Klein-Gordon Equation Calculator
Calculate properties of relativistic spin-0 particles using the Klein-Gordon equation
e.g., pion mass ~ 2.4×10⁻²⁸ kg
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.