Quantum Mechanics
Schrodinger equation
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© The scientific sentence. 2010

 Dirac equation: free particle
1. Dirac's equation
The main idea is to change the D'Alembertian operator in Klein_Gordon
equation into a Lorentz invariant.
Here is KleinGordon equation is:
(1/c^{2})∂^{2}/∂t^{2}ψ  ∇^{2}ψ + (m_{o}^{2}c^{2}/ħ^{2})ψ = 0
or
☐ψ + (m_{o}c/ħ)^{2}ψ = 0
where
☐ = (1/c^{2})∂^{2}/∂t^{2}  ∇^{2}
Like a 4Vector, the wave function ψ(r,t) for which will be applied
the square root of the D'Alembertian must have 4 components. Therefore,
this square root of the D'Alembertian will be transformed by matrices.
Dirac factorised the d'Alembertian as follows:
∇^{2} =  (1/c^{2})∂^{2}/∂t^{2}
= (A∂/∂x + B∂/∂y + C∂/∂z + (i/c)D∂/∂t)
(A∂/∂x + B∂/∂y + C∂/∂z + (i/c)D∂/∂t)
The crossterms of the operator A, B, C, and D will vanish, and their
square must be equal to the operator indentity I. That is:
Their anticommutators {X,Y} = 0 and X^{2} = Y^{2} = I.
(X and Y are any operator A, B, C, or D).
Hence, the Klein_gordon equation:
☐ψ + (m_{o}c/ħ)^{2}ψ = 0 becomes:
[∇^{2}  (1/c^{2})∂^{2}/∂t^{2}]ψ = (m_{o}c/ħ)^{2}ψ
Or
(A∂/∂x + B∂/∂y + C∂/∂z + (i/c)D∂/∂t) ψ = (m_{o}c/ħ) ψ
Let's set:
A = iβα_{1}, B = iβα_{2} , C = iβα_{3}, and D = β
The above equation becomes:
iβ [(α_{1}∂/∂x + α_{2}∂/∂y + α_{3}∂/∂z + (1/c)∂/∂t)] ψ = (m_{o}c/ħ) ψ
Using the symbol ∇ of the gradient operator with the vector matrix α = (α_{1}, α_{2}, α_{3}), we get:
iβ [α.∇ + (1/c)∂/∂t)] ψ = (m_{o}c/ħ) ψ
Multiplying the both sides by ħc, we get:
[iβħc α.∇ + iħ β ∂/∂t)] ψ = (m_{o}c^{2}) ψ
Since β^{2} = 1, we have:
[iħc α.∇ + iħ∂/∂t)] ψ = (βm_{o}c^{2}) ψ
Using the quantum expression of momentum operator P = iħ∇, we get:
[ c α.P + iħ∂/∂t)] ψ = (βm_{o}c^{2}) ψ
We find finally the equation:
(iħ∂/∂t) ψ = (cα.P + βm_{o}c^{2}) ψ
That is the Dirac equation.
Dirac equation:
(iħ∂/∂t) ψ = (c α.P + βm_{o}c^{2}) ψ
α_{1}, α_{2}, α_{3}, and β are 4x4 matrices.
2. Dirac's matrices
Dirac defined the matrices α_{1}, α_{2},
α_{3}, and β as the following:
2. γDirac's matrices
More better, new matrices, known as the gamma matrices are defined:
γ^{0} = β
γ^{i} = β α^{i},
i = 1, 2, 3.
Dirac Equation becomes;
(iħγ^{0}∂/∂t) ψ = (c γ.P + m_{o}c^{2}) ψ
(iħγ^{0}∂/∂t) ψ = (c γ.P + m_{o}c^{2}) ψ
Using the Pauli's matrices:
we can write:

