Another exactly solvable special case is that of a constant magnetic field
B= (0, 0 ,B), for which one possible choice of vector potential is
A 0 = 0, A= (−By, 0 ,0)
- The Pauli-Schr ̈odinger equation (34.3) describes a free spin^12 non-relati-
vistic quantum particle. Replacing derivatives by covariant derivatives
one gets
i
(
∂
∂t
−ieA 0
)(
ψ 1 (x)
ψ 2 (x)
)
=−
1
2 m
(σ·(∇−ieA))^2
(
ψ 1 (x)
ψ 2 (x)
)
Using the anticommutation
σjσk+σkσj= 2δjk
and commutation
σjσk−σkσj=ijklσl
relations one finds
σjσk=δjk+
1
2
ijklσl
This implies that
(σ·(∇−ieA))^2 =
∑^3
j=1
(
∂
∂xj
−ieAj
) 2
+
∑^3
j,k=1
(
∂
∂xj
−ieAj
)(
∂
∂xk
−ieAk
)
i
2
jklσl
=
∑^3
j=1
(
∂
∂xj
−ieAj
) 2
+eσ·B
and the Pauli-Schr ̈odinger equation can be written
i(
∂
∂t
−ieA 0 )
(
ψ 1 (x)
ψ 2 (x)
)
=−
1
2 m
(
∑^3
j=1
(
∂
∂xj
−ieAj)^2 +eσ·B)
(
ψ 1 (x)
ψ 2 (x)
)
This two-component equation is just two copies of the standard Schr ̈oding-
er equation and an added term coupling the spin and magnetic field which
is exactly the one studied in chapter 7. Comparing to the discussion there,
we see that the minimal coupling prescription here is equivalent to a choice
of gyromagnetic ratiog= 1.
- With minimal coupling to the electromagnetic field, the Klein-Gordon
equation becomes
−
(
∂
∂t
−ieA 0
) 2
+
∑^3
j=1
(
∂
∂xj
−ieAj
) 2
−m^2
φ= 0
