Last time we discussed the free particle solutions
of the Dirac equation, and in particular we
could identify the various components of the
conserved current, and also normalised the
wave function accordingly, so that we have
a interpretation for a density as well as
some Lorentz invariant quantity like psi bar
psi. Now, we are prepared to extend this a
free particle solutions to the case of interactions.
And, the simplest interaction to deal with
is the electromagnetic, and that is achieved
by the so called minimal coupling prescriptions.
So, we have to replace the momentum or equivalently
the gradient operator by the simple rule 
that del mu goes to del mu minus e A mu. This
mini times is also called covariant derivative.
We will investigate the consequences of the
electromagnetic coupling into the Dirac equation,
and see what kind of effects it produces.
It is very easy to just include this in the
standard Dirac equation form, and I will write
it using the momentum operators. It says a
bit of notation where the electromagnetic
vector potential is explicitly broken up into
its time component. It produces the A 0, or
the which is equal to phi in this particular
notation, and the space component which are
written as vector components of A.
If you directly do this substitution, the
e phi is actually on the other side of the
equation, but I have just transferred it in
the convenient form. So, the time derivative
defines the evaluation, and this whole term
produces the effective energy which defines
the rate of change in time. So, now, one can
ask how does the electrostatic potential which
is phi change the energy levels, how does
the vector potential change the energy expression.
It is quite easy to see that the electromagnetic
potential acts little differently than the
rest mass from m c square. And in particular
there is a matrix here beta with the rest
mass, but no such matrix with the electrostatic
potential. So, if you look at the whole equation
in the conventional Dirac basis, there will
be positive and negative energy solutions
with beta being diagonal 1 and minus 1.
And, relative to that energy magnitude, the
effect of the electrostatic potential is in
opposite direction for positive energy as
well as negative energy. And so E greater
than 0, and E less than 0 solutions are shifted
in opposite directions by this term e phi.
And this is a consequence of what I have mentioned
earlier, that the charges carried by particle
and the antiparticle are always opposite in
sign. And we will have an explicit example
to solve for a solution to this problem which
is the hydrogen atom problem, and phi is the
coulomb potential. We will come to that later.
Now, let us look at how the vector potential
shifts. Here the matrix which is involved
is the alpha, and in Dirac basis it is of
diagonal matrix. So, we need to do some simplification
to understand the effect of the vector potential
on the energy levels of the particle. And
here it is convenient to take the non relativistic
limit of this equation and see what emerges
out of it. And non relativistic limit basically
means that 
the kinetic energy as well as the potential
energy terms in this equation are much smaller
than m c square, which is the rest mass energy.
And so we expect in the non relativistic limit
the alpha term to be small as well as the
contribution of e phi also to be small. This
is the dominant term. And in that case, we
can now expand the equation into its upper
components and the lower components which
are defined by this matrix beta, and see what
is the next correction in addition to the
m c square.
We can follow the same methodology as I did
last time. And that is, write this equation
in this 2 component notation, phi and chi.
Chi which is going to be much smaller than
phi in the non relativistic limit. Hence,
this off diagonal contribution which is coming
from alpha dot p terms here, is going to be
negligible compared to the rest mass terms.
So, in this notation we have, chi is much
less than phi. And if you now look at the
solution, the total energy can be approximated
by m c square that is a leading term. And
these 2 conditions then give the explicit
ratios from the matrix equation which I mentioned
before; that chi now is approximately sigma
dot p minus e by c A. This is the ratio for
the equation. Actually there is a c times
alpha, so the numerator has the factor of
c and denominator had a factor of energy plus
m c square; and energy is approximately m
c square; and I cancelled a factor of c, so
that is why this is the convenient form.
And now this approximation can be substituted
back into the original equation, and one can
write down the equation motion of phi itself.
So, this term, we will write the equation
for upper 2 components which is essentially
phi. This is diagonal, this term is diagonal,
and this is diagonal. This term produces the
off diagonal corrections, but that is chi,
and we already know which is the expression
for chi, so we can just plug it back in. There
is the same factor ratio which is here, the
appearing over here, and the whole term basically
gets squared.
And, then we have the equation which can be
written as; there is a, this whole thing getting
basically squared and at the factor of c.
So, it is sigma dot p minus e by c A. This
is the whole operator, and what we have is
a square of this operator, divided by 2 m
plus there is a electrostatic part which is
this, and then there is a m c square which
is the rest mass term m, and the whole equation
takes this particular form. So, now, this
equation is satisfied by this 2 component
object phi, and it involves the operators
which are the Pauli matrices.
