If one calculates the energy of a point charge using classical electromagnetism, the result is infinite,
yet as far as we know, the electron is point charge.
One can calculate the energy needed to assemble an electron due, essentially, to the interaction of the electron
with its own field.
A uniform charge distribution with the classical radius of an electron, would have an energy of
the order of
Experiments have probed the electron's charge distribution and found that it is consistent with a point
charge down to distances much smaller than the classical radius.
Beyond classical calculations, the
self energy of the electron calculated in the quantum theory of Dirac is still infinite
but the divergences are less severe.
Electron Self Energy Corrections
At this point we must take the unpleasant position that this (constant)
infinite energy should just be subtracted when we consider the overall zero of energy
(as we did for the field energy in the vacuum).
Electrons exist and don't carry infinite amount of energy baggage so we just subtract off the infinite constant.
Nevertheless, we will find that the electron's self energy may change when it is a bound state and that we
should account for this change in our energy level calculations.
This calculation will also give us the opportunity to understand resonant behavior in scattering.
We can calculate the lowest order self energy corrections represented by the two Feynman diagrams below.
In these, a photon is emitted then reabsorbed.
As we now know, both of these amplitudes are of order
The first one comes from the
term in which the number of photons changes by zero or two
and the second comes from the
term in second order time dependent perturbation theory.
A calculation of the first diagram will give the same result for a free electron and a bound electron,
while the second diagram will give different results because the intermediate states are different
if an electron is bound than they are if it is free.
We will therefore compute the amplitude from the second diagram.
This contains a term causing absorption of a photon and another term causing emission.
We separate the terms for absorption and emission and pull out the time dependence.
The initial and final state is the same
and second order perturbation theory will involve a sum over intermediate atomic states,
and photon states.
We will use the matrix elements of the interaction Hamiltonian between those states.
We have dropped the subscript on
specifying the photon emitted or absorbed leaving a reminder in the sum.
Recall from earlier calculations that the creation and annihilation operators just give a factor of 1
when a photon is emitted or absorbed.
From time dependent perturbation theory, the rate of change of the amplitude to be in a state is given by
In this case, we want to use the equations for the the state we are studying,
, and all intermediate states,
plus a photon.
Transitions can be made by emitting a photon from
to an intermdiate state and transitions can be
made back to the state
from any intermediate state.
We neglect transitions from one intermediate state to another as they are higher order.
(The diagram is emit a photon from
then reabsorb it.)
The differential equations for the amplitudes are then.
In the equations for
we explicitly account for the fact that an intermediate state can make a transition back to the initial state.
Transitions through another intermediate state would be higher order and thus should be neglected.
Note that the matrix elements for the transitions to and from the initial state are closely related.
We also include the effect that the initial state can become depleted as intermediate states are populated
(instead of 1) in the equation for
Note also that all the photon states will make nonzero contributions to the sum.
Our task is to solve these coupled equations.
Previously, we did this by integration, but needed the assumption that the amplitude to be in the initial state was 1.
Since we are attempting to calculate an energy shift, let us make that assumption and plug it into the equations to
verify the solution.
will be a complex number, the real part of which represents an energy shift,
and the imaginary part of which represents the lifetime (and energy width) of the state.
Substitute this back into the differential equation for
to verify the solution
and to find out what is.
Note that the double sum over photons reduces to a single sum because we must absorb the same type of photon that was emitted.
(We have not explicitly carried along the photon state for economy.)
Since this a calculation to order
and the interaction Hamiltonian squared contains a factor of
we should drop the
s from the right hand side of this equation.
We have a solution to the coupled differential equations to order
We should let
since the self energy is not a time dependent thing, however,
the result oscillates as a function of time.
This has been the case for many of our important delta functions, like the dot product of states with definite momentum.
Let us analyze this self energy expression for large time.
We have something of the form
If we think of
as a complex number, our integral goes along the real axis.
In the upper half plane, just above the real axis,
, the function goes to zero at infinity.
In the lower half plane it blows up at infinity and on the axis, its not well defined.
We will calculate our result in the upper half plane and take the limit as we approach the real axis.
