Dirac Equation Solver

Compute energy eigenvalues, 4-component Dirac spinors (positive/negative energy solutions), and visualize the relativistic energy-momentum dispersion relation. Natural units ħ = c = 1 (energy & momentum in MeV).

Electron: 0.511, Proton: 938.27
⚛️ Electron at rest (p=0)
? Electron moving along x (p=1.0)
? Proton at rest (m=938.27)
⚡ Ultrarelativistic (m=0.511, p=100)
? Neutrino-like (m→0, p=2.0)
Local computation: All calculations are performed client-side. No data leaves your device.

The Dirac Equation: Uniting Quantum Mechanics & Special Relativity

The Dirac equation is a relativistic wave equation formulated by Paul Dirac in 1928. It describes spin-½ particles (fermions) such as electrons, quarks, and neutrinos. In natural units (ħ=c=1), the equation reads:

(iγμ∂μ − m)ψ = 0

Where γμ are 4×4 gamma matrices, and ψ is a four-component Dirac spinor. The equation correctly predicts the existence of antimatter, electron spin, and the fine structure of hydrogen. Our solver computes plane-wave solutions ψ(x) = u(p) e−ip·x (positive energy) or v(p) e+ip·x (negative energy), yielding the energy-momentum dispersion E² = p² + m².

Mathematical Formulation & Spinor Construction

For a given three-momentum p = (px, py, pz), the positive-energy spinor u(p) is:

u(p) = N ⎡ χ ⎤
⎣ (σ·p)/(E+m) χ ⎦ ,    N = √(E+m) ,   χ = two-component spinor (↑ or ↓)

For negative-energy solutions (antiparticles), v(p) = N ⎡ (σ·p)/(E+m) χ ⎤
⎣ χ ⎦ with E = +√(p²+m²) and the overall sign convention ensures proper Lorentz transformation. The solver returns the full complex 4-component spinor normalized to u†u = 2E (relativistic invariant).

Conventions and Notations

This solver uses the following standard conventions:

  • Metric signature: (+ − − −) (West Coast convention)
  • Gamma matrices: Dirac representation (standard representation with γ⁰ diagonal)
  • Normalization: u†u = 2E, v†v = 2E (covariant normalization)
  • Pauli matrices: σ₁ = [[0,1],[1,0]], σ₂ = [[0,-i],[i,0]], σ₃ = [[1,0],[0,-1]]
  • Natural units: ħ = c = 1, energies and momenta in MeV
  • Spinor basis: χ↑ = (1,0)T, χ↓ = (0,1)T for spin up/down along z-axis

These conventions follow the standard textbook presentation in Bjorken & Drell's "Relativistic Quantum Mechanics" (1964).

Spin Operators and Helicity

In the Dirac representation, the spin operator for a particle at rest is Σ/2 = (1/2) diag(σ, σ). For moving particles, the spin projection along the momentum direction defines the helicity operator h = (Σ·p)/|p|. In the ultrarelativistic limit (|p| ≫ m), the spin up/down states selected in this solver become approximately helicity eigenstates. For massless particles (m=0), helicity is a conserved quantum number.

Continuity Equation and Probability Interpretation

The Dirac equation leads to a conserved probability current jμ = ψ̅γμψ, where ψ̅ = ψ†γ⁰. The time component j⁰ = ψ†ψ is positive definite, unlike the Klein-Gordon equation. The spinors computed here satisfy the normalization condition u†u = 2E, which ensures the conserved probability density transforms correctly under Lorentz transformations.

Non-relativistic Limit and Comparison with Schrödinger Equation

In the non-relativistic limit (|p| ≪ m), the Dirac equation reduces to the Pauli equation for two-component spinors, with the lower components (3rd and 4th) suppressed by factor |p|/(2m). The Dirac equation naturally incorporates the spin-½ magnetic moment with g-factor g=2, a feature that emerges automatically from relativistic quantum mechanics. The additional factor of 2 in the electron's magnetic moment (g≈2.002319) is explained by quantum electrodynamic corrections.

Step-by-Step Usage

  1. Enter particle mass (MeV/c²) and momentum components (MeV/c).
  2. Choose positive (particle) or negative (antiparticle) energy branch.
  3. Select spin orientation (up/down) to define the Pauli spinor χ.
  4. Click Solve Dirac Equation to obtain energy, γ, β, and the 4‑spinor components.
  5. The dispersion plot displays relativistic energy-momentum relation and marks your state.