Many times this equation is also referred
to as the Pauli equation 
which is a 2 component generalisation of the
Schrodinger equation. Schrodinger equation
just that no sigma; it just p square by 2
m; there was a electrostatic energy; and this
m c square is the shift in the overall scale
which occurred through the rest mass term
in the energy; we can drop it, if you want
to shift the 0 of the energy it does not mean
very much. And the generalisation from Schrodinger
equation to Pauli equation is just by introduction
of this Pauli matrices.
So, instead of having p square, now we have
essentially sigma dot p whole thing square.
And this equation is the non relativistic
limit. And actually it was written down earlier
after Pauli incorporated the spin degree of
freedom into the Schrodinger equation, and
this was the natural generalisation.
We can try to simplify it, and see what are
the various terms appearing in it and that
can be done by using the identities of the
Pauli matrix which is essentially we need
a general expression for multiplying 2 Pauli
matrices which produces delta i j, it is a
chronicle delta, and epsilon i j k times of
the third Pauli matrix. Once you put both
those things inside, one can have now the
simplified version where some of these terms
look more conventional. So, it is the delta
i j part just produces square of the momentum
term.
So, these 3 terms make up exactly the operator
which appears in the Schrodinger equation.
And now we have a mu term which comes from
this epsilon symbol, and that produces a cross
product of these 2 momentum operator. So,
this is the Schrodinger, and this is the new
interaction which is the contributions of
the extra degree of freedom described by Pauli
matrices which happens to be the spin.
So, we can now work out these things in a
very specific potentials term by term, or
some of these things can be written down even
more generally. Now, this is a cross product
of the 2 identical operators, now it will
be 0 if the operators commutate, but they
do not commute, and so the non-trivial effect
actually arises from when the momentum acts
on the vector potential. This is a gradient
operator, the cross product with A easily
produces curl A which is equal to the magnetic
field, and the i gets eaten up by the fact
that converting momentum to gradient we have
to include a factor of i. So, that term is
easy to interpret.
The first term can be written down as the
usual Schrodinger’s equation expansion.
So, it is 
p square, p dot A plus A dot p, and then A
square; then the other stuff is essentially
the same. And this now converted into curl
A, looks like e h cross by 2 m c sigma dotted
with B. And so this is now the more explicit
form; that in addition to having Schrodinger
equation you have an extra sigma dot B term
which produces the Zeeman effect for the spin
angular momentum carried by the electron.
And, it is very easy to see that the gyromagnetic
ratio 
for the spin term is g equal to 2. Remember
that this definition of spin is h cross by
2 times sigma. So, the half which is here
is actually part of the spin, half description
of the problem. On the other hand, the orbital
part 
has a gyromagnetic ratio 
is just equal to 1. And that comes from the
fact that in external magnetic field which
is generally easy to consider a weak field
in some particular direction. And then this
combination simplifies to L dot B when B is
equal to constant.
This is just the standard result for Schrodinger
equation, and you have L dot B plus S dot
B. The only thing is their relative normalisation
is different, and which it explains the experimental
fact that the orbital term is normalised to
this gyromagnetic ratio 1, in that same convention
the gyromagnetic ratio for the spin term is
2. And the convention for the magnetic moment
is then 
the orbital contributions accompanied by the
appropriate gyromagnetic ratio. So, it is
L plus 2 S.
And, this is a major success of Dirac equation
in explaining the properties of the electron
because electron indeed behaves in this way,
that the spin angular momentum couples twice
as strongly to the magnetic field compared
to the orbital angular momentum. So, these
equations are kind of straight forward to
derive, and it was very helpful to get the
spin automatically coming out from the Dirac
formulation and also with the correct properties,
and that is why Dirac called these equation
a theory of electron.
Now, one can look at the same structure in
a covariant language also. Sometimes it is
helpful to understand that as well, and so
we will look at the equation again, but now
using the 4 vector notation in terms of these,
so called covariant derivative. So, we have
the ordinary derivative plus 
the correction from the electromagnetic potential.
With this substitution, now the Dirac equation
can be written as; actually it is only this
operator acting on psi is equal to 0 is the
Dirac equation.
We are just multiplying by another overall
factor to be able to take the limit of non
relativistic approximation easily. And now
it is convenient to just expand out these
terms. There are, these factors of the 2 operators
will just multiply those things together,
and use the gamma matrix algebra where gamma
mu, gamma nu, the anticommutator produces
2 different terms.
So, one can write gamma mu gamma nu as half
the anticommutator, and so half the commutator.
