Klein-Gordon equation | Physics Library

The Klein–Gordon equation is an equation of mathematical physics that describes relativistic spin-0 fields. It is given by

( ) m2c2 □ + ---2- ψ = 0, ℏ

where □ denotes the wave operator, or D’Alembertian, and ψ is the wavefunction or scalar field.

Using the metric-signature convention (+,−,−,−), the D’Alembertian is

2 □ = -1 ∂--− ∇2. c2 ∂t2

The Klein–Gordon equation is Lorentz invariant.

Begin with the relativistic energy–momentum relation

E2 = m2c4 + p2c2.

∂ E → iℏ ∂t, p → − iℏ∇.

Applying these operators to ψ gives

2∂2ψ- 2 4 2 2 2 − ℏ ∂t2 = m c ψ − ℏ c ∇ ψ.

Rearranging,

2∂2 ψ 2 2 2 2 4 ℏ -∂t2 − ℏ c ∇ ψ + m c ψ = 0.

Dividing by ℏ²c² yields

( ) 1--∂2- 2 m2c2- c2∂t2 − ∇ + ℏ2 ψ = 0.

Identifying

1 ∂2 2 □ = -2 --2 − ∇ , c ∂t

we obtain the Klein–Gordon equation

|-------------------| |( m2c2 ) | | □ + ---2- ψ = 0|. --------ℏ------------

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