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00:00:00,590 --> 00:00:02,200
PROFESSOR: OK, time
for us to do one

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00:00:02,200 --> 00:00:04,660
example, a non-trivial
example, which

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is the ionization of
hydrogen. It's a fun example,

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00:00:08,560 --> 00:00:10,450
and let's see how it goes.

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00:00:33,130 --> 00:00:48,310
So ionization of
hydrogen. Ionization

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00:00:48,310 --> 00:00:55,440
of hydrogen. Very good.

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00:00:55,440 --> 00:00:58,110
So we're going to think
of a hydrogen atom

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on its ground state
sitting there,

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00:01:02,220 --> 00:01:09,650
and then you shine an
electromagnetic field,

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00:01:09,650 --> 00:01:14,260
and if the electromagnetic
field is sufficiently strong,

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00:01:14,260 --> 00:01:16,430
the electron is going
to be kicked out

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00:01:16,430 --> 00:01:22,710
by interacting, typically with
the electric field of the wave.

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So the wave comes
in and interacts.

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So we usually think of
this in terms of photons.

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So we'll have a hydrogen
atom in its ground state.

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00:01:42,350 --> 00:01:51,030
And then you have
a harmonic e field,

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00:01:51,030 --> 00:01:56,600
and the electron
could get ejected.

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So typically, you have a
photon in this incident.

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We think of this in terms
of electromagnetism,

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although our treatment
of electromagnetism

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was not going to include
photons in our description.

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But we physically
think of a photon

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that this incident on
this-- here's a proton.

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Here is the electron
going in circles,

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and the photon comes
in, a photon gamma,

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and it kicks the electron out.

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So the photon has energy e
of the photon is h bar omega.

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So when we think of
putting a light beam,

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and we're going to
send many photos,

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we have to think of each
photon, what it's doing--

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whether the energy is very
big, whether the energy is

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very small, and how does it
affect our approximation.

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So half of the story of
doing such a calculation

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is understanding when
it could be valid,

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because we're going to
assume a series of things

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in doing the calculation.

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And the validity
still won't turn out

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to be for a rather
wide range of values.

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But we have to think about it.

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So a couple of quantities
you'll want to consider

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is that the energy of
the electron, energy

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00:03:43,280 --> 00:03:47,480
of the electron
that is ejected is

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00:03:47,480 --> 00:03:52,890
h bar squared k of the
electron that is ejected,

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00:03:52,890 --> 00:03:58,350
k squared over 2m
of the electron.

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00:03:58,350 --> 00:04:04,100
And it's going to be equal
to the energy of the photon

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minus the ground state energy
of the electron in the hydrogen

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00:04:09,470 --> 00:04:11,160
atom.

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00:04:11,160 --> 00:04:18,399
So the hydrogen atom
here is energy equal 0.

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00:04:18,399 --> 00:04:22,450
Ground state is below 0.

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00:04:22,450 --> 00:04:25,720
So if you supply some energy,
the first part of the energy

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has to be to get it
to energy 0, and then

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00:04:28,450 --> 00:04:30,590
to supply kinetic energy.

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So the kinetic energy is the
energy of the photon minus

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what this called
the Rydberg, or Ry.

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00:04:38,320 --> 00:04:45,640
The Rydberg is 13.6
eV with a plus sign.

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00:04:45,640 --> 00:04:51,040
And that's the magnitude
of the depth of the well.

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The Rydberg is e squared over
2a0 or 2 times the Rydberg is e

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00:05:08,780 --> 00:05:09,800
squared over a0.

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00:05:13,750 --> 00:05:18,310
That's what was calculated,
and this is actually

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00:05:18,310 --> 00:05:30,140
equal to h bar c the constant
alpha times h bar c over a0.

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00:05:30,140 --> 00:05:36,080
remember, the constant alpha
was e squared over h bar c.

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So those are some quantities.

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OK.

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00:05:43,090 --> 00:05:46,520
So we need the electron
to be able to go out.

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00:05:46,520 --> 00:05:49,921
The energy of the photon must
be bigger than a Rydberg.