This result is the Minkowski metric. And we
will define this particular quantity to be
a new object it is related to spin, and we
will call this minus i times sigma mu nu,
or equivalently the definition of this object
is i by 2 gamma mu gamma nu. The anticommutator
is antiHermitian object, and that is the reason
for sticking in this factor of i, explicitly.
So, the sigma mu nu, these are Hermitian objects.
And now with this convention we can expand
out the product of 2 d slashes. One of the
term will just produce this g mu nu; that
means, they will be trivially contracted,
and then there will be an extra term coming
out which coefficient sigma mu nu; so with
that convention. And then the cross terms
in the multiplication cancels because m c
by h cross is just a number.
So, then the equation now takes the form that
it is del mu plus i e by h cross c A mu, whole
thing is trivially squared; that is the g
mu nu part. Then there is this extra term
involving sigma mu nu, and it gets multiplied
by the field strength tensor constructed from
this covariant derivative. And then there
is the trivial term of the compton wavelength
inverse, whole thing squared.
The explicit definition of this tensor is,
follows from the definition of covariant derivative.
So, it can be defined as f mu nu is h cross
c divided by i e times the commutator of 2
covariant derivatives; and in terms of explicit
evaluation d mu A nu minus d nu A mu. All
these factors of h cross c by i, just cancel
out, what is the accompanying the definition
of covariant derivative and proportionality
with the vector potential. So, this is now
the equation.
We can now see the extra terms compared to
the Klein Gordon equations. So, the first
this term and the m c h cross square, these
are the terms in the Klein Gordon equation.
And the new contribution is essentially this
sigma mu nu F mu nu. Now, one can break it
up into 2 different parts corresponding to
the electric and magnetic field because the
space component of F mu nu gives magnetic
field, and space time component of F mu nu
gives the electric field.
And, one can break up this object into an
explicit notation. One has to evaluate the
corresponding commutators of gamma mu gamma
nu. When there are 2 of them are spaced like
we get the exactly the spin operator which
we defined earlier. And when one is space
and the other is time, it gives gamma 0 gamma
i which nothing but the alpha matrix as we
had defined earlier.
So, this object simplifies to 2 times what
I had defined as, sigma dot B, and then alpha
dot E. And the convections I have used was,
the spin operator was h cross by 2 times sigma.
So, again you see the effect directly coming
in. This is the Zeeman coupling appearing
to do a spin. It is not part of the Klein
Gordon equation. And then there is this extra
term. So, this is the spin dipole.
And, this alpha dot E term appears many times
in analysis of perturbation theory of centrally
symmetric potential, and it can be rewritten
as it has the spin part buried inside alpha,
but it is off diagonal. And the electric field
which appears here can be related to the magnetic
field in the frame of the moving electron.
So, the static electric field in which electron
is moving becomes a magnetic field in which
the electron is addressed. That is the usual
Lorentz transformation. And this term basically
leads to the coupling between the spin and
a motion of the electron, and that is often
labelled as the spin orbit coupling.
So, both these effects involving spin, directly
a dipole interaction with the external field,
and also a coupling with the orbital angular
momentum. They appear inside this Dirac equation.
They are not part of the Klein Gordon equation
themselves, and we have not used any non relativistic
expansion so far in this analysis. And both
these terms are quite general. One can still
do a non relativistic expansion, and simplify
them further if necessary, but it is already
written in a quite a general form.
So, this is the covariant version of Dirac
equation simplified, so that you can directly
identify the effect of electric and magnetic
field. Now, one can play around with a various
dynamical equations based on this algebra,
how does the particle move; if it is spin
then it will certainly evolve under the external
magnetic field, how does it change and so
on and so forth. And these things can be easily
worked out. Let me illustrate 2 very simple
cases.
So, one is the so called Lorentz force for
a particle moving in some external electromagnetic
field. And it can be obtained as an equation
of motion as a rate of change of momentum.
Except that the momentum which we have to
consider now is the canonical momentum, or
equivalently the one corresponding the covariant
derivative. And the electric and magnetic
field we will define with the standard convention
that E is equal to minus grade phi minus 1
by c del A by del t, and B is equal to curl
of A.
So, then the rate of change of momentum is
easily evaluated by working out the commutator
with the Hamiltonian. So, it is 
this expression. And now, different terms
in the Hamiltonian may or may not commute
with the terms which are involved over here.
And we have to just work it out one by one.
In particular, this operator does not have
any structure in the internal space. There
are no alpha beta matrices involved over here.
