or in terms of the electromagnetic potentials
[ IV-1a ]
[ IV-1b ] Let us now write the total time derivative of a component of the vector potential
so that
[ IV-2 ]
[ IV-3 ] Therefore, we may write the Lorentz force as
[ IV-4 ] We may also write
so that Equation [ IV-4] becomes
[ IV-5 ] which may, in turn, may be written
[ IV-6 ] This last equation may now be compared to the Lagrangian equation of motion -- i.e.
[ IV-7a ] where, in general,
[ IV-7b ] Therefore, we identify
[ IV-8 ] as the Lagrangian for a single charged particle. We may write
[ IV-9a ] which, in light of Equation [ IV-8 ], becomes
Therefore,
[ IV-9b ]
[ IV-9c ] and, using the cannonically conjugate momenta associated Equation [ IV-8 ] -- i.e.,
[ IV-10 ] -- we see that Equation [ IV-9c ] can be written
[ IV-11a ] or finally
[ IV-11b ] We are now in a position to set forth the nonrelativistic Hamiltonian of a single charged particle -- viz.
[ IV-12 ] with the canonical conjugate variables given by
[ IV-13a ]
[ IV-13b ] In principal, we are done, since we may now write the complete Hamiltonian for a many particle material system as
[ IV-14 ] Unfortunately, this form of the Hamiltonian is not the most useful in optical physics since it is expressed in terms of the vector potential and needs to be evaluated at all charge positions. Most annoyingly, it does not yield an interaction term in the form used earlier -- i.e. Equation [ III-3a ] in the lecture set entitled The Interaction of Radiation and Matter: Semiclassical Theory.
THE MULTIPOLE EXPANSION OF THE CLASSICAL HAMILTONIAN:
Before we manipulate the Hamiltonian further, we digress a bit to get a better fix on what we are really look for. From the most basic notions of electrostatics, the energy of interaction with an external transverse field should be expressible as the "energy of assembly" [2]-- viz.
[ IV-15a ] which becomes for a neutral atomwith charge centered at
[ IV-15b ] Electric fields of interest vary only slightly over an atom so that we should be able to expand the external field in a rapidly converging series as follows
[ IV-16 ] Substituting this expansion into Equation [ IV-15b ] and integrating we obtain
[ IV-17a ] Formally, we may cast this expansion for the interaction energy in the form
[ IV-17b ] In the continuum picture, the interaction energy should be expressible in the form
[ IV-18 ] whereis the polarization density. It may be shown quite easily that Equations [ IV-18 ] and [ IV-17 ] are equivalent if the polarization density is expressed in the following multipole expansion
[ IV-19 ] With this background, we return to a consideration of the Hamiltonian in Equation [ IV-14 ]. In what follows, we show that this Hamiltonian is transformed into a form which consistent with Equation [ IV-17b ] if we make an appropriate gauge transformation of the fields. In general, the gauge transformation
[ IV-20 ] where
is an arbitrary scalar gauge function, leaves the electric and magnetic fields unchanged. Motivated by the discussion above, we choose the gauge function
[ IV-21 ] where
is the polarization density in the form shown in Equation [ IV-19 ] and
is the vector potential appearing in Equation [ IV-14 ]. The impact of the transformation on the scalar potential term in Equation [ IV-14 ] is easy to evaluate -- viz.
[ IV-22 ] Dealing with the transformation
[ IV-23 ] is straightforward, but extremely tedious (and not very useful)! Substituting for
from Equation [ IV-19 ] and doing a lot of integrating by parts we could demonstrate that
[ IV-24 ] Making use of these results, the complete Hamiltonian of a atom plus radiation field may be written
[ IV-25 ] In practical terms, the usefulcomplete Hamiltonian is written
[ IV-26a ] where
[ IV-26b ]
[ IV-26c ]
[ IV-26d ]
[ IV-26e ]
[ IV-26f ]
[ IV-26g ] If we take
of the order of magnitude of the Bohr radius
,
of order
, and
of order
then we may estimate the three linear interaction terms[3] -- viz.
Further we note that
!!
