130_notes.dvi

self energy correction, at least when relativistic quantum mechanics is used. Our non-relativistic
calculation gives a qualitative explanation of the effect.

∆En(obs)=

4 α^5

3 πn^3

(

log

(

mc^2

2 ̄hω ̄nj

)

+

11

24

−

1

5

)

mc^2

1.42 The Dirac Equation

Our goal is to find the analog of the Schr ̈odinger equation for relativistic spin one-half particles (See
section 36), however, we should note that even in the Schr ̈odinger equation, the interaction of the
field with spin was rather ad hoc. There was no explanation of the gyromagnetic ratio of 2. One can
incorporate spin into the non-relativistic equation by using theSchr ̈odinger-Pauli Hamiltonian
which contains the dot product of the Pauli matrices with the momentum operator.

H=

1

2 m

(

~σ·[~p+

e

c

A~(~r,t)]

) 2

−eφ(~r,t)

A little computation shows that this gives the correct interaction with spin.

H=

1

2 m

[~p+

e

c

A~(~r,t)]^2 −eφ(~r,t) + e ̄h

2 mc

~σ·B~(~r,t)

This Hamiltonian acts on a two component spinor.

We canextend this concept to use the relativistic energy equation. The idea is to replace
~pwith~σ·~pin the relativistic energy equation.

(

E

c

) 2

−p^2 = (mc)^2

(

E

c

−~σ·~p

)(

E

c

+~σ·~p

)

= (mc)^2

(

i ̄h

∂

∂x 0

+i ̄h~σ·~∇

)(

i ̄h

∂

∂x 0

−i ̄h~σ·∇~

)

φ= (mc)^2 φ

Instead of an equation which is second order in the time derivative, we can make a first order
equation, like the Schr ̈odinger equation, by extending this equation to four components.

φ(L) = φ

φ(R) =

1

mc

(

i ̄h

∂

∂x 0

−i ̄h~σ·∇~

)

φ(L)

Now rewriting in terms ofψA=φ(R)+φ(L)andψB =φ(R)−φ(L)and ordering it as a matrix
equation, we get an equation that can be written as a dot product between 4-vectors.

(

−i ̄h∂x∂ 0 −i ̄h~σ·∇~

i ̄h~σ·∇~ i ̄h∂x∂ 0

)

= ̄h

[(

0 −i~σ·∇~

i~σ·∇~ 0

)

+

( ∂

∂x 4 0

0 −∂x∂ 4

)]

= ̄h

[(

0 −iσi

iσi 0

)

∂

∂xi

+

(

1 0

0 − 1

)

∂

∂x 4

]

= ̄h

[

γμ

∂

∂xμ

]