So, the only term which produces a non 0 answer
is a, when there is a operator inside the
Hamiltonian and which acts non trivially on
the vector potential. If there is nothing
else involved then we do not have any extra
contribution. So, just look at it term by
term. So, the part which acts nontrivially
on this vector potential is just alpha dot
p. The remaining part does not act on A at
all, and so it does not produce any contribution.
So, this is one part.
And, the other part is the terms which depend
on space because it involves the commutator
with the gradient operator buried inside here.
And those terms can now be taken again from
the Dirac equation. There is this alpha dot
A term coming from the momentum part, and
then there is also a static potential which
can get differentiated with the momentum.
The momentum part p itself commutes with p,
and does not contribute over here.
And then to consider a most general situation
where you have the complete time derivative
acting one is here, instead of just being
in a time independent field you have to add
a extra term, and that term is just a partial
derivative directly acting on this term. So,
it actually happens to be; this is the time
derivative, explicit time derivative of this
vector potential. The gradient operator is
not a function of time, and so we add this
extra term to the equation of motion. So,
this is a total result.
And now you have to evaluate the various commutators,
what happens with gradient acts on this term
versus that term, etcetera. And that now can
be done in a straight forward manners; just
let me do that; this is a gradient acting
on A. So, it produces alpha dot gradient acting
on A. The nontrivial part of the commutator
is when p acts on one of these terms; p acting
on anything which follows it does not contribute
to the commutators. So, the first commutator
actually is just this.
The second commutator is the same way where
this p will be acting on this particular terms,
and that produces the term which is here plus
p acting on phi. So, it is actually this part
which gives this; this part which produces
these 2 terms; and then we are still left
with the last partial derivative. So, now,
we have evaluated the commutator. And it is
convenient to put back the electromagnetic
field, inside of the vector potential in this
notation. So, these last 2 terms easily produce
the electric field.
And the, these first two terms gradient acting
on A can be now rewritten as a triple product
which is involves alpha gradient and A, and
that can be written as alpha cross B. So,
if you write alpha cross, curl cross A, and
expand the triple product it will be alpha
dot A, the gradient acting on that, and then
alpha dot grad the whole thing acting on A.
So, this is what becomes of the equation.
And this indeed describes the Lorentz force
with the identification which we have seen
before, that this can be written as 
the velocity operator cross B because velocity
operator was indeed this matrix alpha. So,
the Lorentz force does not survive in exactly
the form where the velocity was, but velocity
has to be replaced by its appropriate operator.
And then the equation works; and then one
can construct the trajectory of the particle
by solving this equation given in certain
external field.
So, this is the modification of what happens
to Lorentz force. There is a equation of motion
not for the coordinates, but for the internal
degree of freedom which is the equation corresponding
to the motion of the spin, which is actually
the spin precession. And the operator here
involved we have already seen before, that
it is this matrix sigma. And we derived the
equation also for this in constructing a angular
momentum operator. So, the total angular momentum
combing L and S was conserved for a free particle,
and that gave a equation of motion of this
object sigma, which was equal to 
minus 2 c divided by h cross times alpha cross
p.
So, this object does not look anything like
the effect of a external field, even when
you substitute this p by covariant derivative.
It does not take a appropriate form required
for the magnetic field, but this alpha is
already the velocity operator which we have
seen before. And to be able to get the equation
of motion which is of the same structure as
in case of classical electrodynamics.
Instead of looking at this operator d sigma
by d t, we will look at its anticommutator
with the Hamiltonian which will have a definite
value in eigen states of Hamiltonian or equivalently
eigen states of energy. So, that is the object.
So, equation of motion is simple for eigen
states of the Hamiltonian not in case of a
general Hamiltonian, or a mixture of various
states.
So, we will see this derivation what happens,
and the object which is convenient to use
in this particular case, it is the anticommutator
we did this same trick in understanding the
velocity operator earlier. And this now can
be written down by inserting the expressions
of the Hamiltonian as well as whatever happens
in sigma dot b n. We have the 2 terms. The
alpha part commutes, the alpha part produces
a non 0 answer with 
the rate of change of the spin operator. And
for the beta part these anticommutator of
alpha and beta is 0, so that, the rest mass
term is not going to contribute.
And, we are only going to look at the effect
of the vector field. So, the electrostatic
field we are going to keep 0. And now, this
can be evaluated by writing down the products
of these alpha matrices. And since alpha has
the Pauli matrices, they obey the same kind
of structure when you have a products of 2
alpha matrices. So, this alpha i alpha j happens
to be delta i j when the 2 ones are equal,
and if they are not equal then we get the
Pauli term. Now, the Pauli term is diagonal
and that is what we have defined as this matrix
sigma k.