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00:05:52,750 --> 00:05:56,810
OK, so conditions for
our approximation.

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One.

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We're going to be using
our harmonic variation.

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We said in our
harmonic variation,

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Hamiltonian delta h was 2h
prime cosine of omega t,

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00:06:12,170 --> 00:06:17,630
and we want this h prime
to be simple enough.

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We want to think of this photon
that is coming into the atom

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as a plane wave,
something that doesn't

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have big spatial
dependence in the atom.

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It hits the whole atom as
with a uniform electric field.

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The electric field
is changing in time.

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It's going up and down, but
it's the same everywhere

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00:06:43,400 --> 00:06:45,180
in the atom.

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So for that, we need that--

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00:06:57,220 --> 00:07:02,450
if you have a wave,
it has a wavelength,

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00:07:02,450 --> 00:07:07,000
if you have an atom
that is this big,

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00:07:07,000 --> 00:07:09,040
you would have the
different parts

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00:07:09,040 --> 00:07:12,130
of the atom are experiencing
different values

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00:07:12,130 --> 00:07:15,890
of the electric field
at the same time.

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00:07:15,890 --> 00:07:20,470
On the other hand, if the
wavelength is very big,

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the atom is experiencing the
same spatially independent

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00:07:25,250 --> 00:07:28,300
electric field at
every instant of time.

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It's just varying up and down,
but everywhere in the atom

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is all the same.

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So what we want for this is
that lambda of the photon

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be much greater than a0.

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So that-- 1.

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So this means that the photon
has to have sufficiently long

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00:07:56,640 --> 00:08:00,150
wave, and you cannot
be too energetic.

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00:08:00,150 --> 00:08:03,130
If you're too energetic,
the photon wave length

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is too little.

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00:08:04,170 --> 00:08:07,260
By the time it becomes
smaller than a0,

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00:08:07,260 --> 00:08:10,150
your approximation is not
going to be good enough.

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00:08:10,150 --> 00:08:12,540
You're going to have to
include the spatial dependence

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00:08:12,540 --> 00:08:14,430
of the wave everywhere.

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It's going to make
it much harder.

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So we want to see
what that means,

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and in the interest of
time, I will tell you

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00:08:24,150 --> 00:08:29,710
with a little bit
of manipulation,

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00:08:29,710 --> 00:08:33,760
this shows that h
omega over a Rydberg

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00:08:33,760 --> 00:08:40,809
must be much smaller than
4 pi over alpha, which

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00:08:40,809 --> 00:08:46,380
is about 1,772.

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So that that's a
condition for the energy.

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The energy of the photon
cannot exceed that much.

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And let's write it here.

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00:09:03,950 --> 00:09:07,760
So that's a good exercise
for you to do it.

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00:09:07,760 --> 00:09:10,220
You can see it also
in the notes just

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00:09:10,220 --> 00:09:12,230
manipulating the quantities.

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00:09:14,780 --> 00:09:25,720
And this actually says h omega
is much less than 23 keV.

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00:09:29,000 --> 00:09:36,650
OK, 23 keV is roughly
the energy of a photon

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whose wavelength is a0.

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That's a nice thing to know.

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OK.

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While this photon
cannot be too energetic,

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it has to be somewhat
energetic as well,

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because it has to
kick out the electron.

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So at least, must be more
energy than a Rydberg.

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00:10:02,760 --> 00:10:07,410
But if it just has
a Rydberg energy,

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it's just basically going
to take the electron out

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to 0 energy, and then you're
going to have a problem.

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People in the hydrogen atom
compute the bound state

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spectrum, and they computer
the continuous spectrum

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in the hydrogen
atom, in which you

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00:10:25,440 --> 00:10:31,230
calculate the plane waves of the
hydrogen atom, how they look.

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They're not all that
simple, because they're

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affected by the hydrogen atom.

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They're very interesting
complicated solutions

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for approximate plane waves in
the presence of the hydrogen

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atom.

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And we don't want
to get into that.

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That's complicated.