Physical Applications & Significance

  • Antimatter prediction: Negative-energy solutions led Dirac to postulate positrons, discovered by Anderson in 1932.
  • Quantum Electrodynamics (QED): The Dirac equation forms the foundation of QED, the most precise theory in physics.
  • Spintronics & Topology: Dirac materials (graphene, topological insulators) exhibit emergent Dirac fermions.
  • Neutrino oscillations: Relativistic spinors describe massive neutrino propagation.
  • Magnetic moment of the electron: The Dirac equation predicts the electron's magnetic moment with g=2, explaining the fine structure of hydrogen.
  • Particle physics: Dirac spinors are fundamental in the Standard Model for describing quarks and leptons.
Case Study 1: Relativistic Electron in a Magnetic Field

Consider an electron (m=0.511 MeV) with momentum p = (1.0, 0, 0) MeV/c. The Dirac solver yields E ≈ √(1.0²+0.511²) ≈ 1.123 MeV, γ ≈ 2.197, β ≈ 0.890. The spinor components reveal that the lower components are suppressed by a factor ∼ |p|/(E+m) ≈ 0.612. This suppression is key to understanding why the electron's magnetic moment deviates from the Dirac value by quantum corrections (g-2).

Case Study 2: Verification of Numerical Accuracy

This solver has been extensively tested against analytical limits:

  • Rest frame (p=0): For m=0.511 MeV, spin up: ψ = √(E+m)[1,0,0,0]ᵀ with E=m.
  • Massless limit (m=0): For p=(0,0,p), the spinor simplifies to helicity eigenstates, with upper and lower components related by (σ·p̂)χ.
  • Non-relativistic limit: For |p| ≪ m, the lower components are proportional to |p|/(2m) times the upper components.
  • Normalization: The computed spinor always satisfies u†u = 2E (or v†v = 2E) to machine precision.

These verifications ensure the solver provides physically accurate results for educational and research purposes.

Common Pitfalls & Clarifications

  • Zero mass limit: For m=0, the expression (σ·p)/(E+m) simplifies to (σ·p̂) (helicity eigenstates). Our solver handles m→0 numerically with a safe limit.
  • Spin orientation: In the ultrarelativistic limit, spin up/down correspond to positive/negative helicity projections.
  • Normalization: The spinors satisfy u†u = 2E, which is Lorentz invariant up to a boost factor.
  • Energy interpretation: Negative-energy solutions correspond to antiparticles with positive energy in the modern Feynman-Stückelberg interpretation. Our solver directly computes the negative-energy spinor v(p) for completeness.

Authoritative References & Further Reading

  • Dirac, P.A.M. (1928). "The Quantum Theory of the Electron". Proceedings of the Royal Society A.
  • Peskin, M.E. & Schroeder, D.V. (1995). An Introduction to Quantum Field Theory. Westview Press.
  • Bjorken, J.D. & Drell, S.D. (1964). Relativistic Quantum Mechanics. McGraw-Hill. (Primary reference for the conventions used in this solver.)
  • Thaller, B. (2013). The Dirac Equation. Springer Science & Business Media.
  • Itzykson, C. & Zuber, J.B. (1980). Quantum Field Theory. McGraw-Hill.
  • NIST Reference on Constants (for particle masses and fundamental constants)

Quantum foundations & rigorous implementation – This solver implements exact algebraic solutions of the Dirac equation in the momentum representation. The code follows the conventions of Bjorken & Drell (Relativistic Quantum Mechanics). All expressions are derived from first principles and validated against known analytical limits. The implementation uses the Dirac representation of gamma matrices, metric signature (+ − − −), and covariant normalization u†u = 2E. The solver has been extensively tested for numerical stability and physical consistency. Reviewed by the GetZenQuery Tech team, updated April 2026.

Frequently Asked Questions

The term (σ·p) introduces complex numbers when momentum has multiple components. This reflects the quantum mechanical nature of spin and phase factors under rotations. The complex numbers are essential for representing the spin degrees of freedom and ensuring proper transformation properties under Lorentz transformations.

In Dirac's hole theory, negative-energy states are fully occupied (the Dirac sea); holes correspond to antiparticles. In modern quantum field theory, negative-energy solutions correspond to antiparticles with positive energy, traveling backwards in time (Feynman-Stückelberg interpretation). Our solver directly computes the negative-energy spinor v(p) for completeness.

Yes, set m to a tiny value (e.g., 0.0001 MeV) to approximate massless neutrinos. The spinor then becomes helicity eigenstate. For actual neutrino phenomenology, one must consider neutrino mixing and oscillations, which require superposition of mass eigenstates.

No, this version solves the free Dirac equation. For interacting systems (e.g., hydrogen atom), a more advanced solver is needed. The Dirac equation with electromagnetic interaction is obtained via minimal substitution: ∂μ → ∂μ + ieAμ, leading to the complex system for bound states.

Dirac spinors (4 components) describe massive spin-1/2 particles with distinct antiparticles. Weyl spinors (2 components) describe massless fermions or chiral projections. Majorana spinors (4 components) describe particles that are their own antiparticles. The solutions computed here are Dirac spinors, which can be decomposed into left- and right-handed Weyl spinors for massless particles.