This is well behaved everywhere except at
The second term goes to
A little further analysis could show that the second term is a delta function.
the real part of
corresponds to an energy shift in the state
and the imaginary part corresponds to a width.
All photon energies contribute to the real part.
Only photons that satisfy the delta function constraint contribute to the imaginary part.
Moreover, there will only be an imaginary part if there is a lower energy state into which the state in question can decay.
We can relate this width to those we previously calculated.
The right hand side of this equation is just what we previously derived for the decay rate of state
over all final states.
The time dependence of the wavefunction for the state is modified by the self energy correction.
This also gives us the exponential decay behavior that we expect,
keeping resonant scattering cross sections from going to infinity.
So, the width just goes into the time dependence as expected and we don't have to worry about it anymore.
We can now concentrate on the energy shift due to the real part of
In our calculation of the
total decay rate summed over polarization and integrated over photon direction
we computed the cosine of the angle between each polarization vector and the (vector) matrix element.
Summing these two and integrating over photon direction we got a factor of
the polarization is eliminated from the matrix element.
The same calculation applies here.
Note that we wish to use the
electric dipole approximation which is not valid for large
It is valid up to about 2000 eV so we wish to cut off the calculation around there.
While this calculation clearly diverges, things are less clear here because of the eventually rapid oscillation of the
term in the integrand as the E1 approximation fails.
Nevertheless, the largest differences in corrections between free electrons and bound electrons occur in the region in which the E1 approximation
For now we will just use it and assume the cut-off is low enough.
It is the difference between the bound electron's self energy and that for a free electron in which we are interested.
Therefore, we will start with the free electron with a definite momentum
The normalized wave function for the free electron is
It easy to see that this will go to negative infinity if the limit on the integral is infinite.
It is quite reasonable to cut off the integral at some energy beyond which the theory we are using is invalid.
Since we are still using non-relativistic quantum mechanics,
the cut-off should have
For the E1 approximation, it should be
We will approximate
since the integral is just giving us a number and
we are not interested in high accuracy here.
We will be more interested in accuracy in the next section when we compute the difference between free electron and bound electron
self energy corrections.
If we were hoping for little dependence on the cut-off we should be disappointed.
This self energy calculated is linear in the cut-off.
For a non-relativistic free electron the energy
decreases as the mass of the electron increases,
so the negative sign corresponds to a positive shift in the electron's mass,
and hence an increase in the real energy of the electron.
Later, we will think of this as a renormalization of the electron's mass.
The electron starts off with some bare mass.
The self-energy due to the interaction of the electron's charge with its own radiation field
increases the mass to what is observed.
Note that the correction to the energy is a
constant times , like the non-relativistic formula for the kinetic energy.
If we cut off the integral at , the correction to the mass is only about 0.3%,
but if we don't cut off, its infinite.
It makes no sense to trust our non-relativistic calculation up to infinite energy,
so we must proceed with the cut-off integral.
If we use the Dirac theory, then we will be justified to move the cut-off up to very high energy.
It turns out that the relativistic correction diverges logarithmically (instead of linearly)
and the difference between bound and free electrons is finite relativistically (while it diverges logarithmically for
our non-relativistic calculation).
Note that the self-energy of the free electron depends on the momentum of the electron,
so we cannot simply subtract it from our bound state calculation.
would we choose?)
Rather me must account for the mass renormalization.
We used the observed electron mass in the calculation of the Hydrogen bound state energies.
In so doing, we have already included some of the self energy correction and we must not double correct.
This is the subtraction we must make.
Its hard to keep all the minus signs straight in this calculation, particularly if we consider the bound and
continuum electron states separately.
The free particle correction to the electron mass is positive.
Because we ignore the rest energy of the electron in our non-relativistic calculations,
This makes a negative energy correction to both the bound (
Bound states and continuum states have the same fractional change in the energy.
We need to add back in a positive term in
to avoid double counting of the self-energy correction.
Since the bound state and continuum state terms have the same fractional change,
it is convenient to just use
for all the corrections.
Because we are correcting for the mass used to calculate the base energy of the state
our correction is written in terms of the electron's momentum in that state.