Physical question
How are relativistic dispersion, spin and probability connected?
Relativistic quantum dynamics must respect a quadratic energy–momentum relation while describing a wave function that evolves consistently in time. The Dirac construction linearizes the Hamiltonian using matrices, introduces spinor degrees of freedom and gives a positive density ψ†ψ. At a fixed momentum, the distinction between a single positive-energy state and a coherent mixture of energy sectors becomes visible through time-dependent velocity.
Computed relativistic observables
Energy, current and spinor probabilities
Relativistic mass shell
Shared Klein–Gordon and Dirac plane-wave energies
Both relativistic wave equations admit plane-wave frequencies ±√(1+p²) in the chosen units. Their spin content, conserved currents and interpretations differ. The red points show energies at your selected momentum.
The nonrelativistic limit
How far can the kinetic-energy expansion be trusted?
Blue: exact √(1+p²)−1. Orange dashed: p²/2. Teal dashed: p²/2−p⁴/8. The expansion is asymptotic for |p|/(mc)≪1; the quartic approximation becomes unreliable at relativistic momenta.
Time-dependent probability current
What velocity does the Dirac state carry?
Blue: 〈α〉 = 〈σx〉, the Dirac current velocity divided by c. Orange dashed: positive-energy group velocity p/E. A pure positive-energy eigenstate gives constant velocity; an upper-component state at finite momentum contains energy-sector interference.
Derive the relativistic spectrum
Linearizing the mass–shell equation
Relativistic energy satisfies E²=p²c²+m²c⁴. The Klein–Gordon equation substitutes E→iℏ∂t directly into this quadratic relation. Dirac instead chooses matrices satisfying anticommutation constraints so that a Hamiltonian first order in time squares to the mass–shell operator.
In a one-dimensional momentum/spin block we may take α=σx, β=σz. In units m=c=ℏ=1, H=pσx+σz, so H²=(p²+1)I. The eigenenergies are ±E, E=√(p²+1).
For a normalized positive-energy spinor the weights of the upper and lower components are (E+1)/(2E) and (E−1)/(2E); the lower component is suppressed at low momentum. The Dirac probability current uses j⁰=ψ†ψ≥0 for every spinor.
Derive the nonrelativistic expansion
For |u|=|p|/(mc)≪1, √(1+u²)−1=u²/2−u⁴/8+u⁶/16−⋯. Multiplying by mc² yields p²/(2m)−p⁴/(8m³c²)+⋯. The first term is the Schrödinger kinetic energy; the next correction is negative. The approximation is controlled by u², which can be inspected directly in the graph.
Calculate the oscillatory velocity
For a fixed dimensionless p, prepare the upper component (1,0)ᵀ. The time propagator is U(τ)=cos(Eτ)I−i sin(Eτ)(pσx+σz)/E. Since the velocity operator equals cα=cσx, its expectation in this state is 〈v〉/c=(p/E²)[1−cos(2Eτ)]. A positive-energy eigenstate instead has 〈v〉/c=p/E at all times. Oscillations require coherence between energy sectors in this simplified free-particle model.
Why distinguish Klein–Gordon and Dirac currents?
The complex Klein–Gordon field has a conserved current whose temporal component takes opposite signs for positive- and negative-frequency plane waves at equal amplitude, so it is naturally interpreted as a charge current within field theory. For Dirac spinors, ψ†ψ is nonnegative and obeys a first-order continuity equation, with spatial current cψ†αψ. Neither the existence of a negative-energy branch nor this two-component toy dynamics eliminates the need for quantum field theory when antiparticles and pair creation matter.
Guided explorations
Separate spectral physics from state preparation
- Low momentum. Choose |p|/(mc)=.15. Compare exact kinetic energy with p²/2−p⁴/8 and estimate the order of the neglected term.
- Stationary velocity. Choose a positive-energy eigenstate, then vary time. The upper/lower probabilities and velocity remain constant even though the wave function accumulates a phase.
- Energy-sector interference. With p=1.25, select the upper-component preparation. Inspect the velocity oscillations and compare their frequency with twice the relativistic energy.
Self-check: What is the group velocity of a positive-energy relativistic plane wave?
Interpretation and reading
The interactive dynamics is a fixed-momentum, two-component reduction of the free Dirac equation. This makes energy-sector interference and the one-dimensional velocity operator accessible without performing a full four-spinor wave-packet simulation. The full three-dimensional Dirac theory uses four components and physical antiparticles require second quantization. Klein–Gordon plane waves are compared at the level of their dispersion and current-sign interpretation.
Read Klein–Gordon and Dirac equations and free Dirac spinors. Connections: Pauli spin operators (Chapter 3) and relativistic perturbative correction (Chapter 4).