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M3L15a.txt
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#
# File: content-mit-8-421-3x-subtitles/M3L15a.txt
#
# Captions for 8.421x module
#
# This file has 114 caption lines.
#
# Do not add or delete any lines.
#
#----------------------------------------
--the result, which we obtained on Wednesday,
for spontaneous emission for the Einstein A coefficient
regarded as an accomplishment as a highlight of the cause.
We've worked hard to talk about atoms
and electromagnetic fields.
And ultimately, to deal with spontaneous emission,
it was not enough to put a semi-classical light atom
interaction, the dipole Hamiltonian, Rabi oscillation,
and such to put that into the picture.
We really needed a quantized version
of the electromagnetic field.
And this is the result when an atom is excited and interacts
with all of the empty modes of the vacuum.
And we summed up the probability that a photon is
emitted in any of those modes.
And by doing all the ever reaching
with the density of state for all
of the possibility of actions, we obtained the famous result
for the Einstein A coefficient, which
is also the natural line width of the atomic excited state.
Do you have any questions about the derivation
or what we did last week?
Then I think I will just continue and interpret
the result.
So we got a result for an Einstein A coefficient.
And for now, the question is, how big is it?
Well, it has a number of constants.
Let's discuss it now in atomic units.
Well, if we assume the frequency or the energy
is on the order of a Rydberg, if it's
sort of the measure for an electronic excitation
in the atom, we assume the dipole matrix element is one.
That means one per radius.
Since we have, pretty much, set everything one
and expressed everything in atomic units,
it means that the speed of light is-- remember?
The velocity of the atom in any [INAUDIBLE]
was alpha times smaller than the speed of light.
But the velocity of the atom is one atomic unit.
So therefore, the speed of light in atomic units
is one over alpha.
And that means that, if you look at the formula,
there is the speed of light to the power of 3
in the denominator.
And that means that, in atomic units,
the Einstein A coefficient is alpha
to the 3, which is 3 times 10 to the minus 7.
So that means that the ratio of the spontaneous emission rate,
which is also the inverse lifetime, and therefore
the natural alignments of the excited
state, relative to the transition frequency--
so the denting of the harmonic oscillator,
or the two-level system, relative to the [INAUDIBLE]
of the oscillator, is small.
It's actually alpha cube.
So if you take this 3 times 10 to the minus 7
and multiply it with the atomic unit of frequency,
which is two Rydbergs, we obtain on the order of 10 to the nine.
And that's a rate of 10 to the nine per second.
And that means that the lifetime of a typical atomic level
is on the order of one nanosecond.
Well, often, it's 10 to 100th nanosecond,
because mainly, transition frequencies
are smaller by quite a factor than the atomic unit
of the transition frequency.
Remember, the Rydberg frequency would be deep in the UV.
But a lot of atoms have transitions in the visible.
I highlighted already when I derived it,
that the spontaneous emission has this famous omega cube
dependence.
And that is actually important to understand
why lower-lying levels, excited hyperfine levels,
do not radiate.
So let me just kind of formalize it.
If I would now estimate what is the radiative lifetime
for transition-- which is not, as I just assumed, in the UV
or in the visible-- let me estimate
what is the radiative lifetime to emit
a microwave for a few gigahertz.
Well, then microwave frequency, roughly gigahertz,
10 to the nine, is five orders of magnitude
smaller than the frequency 10 to the 14
of an optical transition.
So therefore, this is 10 to the 15 times longer.
And if you have, typically, 1 or 10
nanoseconds for an electronic transition,
that means that this spontaneous lifetime for microwave
transition is several months.
If in addition, we factor in that hyperfine transitions have
an operator, which is the Bohr magneton, a magnetic dipole
operator, not an electric dipole,
and we discuss that the Bohr magneton is actually--
when we discuss multipole transitions,
we discuss that the Bohr magneton is alpha times smaller
than a typical electric dipole moment,
so therefore, a magnetic dipole transition
is alpha times weaker than an electronic dipole transition.
And that means now, if we multiply months, which
we obtained by the frequency scaling,
again, by alpha square for the weakness
of the magnetic dipole, we find that atomic hyperfine levels
have a lifetime which is on the order of 1,000 years.
And this is why it's very safe to neglect
those transition in the laboratory
and assume that all hyperfine states in the ground state
manifold, pretty much, don't decay and are long-lived.
Questions?