Zitterbewegung — The Trembling Dirac Electron

Free Dirac Electron

Mass m 1.00

Momentum p_x 0.60

Negative-energy admixture β 0.50

Live State

Energy E = √(p²+m²)

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ω_Z = 2E (natural units)

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Group velocity v_g=p/E (·c)

0.00

Instantaneous |v| (·c)

0.00

How it works

A free electron obeys the 2D Dirac equation (ħ=c=1, momentum along x only), with 2-component pseudospinor ψ and Hamiltonian built from Pauli matrices:

H = p σ_x + m σ_z,  E = √(p² + m²)
H² = E² · I  (exact, no approximation)

Because H² is a multiple of the identity, the exact time evolution has a closed form — no numerical integration error, ever:

ψ(t) = cos(Et)·ψ(0) − i·sin(Et)·(H/E)ψ(0)

The velocity operator is v = σ (Heisenberg eq. of motion), whose eigenvalues are always ±c — never the smooth group velocity p/E. A state prepared as a mix of the positive-energy eigenstate |+⟩ (eigenvalue +E) and the negative-energy eigenstate |−⟩ (eigenvalue −E),

ψ(0) = √(1−β)|+⟩ + √β·e^(iπ/2)|−⟩

evolves so that ⟨v(t)⟩ = v_g + an oscillating term at the beat frequency ω_Z = 2E/ħ between the two energy branches — the electron's trajectory trembles around its classical straight-line drift. This is Zitterbewegung, first derived by Schrödinger in 1930. Set β=0 for a pure positive-energy state (no negative-energy admixture, no trembling — a straight line); increase β to mix in the negative-energy branch and watch the trembling amplitude grow. This exact 2-level model is the same one realised experimentally with trapped ions (Gerritsma et al., 2010) as a quantum simulation of the Dirac equation.

Mass m — heavier electron → larger E → faster, smaller-amplitude trembling.
Momentum p_x — sets the classical drift velocity v_g=p/E, always below c.
Admixture β — fraction of negative-energy component; controls trembling amplitude directly.

Electron (exact spinor evolution) Classical drift guide

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