1.3.ELECTRON IN ELECTROMAGNETIC FIELD 15
1.3 Electron in Electromagnetic Field
After we have introduced the Dirac equation for a free fermion,we are now going to
consider the interaction of a charged fermion,like e.g.an electron,with an external elec
tromagnetic eld.Again we will give a short review on a covariant description of the
electromagnetic elds and of the interaction of charged particles with such an electromag
netic eld.
1.3.1 Maxwell Equations
The basic equation of classical electrodynamics are the Maxwell equations.Here we will
use the HeavisideLorentz units to dene charges,currents and elds.Applying this
scheme the two homogenous Maxwell equations can be written
div
~
B =
~
r
~
B = 0
rot
~
E +
@
~
B
@t
=
~
r
~
E +
@
~
B
@t
= 0:(1.55)
The rst of these two equations implies that the magnetic eld
~
B can be obtained from
a vector potential
~
A by
~
B =
~
r
~
A = rot
~
A;(1.56)
which means that the second of the homogenous Maxwell equations can be rewritten to
rot
~
E +
@rot
~
A
@t
= rot
"
~
E +
@
~
A
@t
#
= 0:
This implies that the sum of the electric eld and the time derivative of the vector eld
~
A does not exhibit vortices,which means that it can be written as a gradient of a scalar
eld (~r;t)
~
E +
@
~
A
@t
= Grad =
~
r;
or
~
E =
~
r+
@
~
A
@t
:(1.57)
Due to the validity of the homogenous Maxwell equations,the electric eld
~
E and the
magnetic eld
~
B can be determined from potential elds and
~
A using (1.56) and (1.57).
In fact,these potential elds are not uniquely dened:Dierent combinations of and
~
A yield the same electromagnetic elds.One can reduce this freedom in the evaluation
of the potential elds by requiring a Gauge condition.A possible Gauge condition
requires for example
0 = div
~
A+
d
dt
= @
A
:(1.58)
This is the socalled LorentzGauge.It demonstrates that the elds and
~
A can be
cast together in a contravariant Lorentz vector using the denition
A
=
A
0
=
~
A
:(1.59)
16 CHAPTER 1.ELECTROMAGNETIC INTERACTION BETWEEN FERMIONS
The gauge condition (1.58) demonstrates that this is indeed a Lorentz vector as its product
with the covariant vector @
yields the value zero,which is obviously a scalar under any
Lorentz transformation.
One may determine a dierent set of potential elds,which do not obey the Lorentz gauge
but lead to the same electric and magnetic elds,by
e
= +
d (~r;t)
dt
and
e
~
A =
~
A
~
r;
for an arbitrary scalar eld .These conditions for changing the gauge can be rewritten
in the form
e
A
:=
e
e
~
A
!
= A
+@
;(1.60)
which shows that also the new potential vector
e
A forms a Lorentz vector even if it does
not respect the Lorentz gauge condition.
Using such a potential vector A
one can dene a Lorentz tensor of rank two by the
denition
F
:= @
A
@
A
:(1.61)
By construction this object transforms under a Lorentz transformation like a tensor of
rank two.This means:If we want to calculate this object in a dierent frame of reference
(call it
F) and we know that this change of references leads to a transformation of Lorentz
vectors applying the transformation T
as described in (1.10) we know that we must
determine
F from the original F according to
F
= T
T
F
:(1.62)
This Lorentz tensor F is not of interest only because it has such interesting transformation
properties.One can convince oneself that e.g.
F
12
= @
1
A
2
@
2
A
1
=
dA
y
dx
+
dA
x
dy
= B
z
:
So if we write the F
in form of a 4 4 matrix where the rst index +1 refers to the
row and the second index +1 to the column we see that second line and third column of
this matrix should contain the element B
z
the negative value of the zcomponent of the
magnetic eld.Evaluating the other entries of the matrix in a similar manner one nds
F =
0
B
B
@
0 E
x
E
y
E
z
E
x
0 B
z
B
y
E
y
B
z
0 B
x
E
z
B
y
B
x
0
1
C
C
A
:(1.63)
Because of this representation the tensor F
is called the electromagnetic eldstrength
tensor.From its denition (1.61) it is obvious that it is antisymmetric,which means that
there is a change of sign when column and row indices are interchanged.This is also the
reason why the diagonal matrix elements must be zero.
1.3.ELECTRON IN ELECTROMAGNETIC FIELD 17
Up to this point we have only discussed the homogenous Maxwell equations.They are sup
plemented by the inhomogeneous Maxwell equation,which,using the HeavisideLorentz
units,can be written
div
~
E =
~
r
~
E =
rot
~
B
d
~
E
dt
=
~
j;(1.64)
with (~r;t) and
~
j(~r;t) the electric density and charge current,respectively.Using the
representation of the electromagnetic eldstrength tensor in (1.63) one nds,that these
inhomogeneous Maxwell equations could also be written in the form
@
F
= j
;with j
=
~
j
:(1.65)
This equation shows,that the Maxwell equations are covariant under any Lorentz trans
formation,which means that they are compatible with special relativity.
