Klein-Gordon Equation Calculator

Calculate properties of relativistic spin-0 particles using the Klein-Gordon equation

Particle Mass (kg)

e.g., pion mass ~ 2.4×10⁻²⁸ kg

Momentum (kg m/s)

Position x (m)

Time t (s)

Energy-Momentum Relation

Energy E = √(p²c² + m²c⁴)

134.6519 MeV

2.1571e-11 J

Rest Energy mc²

134.6517 MeV

Kinetic Energy

2.0833e-17 J

Wave Properties

Angular Frequency ω

2.0455e+23 rad/s

Wave Number k

9.4825e+11 m⁻¹

de Broglie Wavelength

6.6261e-12 m

Compton Wavelength

1.4657e-15 m

Compton Frequency

2.0455e+23 rad/s

Velocities

Phase Velocity vₚ = ω/k

719.5207c

2.1571e+11 m/s

Always ≥ c (superluminal)

Group Velocity vg = dω/dk

0.001390c

4.1667e+5 m/s

Always < c (subluminal)

Wave Function at (x, t)

Real Part Re(ψ)

1.0000

Imaginary Part Im(ψ)

0.0009

Positive and Negative Energy Solutions

Positive Energy (Particle)

+2.1571e-11 J

Negative Energy (Antiparticle)

-2.1571e-11 J

About the Klein-Gordon Equation

The Klein-Gordon equation (∂²/∂t² - c²∇² + m²c⁴/ℏ²)ψ = 0 is the relativistic wave equation for spin-0 particles (scalar fields) like pions. It was the first attempt at a relativistic quantum wave equation. It predicts both positive and negative energy solutions, which are now interpreted as particles and antiparticles.