mean velocity on its own scale. Gold points: sample means with one-standard-error bars.

04 / ENERGY BRANCHES

Two branches, one spinor.

Momentum horizontally, energy vertically. Dot size shows initial spectral weight on each branch; it is a visual scale, not a separate probability axis.

0

$pc/E_0$ · $E/E_0$

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

P_\pm(p)=\frac12\left(I\pm\frac{p\sigma_x+\mu\sigma_z}{\sqrt{p^2+\mu^2}}\right)

Exact free evolution for each Fourier mode. The initial Gaussian has the selected spinor mixing and phase. Projection is applied before evolution and renormalized; it changes the prepared state rather than deleting a visual layer.

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 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: No force. Still, it trembles.
One-sentence intuition: Energy branches are momentum-dependent spinors. Their interference contributes an oscillatory term to the mean velocity.

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}

Projecting to one branch at each momentum changes the initial state. It is not hiding one spinor component.

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 dimensionless one-dimensional quantum simulator. The two spinor components are not the two energy branches. Position density and current come from the same evolved state. No real electron trajectory, relativistic field quantization or pair creation is simulated.

Run the experiment

01
Predict first
Play Let it tremble and watch the arrows change direction.
What to observe: The center and mean velocity oscillate even without a potential.
02
Change the cause
Switch to One energy branch, then Remove the mass.
What to observe: A single branch has a constant mean velocity. A massless chiral packet translates without trembling.
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.