Dirac dynamics / Two energy branches

Dirac Trembling Motion

Why does a free Dirac wave packet tremble when no force acts on it? Because it mixes positive- and negative-energy parts. The interference term between them makes the packet's average velocity oscillate; this is Zitterbewegung. Build the packet from positive energies only and the term disappears, taking the trembling with it.

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

This is a dimensionless quantum simulator in one dimension. The two components of the spinor are not the two energy branches. The position density and the current come from the same evolved state. The Lab does not simulate a real electron trajectory, relativistic field quantization or pair creation.

FLOW RIGHTFLOW LEFT

MEAN VELOCITY

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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

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.

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$pc/E_0$ · $E/E_0$

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.

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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

Does the oscillation survive one energy branch?

Try: Compare Let it tremble with One energy branch. Scrub through the evolution, then measure position and velocity, each on its own set of fresh preparations.
Read: Compare the sample means with the model’s expectation values. Remove the interference between the energy branches and the free mean velocity stops oscillating. The two spinor components and the two energy branches are different things.
Scope: Speeds are dimensionless, with the unit speed set to one. This one-dimensional, two-component analogue complements the article’s four-component theory of the electron. It does not observe a real electron’s trajectory, solve a hydrogen spectrum or calculate a magnetic anomaly.

\begin{aligned}H_0&=p\sigma_x+\mu\sigma_z\\v&=\sigma_x\end{aligned}

Model reference: Gerritsma et al. · Dirac simulation (2010)

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

Start with the essentials

Focus question: Why does a free Dirac wave packet tremble when no force acts on it?
One-sentence intuition: Each energy branch has its own spinor, which changes with momentum. When the two branches interfere, the mean velocity picks up an oscillating term.

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}

Keeping only one branch at each momentum makes a different initial state. It is not the same as hiding one spinor component.

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 dimensionless quantum simulator in one dimension. The two components of the spinor are not the two energy branches. The position density and the current come from the same evolved state. The Lab does not simulate a real electron trajectory, relativistic field quantization or pair creation.

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 conditions
Switch to One energy branch, then Remove the mass.
What to observe: With a single branch, the mean velocity stays constant. A massless chiral packet slides along without trembling.
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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.