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© The scientific sentence. 2010

Klein-Gordan's equation: free particle



The idea here is to take into account the relativistic effect on a fast moving free particle, in the Schrodinger's equation. Klein-Gordan's equation is just the time-dependent relativistic Schrodinger's equation.

The classical expression of the total energy E of a free particle of mass at rest m and momentum p is:

E = p2/2m

The relativistic expression of the total energy E of a free particle of mass at rest mo and momentum p is:

E2 = p2c2+ mo2c4

The quantum expression of the momentum is:

P = - i ħ ∇

The time-dependent non-relativistic Schrodinger's equation for this particle is:

(p2/2m) ψ = E ψ = iħ∂ψ/∂t

We replace the momentum p by its quantum mechanical operator P = - i ħ ∇, and E by its relativistic expression : [p2c2+ mo2c4]1/2, and get:

iħ∂ψ/∂t = [(- i ħ ∇)2c2+ mo2c4]1/2 ψ

To get rid of the square root operation, we apply the operator iħ∂ψ/∂t two times and get:

[iħ∂ /∂t]2 ψ = [(- i ħ ∇)2c2+ mo2c4] ψ

That is:
- ħ22ψ/∂t2ψ = - ħ22c2ψ + mo2c4ψ
or
2ψ/∂t2 - ∇2c2ψ + (mo2c42)ψ = 0

(1/c2)∂2 /∂t2 ψ - ∇2ψ + (mo2c22)ψ = 0

The operator ☐ = (1/c2)∂2/∂t2- ∇2
is called d'Alembertian operator.


The above expression can be written as:

☐ψ + (mo2c22)ψ = 0

Klein-Gordan equation is:

(1/c2)∂2/∂t2 ψ - ∇2ψ + (mo2c22)ψ = 0

or

☐ψ + (moc/ħ)2ψ = 0

with

☐ = (1/c2)∂2/∂t2- ∇2






  


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