Elements of Modern Physics
122
zero of the energy. Equations (4.89) and (4.90) introduce what is known as the
minimal electromagnetic interaction. With these prescriptions, the Schrödinger
equation for a spin 1/2 particle in the presence of electromagnetic fields, comes
out as
i t
∂ψ
∂
=
2
2
2 [(
) ]
− ∇ −
⋅
ψ + φψ
i
q
q
m
A S
(4.91)
where
ψ = ψ 1 (r, t) α + ψ 2 (r, t)β
(4.92)
Using Eqs. (4.80), (4.85) and (4.86), this reduces to
i t
∂ψ
∂
=
(
)
2
1
. (
)
2
− ∇ −
ψ
∇ × + ×∇ ψ + φψ
q
i
q
q
m
m
A
S
A A
(4.93)
Finally, noting that ∇ in ∇ × A operates on A as well as on ψ, and writing
V for qφ, gives
i t
∂ψ
∂
=
(
)
2
1
2
− ∇ −
ψ −
⋅ ψ + ψ
q
i
q
V
m
m
A
S B
(4.94)
This expression for the energy contains a term which corresponds to the
interaction of a particle with magnetic moment
µ =
q
m
S
(4.95)
with the external magnetic field B. Thus, the particle which satisfies Eq. (4.91),
has an intrinsic spin S and an associated magnetic moment given by Eq. (4.95).
For the electron q = –|e|. These results are in conformity with the experimental
observations discussed in Sec. 4.2.
4.8 DIRAC EQUATION
The spin and the magnetic properties of an electron can be discussed in terms
of the Schrödinger equation given in Eq. (4.91). However, it has been pointed
out earlier that the fine structure is essentially of relativistic origin. Therefore, a
relativistic description of the spin 1/2 particle has to be considered to provide a
satisfactory explanation of the fine structure.
The relativistic equation for a spin 1/2 particle of mass m may be obtained
from the relation
E
2
=
2
2 4
2
4 (
)(
)
⋅
⋅ +
c
m c
p S p S
(4.96)
which is equivalent to E
2
= p
2
c
2
+ m
2
c
2
. On taking the momentum term to the
left hand side and factorizing, it leads to the equation