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M2L11j.txt
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M2L11j.txt
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#
# File: content-mit-8-421-2x-subtitles/M2L11j.txt
#
# Captions for 8.421x module
#
# This file has 157 caption lines.
#
# Do not add or delete any lines.
#
#----------------------------------------
The second point of discussion is the concept
of the oscillator strengths.
So what I'm teaching you in the next five minutes
is so old fashioned that I sometimes wonder,
should I still teach it or not.
On the other hand, you find it in all the textbooks.
You also want to understand a little bit the tradition,
and at least I'm giving you some motivation
to learn about it, that if I parametrize
the matrix element within oscillator strengths,
and most of your atoms-- most of the alkali
atoms-- have an oscillator strength for the s
to p transition for the d lines, which is unity.
You can actually write down what is the matrix element, what
is the spontaneous lifetime of your atom without knowing
anything about atomic structure, just memorizing that f equals
1-- the oscillator strength is 1-- is pretty much all you have
to know about your atom, and the rest-- the only other thing you
have to know is, what is the resonant frequency
of your laser-- 780 nanometer, 589 nanometer, 671 nanometer.
So the modern motivation for this old-fashioned concept
is, for simple atoms where the oscillator strength
is close to 1, this is probably the parametrization you
want to use, because you can forget
about the atomic structure.
But the derivation would go as follows.
I want to compare our result, how an atom
responds to an electric field.
I want to compare this result to the result
of a classical oscillator, compare our result for the AC
polarizability to a classical harmonic oscillator.
So I assume this classical harmonic oscillator
has a charge, a mass term, and a frequency.
And both the atom and the classical harmonic oscillator
are driven by the time-dependent electric field,
which we have already parametrized by cosine omega t.
If you look at the classical harmonic oscillator,
you find that we derive it at omega
and ask what is the time-dependent dipole moment.
It's driven by the cosine term, and what I mean, of course,
is dipole moment of a classical harmonic oscillator
is nothing else than charge times displacement.
And, well, if you spend one minute
and solve the equation for the driven harmonic oscillator,
you find that the response, the amplitude-- the steady state
amplitude zk of the harmonic oscillator--
is cosine omega t times a prefactor, which
I'm writing down now.
There is the resonant behavior.
So that's the response of the classical harmonic oscillator.
So this is just classical harmonic oscillator physics,
and I now want to define a quantity, which
I call the oscillator strength of the atom.
I'm just jumping now from the harmonic oscillator to the atom
and then I combine the two, and the oscillator strength
is nothing else than a parametrization
of the matrix element between two different states,
but it's dimensionless, and it's made dimensionless
by using the mass, by using h bar,
and by using the transition frequency.
So the atomic-- let me just make sure we keep track of it.
This is the result for the classical harmonic oscillator,
and this is now the result for the atom
that we have found already the result for the atom before,
and now I'm re-writing it simply by expressing
the matrix element square by the oscillator strengths,
and this here is just another expression
for the polarizability alpha.
Well, let's now compare the result
of a quantum mechanical atom, exactly
described time-dependent perturbation theory,
to the result of a classical harmonic oscillator.
The frequency structure is the same,
so if I would now say we have an ensemble
of harmonic oscillators, and the harmonic oscillators may have
different frequencies and different charges,
then I have made those formulas exactly equal.
And I can now formulate that the atom reacts
to an electric field-- to time-dependent
electric field-- exactly as an ensemble
of classical oscillators.
If I would say I have an ensemble of oscillators
with effective charge, then the response
of the atom and the response of the ensemble
of classical oscillators is absolutely identical.
For the atom response as a set of classical oscillators
with an effective which is given here.
So therefore you don't have to go
further if you want to have any intuition how an atom reacts
to light.
The classical harmonic oscillator
is not an approximation.
It is exact.
That result is relevant, because it
allows us to clearly formulate the classical correspondence.
The second thing is, is you can easily
show with basic commutator algebra,
there is the Thomas-Kuhn sum rule, which
is discussed in all texts in quantum physics, which
says that the sum over all oscillator strengths is 1,
so therefore we know if we have transitions from the ground
state to different states, the sum of all the oscillator
strengths to all the states can only be 1.
And another advantage of the formulation
with oscillator strengths is that it
is a dimensionless unit.
It's a dimensionless parameter, which
tells us how the atom corresponds
to an external electromagnetic field.
If you have hydrogen, the 1s to 2p transition
is the strongest transition, and it has a matrix element
which corresponds to an oscillator strength of about
0.4, so the rest comes from more highly excited states.
However, for alkali atoms, the d line, the s to p transition,
has an oscillator strength-- I didn't write down
the second digit, but it's with excellent approximation--
1.98 or something-- so not just qualitative,
the almost quantitative, and you capture
the response from the atom by saying f equals 1.
So if you use for the alkali atoms f equals 1,
then simply the transition frequency of the d line
gives you the polarizability alpha
and, as we will see later-- because we haven't introduced
this-- but it will also give you gamma, the natural line width,
because all the coupling to the electromagnetic field--
to an external field-- is really captured
by saying what the matrix element is,
and f equals 1 is nothing else than saying the matrix
element is such and such.
And indeed, if I now use the definition of the oscillator
strengths in reverse, the matrix element between two
transitions, between two states, is the matrix element squared
is the oscillator length times 1/2,
and if you just go back and look at the formula,
you find that this is now-- well, this is dimensionless,
so we need now, because the left-hand side is the length
squared, we need two lengths.
One is the Compton wavelength, which is h bar over 2 times
the mass of the electron, and lambda bar
is the transition wavelength lambda divided by 2 pi.
So therefore, I haven't found it anywhere in textbooks,
but this is my sort of summary of this,
for if you have a strong transition,
and strong means that the oscillator length is
close to 1, then the matrix element for the transition
is approximately the geometric mean of the Compton wavelengths
of the electron and the reduced wavelengths
of the resonant transition.
So now if you want to know what is
the matrix element for the d line of rubidium,
take the wavelength of 780 nanometer,
take the Compton wavelength of the electron,