We now wish to compute the Hamiltonian in terms of the coefficients
This is an important calculation because we will
use the Hamiltonian formalism to do the quantization of the field.
We will do the calculation using the covariant notation (while Sakurai outlines an alternate calculation using 3-vectors).
We have already calculated the Hamiltonian density for a classical EM field.
Now lets compute the basic element of the above formula for our decomposed radiation field.
We have all the elements to finish the calculation of the Hamiltonian.
Before pulling this all together in a brute force way,
its good to realize that almost all the terms will give zero.
We see that the derivative of
is proportional to a 4-vector, say
and to a polarization vector,
The dot products of the 4-vectors, either
with itself or
Going back to our expression for the Hamiltonian density, we can eliminate some terms.
The remaining term has a dot product between polarization vectors which will be nonzero if the polarization
vectors are the same.
(Note that this simplification is possible because we have assumed no sources in the region.)
The total Hamiltonian we are aiming at, is the integral of the Hamiltonian density.
When we integrate over the volume only products like
will give a nonzero result.
So when we multiply one sum over
by another, only the terms with the same
will contribute to the integral,
basically because the waves with different wave number are orthogonal.
This is the result we will use to quantize the field.
We have been careful not to commute and here in anticipation of the fact that they do not commute.
It should not be a surprise that the terms that made up the Lagrangian gave a zero contribution because
and we know that E and B have the same magnitude in a radiation field.
(There is one wrinkle we have glossed over; terms with