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 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 so-called Lorentz-Gauge.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 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 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 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 denition 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

~

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 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 dierential 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 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 g-factor of 2.Therefore the Dirac

equation in the non-relativistic 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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