Electric dipole approximation
Suppose that an electromagnetic wave
is travelling through an atom, and affecting the electrons.
The general Hamiltonian of an electron in an electromagnetic field is:
Where is the electron’s charge,
is the canonical momentum operator,
is the magnetic vector potential,
and is the electric scalar potential.
We start by fixing the Coulomb gauge
such that and commute;
let be an arbitrary test function:
Furthermore, we assume that is so small that is negligible,
so the Hamiltonian is reduced to:
We now split like so,
where can be regarded as a perturbation to the “base” :
In an electromagnetic wave, is oscillating sinusoidally in time and space:
Mathematically, it is more convenient to represent this with a complex exponential,
whose real part should be taken at the end of the calculation:
The corresponding perturbative electric field is then given by:
Light in and around the visible spectrum
has a wavelength ,
while an atomic orbital is several Bohr radii ,
so is very small. Therefore:
This is the electric dipole approximation:
we ignore all spatial variation of ,
and only consider its temporal oscillation.
Also, since we have not used the word “photon”,
we are implicitly treating the radiation classically,
and the electron quantum-mechanically.
Next, we want to rewrite
to use the electric field instead of the potential .
To do so, we use that momentum
and evaluate this in the interaction picture:
Taking the off-diagonal inner product with
the two-level system’s states and gives:
Where is the resonance of the energy gap,
close to which we assume that and are oscillating, i.e. .
Therefore, , so we get:
the transition dipole moment operator of the electron,
hence the name electric dipole approximation.
Finally, we take the real part, yielding:
If this approximation is too rough,
can always be Taylor-expanded in :
Taking the real part then yields the following series of higher-order correction terms:
- M. Fox,
Optical properties of solids, 2nd edition,
- D.J. Griffiths, D.F. Schroeter,
Introduction to quantum mechanics, 3rd edition,