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M2L11b.txt
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M2L11b.txt
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#
# File: content-mit-8-421-2x-subtitles/M2L11b.txt
#
# Captions for 8.421x module
#
# This file has 152 caption lines.
#
# Do not add or delete any lines.
#
#----------------------------------------
So today I want to talk to you briefly
what happens when we go beyond perturbation theory,
when we go beyond the quadratic Stark effect,
and that leads us to a discussion of stability
of atoms in strong electric field and field ionization.
I like to sort of feature this, because it allows
me to tell you something about the peculiar properties
of Rydberg atoms and also the ionization of Rydberg atoms
for electric field.
This is how people in our field create cold plasmas,
and it's also a way to do a very sensitive detection of atoms.
So what I'm telling you today is interesting for its own sake
but also because it's an important tool
for manipulating atoms, creating plasmas,
or sensitive detection.
So this will probably only take 10 or 20 minutes,
and then we want to go from DC electric fields
to AC electric fields.
So we then discuss the AC polarizability and, well,
that will take us from perturbation theory
in a time-independent way, what we have done now for DC fields
to time-dependent perturbation theory.
So all the topics are a lot of basic aspects of quantum
physics but, as usual, I try to give you
some special perspectives from the atomic physics side.
So in perturbation theory, we have a mixture of other states,
and the said mixture is done with the matrix element,
and in perturbation theory we always
have an energy denominator-- intermediate state
to ground state-- and if the electric field is smaller
than this value, then we add mixture of other states
until the ground state is smaller.
So usually when we estimate the validity of perturbation
theory, we look for the state which
is closest to the ground state for which the energy
denominator is smallest, so therefore "i"
here is the nearest state, but-- and this
is important-- of opposite parity.
Otherwise, because of the parity selection rule,
the matrix element would be 0, and the electric field
does nothing.
I made a comment on Friday, but let me do it again.
That, of course, means that we apply electric field
to our favorite atoms, we don't have
to worry about the other hyperfine states.
The other hyperfine states have the same spatial wave function,
have the same parity, so we really
are talking about the first excited state,
in a concrete example, for those of you who work with-- alkalize
with an S ground state.
The relevant energy scale here is the excitation energy
to the first P state.
So let's just estimate for a single electron atom.
This is a hydrogenic estimate.
The excitation energy to the first excited
state, 1 is to 2 p, is about 1 Rydberg,
and the Rydberg is nothing-- or Hartree--
and it's nothing else than e square
over a naught, its electric potential of two charges
separated by Bohr radius.
And if we estimate that the matrix element
for strong transition is on the order of the Bohr radius,
there is no other length scales in the problem,
we find that the value for the electric field in atomic units
is the charge divided by the Bohr radius squared,
and this is really high.
It is on the order of 5 times 10 to the 9 volt per centimeter,
and this is 1,000 times larger than laboratory
electric fields.
Those fields would just create sparking along the electrodes.
You cannot apply such high electric fields
in a laboratory.
So therefore nothing to worry about.
If you have ground state atoms, the Stark effect
and perturbation theory is all you need.
Actually, it would be a little bit more precise when
I use the Rydberg and a naught.
I made a little bit over estimate,
so that typically the critical electric field
which would cause a breakdown of perturbation theory
for the ground state is around 10
to the 9 volt per centimeter.
So we are safe when we talk about ground states,
but once we go to the excited state,
we have degeneracy-- P states with 3, 4 degeneracy--
and they are states with opposite parity.
So we can have mixing there and actually,
as you will see for excited states,
we have already a breakdown of perturbation theory
at very, very small electric field.
So let us discuss hydrogenic orbits with principal quantum
number n, and if we estimate what
is the size of the matrix element,
well, it's not just a naught.
There is a scaling with n, which is n square.
The matrix element in higher and higher excited state scales
with n square.
Well, how does the energy separation scale?
Well, let's not discuss hydrogen here,
because in hydrogen energy levels are degenerate,
and we would immediately get a breakdown
of perturbation theory.
Let's rather formulate it for general atoms,
and we had this nice discussion about the quantum defect,
so if you compare the energy of two l states, they scale as 1
over n square.
But for different l states, we have different quantum defects.
Delta l plus 1, and here we have delta l,
so therefore doing an expansion in n,
which we assume to be large, we find that the energy difference
is proportional to the difference between the quantum
defects for the two states we want
to mix with the electric field and then, again,
the scaling with the principal quantum number is n cube.
So therefore we find for the critical field
using the criterion I mentioned above, we take the energy
splitting-- the energy denominator which appears
in perturbation theory-- we divide
by the value of the matrix element.
Well, we had the Rydberg constant,
or 2 times the Rydberg constant is nothing else than b
square over a naught.
The matrix element was 1 over a naught,
and then we have the difference between quantum defects.
And now-- and this makes it really so dramatic--
we had an n square scaling of the matrix element
and we have n to the minus 3 scaling of energy differences,
so that means the critical field scales by n to the 5.
Go to an excited state with n equals 10,
and the breakdown of perturbation theory
happens 100,000 times earlier.
So some of the scaling in atomic physics
is very, very dramatic when you to higher-- more highly excited
state.
If you throw in that quantum defects become
very small once you go beyond S and P states,
the higher states just don't penetrate into the nucleus,
so if l is larger than 2, if you have more complicated atoms,
you may add the angular momentum of the core here.
So if you put n to the 5 scaling and the small quantum defect
together, you find that critical electric fields
are smaller than 1 volt per centimeter,
already for principal quantum numbers as low as 7.
So it means bring a 1.5 volt battery close to your atom,
and you drive it crazy, you drive it out
of perturbation theory.