We base our discussion on the
picture of discretized field - that means that there would be a
"countable" system of eigenmodes for any field system
(for N coupled oscillators there are N eigenmodes). For electromagnetic
field there is a total field energy
expression (as there would be a total energy expression in terms of
displacements for mechanical oscillators). We
see that as it is the E and B are sort of mixed both displacement and
momenta (this gets sort of more
understandable if we use the vector potential "in the radiation gauge",
then E looks more like momentum).
The expression for A taken from
the litterature contains a mystery constant.
This constant comes from the
requirement that the "classical expression" when inserted the
expression for A gives the same result as the
algebraic (creation-annihilation) expression with the number of
excitation quanta times hbar omega (in the slide we
have a problem with factor 2 - it might originate from a different
expression in SI and Gaussian systems - but
the rest of the constant is clearly shown
Thus we have an algebraic
description - number of photons - based for the electromagnetic fields, as we need in our "unperturbed"
energy operator Density of
Modes
Next topic is "density of modes"
or in the Golden rule expression - the density of states This is always based on
discretization - quantization in a large box with edges L x L x L
D010-density_of_modes_and_states.png
D010-density_of_modes_and_states.png
D020-density_of_modes_and_states.png
D020-density_of_modes_and_states.png
D030-density_of_modes_and_states.png
D030-density_of_modes_and_states.png
The
atom - field interaction term
The interaction between the atom
and the field is obtained from taking into account how a charged
particle gets energy from the field. In
classical physics we would also need to consider how the field gets
energy from a moving charge (and there it
would be the acceleration which is usually used...)
In the Hamilton-like formulation
it will be enough just to take the "action" - the "reaction" should be
automatically included - we do not investigate
this in detail - there is enough to do here for the Lagrangian
E020_Elmag_Field--Particle-Interaction.png
E020_Elmag_Field--Particle-Interaction.png
Particle in electric and magnetic field - Lagrangian and
Hamiltonian - previously shown in ../2010.11.04/charge-in-elmag-derivation-jpg/
We are working on a new text - which has been used in the slides below
a preliminary part (which will be replaced later) is as a pdf here: preliminary_charge_in_elmag.pdf
L010-Lagrangian_for_particle_in-ELMAG.png
L010-Lagrangian_for_particle_in-ELMAG.png
L020-transform-Lagrangian__in-ELMAG.png
L020-transform-Lagrangian__in-ELMAG.png
L030-transform-Lagrangian__in-ELMAG.png
L030-transform-Lagrangian__in-ELMAG.png
Here is the charged particle in
electromagnetic field Lagrangian transformed to a Hamiltonian as is
usual in the literature (Bransden and
Joachain book)
This completes all the components for application of the Golden rule -
to evaluate the transition rate from excited atom to atom in
lower energy state and one emitted photon.