And, with this the product of 2 alpha can
be written in a simple fashion, and the cross
products can be written again with a epsilon symbol. So, this object then can be simplified to all the various 
factors of i are going to appear from this commutator as
well as this term minus sign is from this
part. And the 2 products of alpha are now
quite explicitly written. There is a cross
product here, which I am going to use this
epsilon L m n to expand.
And then there is a epsilon here in the products
of alpha which I will write using epsilon
i l j sigma j. And then now we have the remaining
part which is these 2 terms. One of them has
the same index as i which comes from the product
of alpha, and second one comes from these
cross products. And it will have the index
m. And the combination of these 2 terms becomes
a commutator instead of an anticommutator.
And the reason for being that is the cross
product over here. So, when you change the
order of the alpha matrices and do the simplification,
you have to flip around the order of this
indices as well, and that can be taken inside
by relabeling, and this object then reduces
to this.
Anticommutator, this object is quite familiar
and one can work it out as has been the case
many times in non relativistic quantum mechanics.
It is just a commutator of 2 covariant derivatives,
and we have seen it what it produces. It produces
the f mu nu or electromagnetic field. In this
case these are all space components. So, it
produces a magnetic field. And the result
of that simplification is now written down
explicitly in terms of all these various tensors;
the one more epsilon symbol from this commutator
and the magnetic field.
So, now, the whole structure is in the form
where the only nontrivial operator it has
the spin operator and the magnetic fields.
And these products of all epsilon can be simplified
in terms of chronicle deltas. And once one
does that, for instance, they are product
of the first 2, I can just write it explicitly.
So, this is gives delta i l delta n k, and
the one which is the reverse order. This middle
one leave it as it is, so it involves this
sigma terms in the quite the appropriate fashion.
And, now this chronicle delta simplify the
various indices put everything in place. And
this term becomes sigma cross B. And this
is indeed the expression for spin precession
which is also referred to as Larmor precession
of a magnetic dipole in an externally applied
magnetic field. The factor of 2 came here
because I took the anticommutator, but did
not divide by a factor of 2. So, there is
a, this term actually represents 2 times energy
multiplied by d sigma by d t.
And, if you take all these thing, this is
a analogue of the classical equation of a
d s by d t. You have to take out these 2 from
here, and insert the factor of h cross by
2, but that is the sigma is there on both
sides of equation. It just does not matter.
And h will give a non relativistic limit.
It will become close to m c square. So, this
produces the usual Bohr Magneton constant,
well upto h cross and S cross B. So, with,
again the important point is the gyromagnetic
ratio has a value 2. And we have approximated
the value of the Hamiltonian by m c square
in taking out the factor of the energy. So,
this is the way the spin precision also appears
with the correct gyromagnetic ratio in the
external field.
So, not only we have the energy eigen states,
we also have dynamical evolution equations
in presence of electromagnetic field. And
they can be used in many different situations
to calculate many kinds of effects. Of course,
all these analysis was nice, and it helps
to understand the effects which are mostly
seen as a small corrections to the dominant
part. And in general, they are evaluated using
perturbation theory and external fields.
The dominant part was actually solution of
the coulomb potential problem, and that was
the crucial component of the atomic physics,
and all these things were extra corrections.
What we have seen so far is that the extra
corrections do work out as expected, the spin
degree of freedom is there it has the correct
normalisation, etcetera. But the benchmark
is the solution of the complete hydrogen atom
problem. That is where everything has to be
checked.
And, Dirac’s equation got the approval once
the hydrogen atom eigen spectrum came out
correct. It differs from the spectrum which
I derived in earlier lectures from the Klein
Gordon equation, and it matches the formula
which was obtained by Sommerfeld without knowing
anything about the spin. Sommerfeld’s formula
agrees with experiment, and so does Dirac’s
answer. And we will work this explicit solution
out in the next class.
And, this is the hydrogen atom problem, sometimes
also referred to as the Kepler problem, the
one over r potential. And we have to use a
standard normalisation which is a conveniently
taken to be z e by r, and this solution has
certain symmetries. So, one is a rotational
symmetry which means that angular momentum
is conserved, the other is a time independence
which means energy is conserved. And both
these objects correspond to well known quantum
numbers for the hydrogen atom, both in relativistic
theory as well as the non relativistic theory.
And, we want to obtain the general solution
in terms of those quantum numbers for the
Dirac equation which means you have to solve
it. Find an operators corresponding to these
conserved numbers and then express the solution
in terms of these conserved quantum numbers
to get the complete expression for energy
eigen states and the wave function, etcetera.