Using the denition of the eldstrength tensor in (1.61) the inhomogeneous Maxwell
equations in (1.65) may also be written in the form
j
= @
@
A
@
@
A
= A
@
(@
A
);(1.66)
with the D'Alembert operator.If the potential elds obey the Lorentz gauge condition
(1.58) this equation reduces to
A
= j
:(1.67)
After this short discussion of the Maxwell equations,which allow us to determine the
electromagnetic elds from the electric charge and current distribution,we no turn to
the forces,which these electromagnetic elds impose on a particle with charge q.The
electric eld leads to the Coulomb force,while the magnetic eld results in the Lorentz
contribution to the total force
~
K
~
K = q
~
E +q~v
~
B:(1.68)
The Lorentz force depends on the velocity of the particle.Therefore it cannot simply
be described in terms of a mechanical potential.One can show,however,that the con
sequences of the electromagnetic elds on the motion of the charged particle can most
easily be incorporated by means of the socalled minimal substitution.This means that
one should consider a Hamilton function in which the threemomentum of the particle is
replaced by
~p!~p q
~
A;
while the potential should be multiplied with the charge q and added to the Hamilton
function.This would lead to a Hamilton function of the form
H =
~p q
~
A
2
2m
+q:(1.69)
If we recall,that for a system without constraining conditions the Hamiltonian corre
sponds to the energy,i.e.the zero component of the momentumvector,we can reformulate
the rule of the minimal substitution in a covariant way by
p
!p
qA
:(1.70)
18 CHAPTER 1.ELECTROMAGNETIC INTERACTION BETWEEN FERMIONS
1.3.2 Dirac Equation for the Electron
The rule for the minimal substitution can now directly be applied to the Dirac equation
for the free particle.This leads us to the Dirac particle for a particle with mass m and
charge q in the form
(^p
qA
) (~x;t) = 0:(1.71)
We see that this rule of the minimal substitution ensures the invariance of the Dirac
equation under a Lorentz transformation as also
A
forms a scalar under such a trans
formation.We can write the products of the matrices more explicitly
h
0
(i@
t
q) ~
i
~
rq
~
A
m
i
= 0;
and use the ansatz for a plane wave state for a particle with momentum ~p
(x;t) =
e'
e
e
i~p~x
:(1.72)
If we insert this Ansatz into the Dirac equation (1.72) and multiply the whole equation
from the left with
0
we obtain
i@
t
e'
e
=
~p q
~
A
0 ~
~ 0
e'
e
+q
e'
e
+m
1 0
0 1
e'
e
:(1.73)
It will be our aim to inspect the nonrelativistic limit of this Dirac equation and its
solution.Therefore we will in the next step factorize that timedependence of the states,
which is due to the appearance of the restmass m in the energy of the particle.So we
will consider the Dirac spinor in the form
e'
e
= e
imt
'
:
Therefore the lefthand side of (1.73) can be written
me
imt
'
+ie
imt
@
t
'
:
If we multiply the whole equation (1.73) we obtain
i@
t
'
=
~p q
~
A
~
'
+q
'
2m
0
:(1.74)
This vector equation represents actually two dierential equations for the unknown func
tions and'.In the nonrelativistic limit,the amplitudes of the large components in
this Dirac spinor,represented by',shall be large as compared to the small ones .This
means that in the equation written as the second row of (1.74) we can ignore the terms
with i@
t
and q as compared to the term involving'and the one 2m which is large
as it contains twice the rest mass.Therefore this equation of the second line in (1.74)
reduces to
0 =
~p q
~
A
~'2m;
1.3.ELECTRON IN ELECTROMAGNETIC FIELD 19
which leads us to
=
~
~p q
~
A
2m
':(1.75)
This representation for the small component can be inserted into the equation of the
rst line of (1.74) leading us to
i@
t
'=
~
~p q
~
A
~
~p q
~
A
2m
'+q':(1.76)
If we identify ~$ = (~p q
~
A) we can use (1.49) to evaluate the term in the numerator on
the right hand side of this equation to
(~ ~$) (~ ~$) = ~$ ~$11 +i~ (~$ ~$):
Applying this relation,we have to realize,however,that ~$ contains the momentum
operator ~p = i
~
r and therefore the vectorproduct
~$ ~$ = iq
~
r
~
A = iq
~
B:
Inserting this result into (1.76) we obtain
i@
t
'=
2
6
4
~p q
~
A
2
2m
q
2m
~
~
B +q
3
7
5
'(1.77)
which corresponds to the Schrodinger equation (1.69) for a particle with mass m and
charge q in an electromagnetic eld.There is only one extra term
q
2m
~
~
B =
q~
2m
2
~s
~
~
B;(1.78)
which corresponds to the energy of a magnetic moment in a magnetic eld
~
B.The
magnetic moment is obtained a
~ =
q~
2m
2
~s
~
;
as the product of the Bohr magneton (if we consider charge and mass of an electron)
multiplied with the spin ~s = ~=2 (in units of ~) and a gfactor of 2.Therefore the Dirac
equation in the nonrelativistic limit does not only reduce to the Schrodinger equation,
it also yields the feature of the spin with an anomalous magnetic moment described by
g = 2.
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