1.3 Electron in Electromagnetic Field

manyhuntingΠολεοδομικά Έργα

16 Νοε 2013 (πριν από 3 χρόνια και 9 μήνες)

77 εμφανίσεις

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 Heaviside-Lorentz units to dene 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 dened:Dierent 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 so-called Lorentz-Gauge.It demonstrates that the elds  and
~
A can be
cast together in a contravariant Lorentz vector using the denition
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 dierent 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 dene a Lorentz tensor of rank two by the
denition
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 dierent 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 one-self 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 z-component 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 eld-strength
tensor.From its denition (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 Heaviside-Lorentz
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 eld-strength 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 denition of the eld-strength 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 so-called minimal substitution.This means that
one should consider a Hamilton function in which the three-momentum 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
~
rq
~
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 non-relativistic limit of this Dirac equation and its
solution.Therefore we will in the next step factorize that time-dependence of the states,
which is due to the appearance of the rest-mass m in the energy of the particle.So we
will consider the Dirac spinor in the form

e'
e

= e
imt

'


:
Therefore the left-hand 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 dierential equations for the unknown func-
tions  and'.In the non-relativistic 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 vector-product
~$ ~$ = 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 Schrodinger 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 g-factor of 2.Therefore the Dirac
equation in the non-relativistic limit does not only reduce to the Schrodinger equation,
it also yields the feature of the spin with an anomalous magnetic moment described by
g = 2.