M2L10zg.txt

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Let's talk about atoms in electric field.
We put the atoms in a uniform electric field.
Again, we assume that it points in the Z direction,
and its magnitude is epsilon, and we
want to ask, what is the electrostatic energy
in this electric field?
And we're using the fact that electrostatic energy
can be expended in a multipole expansion.
We have a monopole term, we have a dipole term,
and we have a quadratic term.
So the charge, of course, is the atom
itself is a neutral atom, so there is no monopole term.
The linear term in the electric field
would correspond to a permanent dipole moment,
and I will remind you in a moment that this is 0.
And then the term which provides us with a Stark effect
with the energy shift of atoms in electric field
will be the third term here, which is characterized
by the polarizability alpha, and it corresponds
to an induced dipole moment that there is an induced dipole
moment, which is alpha times epsilon,
and then the induced dipole moment interacts
with the electric field, and that gives then epsilon times
epsilon, epsilon squared.
So this would be a classical multipole expansion,
and we will now derive results quantum mechanically.
The perturbation operator for us is the dipole operator
and that could, in principle, include
a permanent or an induced dipole moment that
would take care of the second and third term,
the dipole operator and its projection on the z-axis.
So the dipole operator is the charge
of the electron times the position,
and as long as the polarizability
in this situation is isotropic-- the [INAUDIBLE] is a minus
sign, minus e is a charge.
If you apply an electric field in the Z direction,
all the relevant dipole moments are in the Z direction.
For anisotropic materials, you could
have an electric field in the Z direction,
and the dipole moment points at an angle,
but we do not have such a situation for our atoms.
So the operator is then simply charge of the electron,
the Z coordinate times the electric field,
and this has odd parity, and that
leads us immediately to the result
when we have an atom in an eigenstate n.
And we ask, what is the expectation value of H prime?
It is 0, because of parity, so the answer
is, we have no permanent dipole moment until we
have degenerate energy levels.
If n is a non-degenerate level, this matrix element
is 0 by the parity selection rule.
Now we want to do perturbation theory,
so our perturbation operator is this.