130_notes.dvi

36.2 The Schr ̈odinger-Pauli Hamiltonian

In the homework on electrons in an electromagnetic field, we showedthatthe Schr ̈odinger-Pauli
Hamiltonian gives the same result as the non-relativistic Hamiltonian we have been
using and automatically includes the interaction of the electron’s spin with the magnetic
field.

H=

1

2 m

(

~σ·[~p+

e

c

A~(~r,t)]

) 2

−eφ(~r,t)

The derivation is repeated here. Recall that{σi,σj}= 2δij, [σi,σj] = 2ǫijkσk, and that the momen-
tum operator differentiates bothA~and the wavefunction.

(

~σ·[~p+

e

c

A~(~r,t)]

) 2

= σiσj

(

pipj+

e^2

c^2

AiAj+

e

c

(piAj+Aipj)

)

=

1

2

(σiσj+σjσi)

(

pipj+

e^2

c^2

AiAj

)

+

e

c

σiσj

(

Ajpi+Aipj+

̄h

i

∂Aj

∂xi

)

= δij

(

pipj+

e^2

c^2

AiAj

)

+

e

c

(σiσjAjpi+σjσiAjpi) +

e ̄h

ic

σiσj

∂Aj

∂xi

= p^2 +

e^2

c^2

A^2 +

e

c

{σi,σj}Ajpi+

e ̄h

ic

1

2

(σiσj+σiσj)

∂Aj

∂xi

= p^2 +

e^2

c^2

A^2 +

e

c

2 δijAjpi+

e ̄h

ic

1

2

(σiσj−σjσi+ 2δij)

∂Aj

∂xi

= p^2 +

e^2

c^2

A^2 +

2 e

c

A~·~p+e ̄h

ic

(iǫijkσk+δij)

∂Aj

∂xi

= p^2 +

e^2

c^2

A^2 +

2 e

c

A~·~p+e ̄h

ic

iǫijkσk

∂Aj

∂xi

= p^2 +

e^2

c^2

A^2 +

2 e

c

A~·~p+e ̄h

c

~σ·B~

H =

p^2

2 m

+

e

mc

A~·~p+ e

2

2 mc^2

A^2 −eφ+

e ̄h

2 mc

~σ·B~

H =

1

2 m

[~p+

e

c

A~(~r,t)]^2 −eφ(~r,t) + e ̄h

2 mc

~σ·B~(~r,t)

We assume the Lorentz condition applies. This is a step in the right direction. Thewavefunction
now has two components(a spinor) and the effect of spin is included. Note that this form of
the NR Hamiltonian yields the coupling of the electron spin to a magneticfield with thecorrectg
factor of 2. The spin-orbit interaction can be correctly derived from this.