Klein-Gordon equationequation
Quantum particles without spin, such as hypothetical mesons, are described by waves that must respect Einstein's speed limit. The Klein-Gordon equation is the relativistic version of wave mechanics: it dictates how 'quantum ripples on water' spread and move if they couldn't outrun light.
How it works
The equation describes free spinless particles — for example, pions or the Higgs boson before spontaneous symmetry breaking. In quantum field theory, it becomes the basis for describing scalar fields, where particles are quanta of field excitation.
💡 This is the first equation where quantum physics met special relativity. Schrödinger almost discovered it, but he deemed it unsuccessful because it didn't describe electron spin.
\left( \Box + \frac{m^2 c^2}{\hbar^2} \right) \psi = 0
ψ — wave function (scalar field); □ — d'Alembert operator, □ = ∂_μ ∂^μ = (1/c²) ∂²/∂t² − ∇²; m — rest mass of the particle; c — speed of light in vacuum, c ≈ 3.0×10⁸ m/s; ħ — reduced Planck constant, ħ ≈ 1.0546×10⁻³⁴ J·s
\frac{1}{c^2} \frac{\partial^2 \psi}{\partial t^2} - \nabla^2 \psi + \frac{m^2 c^2}{\hbar^2} \psi = 0
ψ — wave function (scalar field); c — speed of light in vacuum, c ≈ 3.0×10⁸ m/s; ∂²ψ/∂t² — second partial derivative with respect to time; ∇² — Laplace operator (sum of second derivatives with respect to spatial coordinates); m — rest mass of the particle; ħ — reduced Planck constant, ħ ≈ 1.0546×10⁻³⁴ J·s