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00:10:49,440 --> 00:10:52,860
We want to consider
cases where the electron,

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00:10:52,860 --> 00:10:58,920
once it escapes the proton,
it's basically a plane wave.

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00:10:58,920 --> 00:11:06,070
So that requires that
the energy of the photon

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00:11:06,070 --> 00:11:10,570
is not just a little bit
bigger than a Rydberg,

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00:11:10,570 --> 00:11:12,420
but it's much bigger a Rydberg.

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And saw the electron,
the free electron

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does not feel the Coulomb
field of the proton.

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00:11:43,690 --> 00:11:48,090
And it's really a plane wave.

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00:11:48,090 --> 00:11:53,090
So here, it's a point
where you decide,

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00:11:53,090 --> 00:11:56,440
and let's be
conventional and say

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that 1 is much less than 10.

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00:12:02,140 --> 00:12:05,600
That's what we mean
by much less 1/10.

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00:12:05,600 --> 00:12:10,810
So with this
approximation, h omega

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00:12:10,810 --> 00:12:15,040
must be much bigger than 13.6.

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00:12:15,040 --> 00:12:20,980
So it should be
bigger than 140 eV.

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That's 10 times that.

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00:12:24,450 --> 00:12:29,540
And it should be smaller,
much smaller and 23 keV,

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00:12:29,540 --> 00:12:32,940
so that's 2.3 keV.

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So this is a range, and
we can expect our answer

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to make sense.

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If you want to do better,
you have to work harder.

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You can do better.

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People have done this
calculation better and better.

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00:12:54,780 --> 00:12:57,450
But you have to
work much harder.

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I want to emphasize one
more thing that is maybe I

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00:13:01,080 --> 00:13:03,770
can leave you this an exercise.

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00:13:03,770 --> 00:13:07,130
So whenever you have a
photon in this range,

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00:13:07,130 --> 00:13:12,740
you can calculate the
k of the electrode,

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00:13:12,740 --> 00:13:17,210
and you can calculate how does
the k of the electron behave.

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00:13:17,210 --> 00:13:21,920
And you find that k of
the electron times a0

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00:13:21,920 --> 00:13:31,030
is in between 13 and
3 for these numbers.

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00:13:31,030 --> 00:13:34,210
When the energy of the photon
is between those values,

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00:13:34,210 --> 00:13:38,530
you can calculate the
momentum of the electron, k

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00:13:38,530 --> 00:13:39,520
of the electron.

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00:13:39,520 --> 00:13:43,120
I sometimes put an e to
remind us of the electron.

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00:13:43,120 --> 00:13:44,650
But I'll erase it.

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00:13:44,650 --> 00:13:46,300
I think it's not necessary.

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00:13:48,970 --> 00:13:53,121
And that's the range.

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OK.

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00:13:53,620 --> 00:13:55,390
So we're preparing the grounds.

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00:13:55,390 --> 00:13:58,990
You see, this is our
typical additive.

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00:13:58,990 --> 00:14:01,870
We're given a problem,
a complicated problem,

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00:14:01,870 --> 00:14:04,120
and we take our
time to get started.

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00:14:04,120 --> 00:14:08,290
We just think when will it
be valid, what can we do.

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00:14:08,290 --> 00:14:11,980
And don't rush too much.

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00:14:11,980 --> 00:14:14,590
That's not the
attitude in an exam,

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00:14:14,590 --> 00:14:19,430
but when you're thinking about
the problem in general, yes,

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00:14:19,430 --> 00:14:24,190
it is the best attitude.

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00:14:24,190 --> 00:14:29,980
So let's describe what the
electric field is going to do.

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00:14:32,680 --> 00:14:35,590
That's the place
where we connect now

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00:14:35,590 --> 00:14:37,840
to an electric
field that is going

188
00:14:37,840 --> 00:14:39,520
to produce the ionization.

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00:14:39,520 --> 00:14:43,540
So remember the perturbation
of the Hamiltonian,

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00:14:43,540 --> 00:14:48,610
now, it's going to be the
coupling of the system

191
00:14:48,610 --> 00:14:50,120
to an electric field.

192
00:14:50,120 --> 00:14:52,060
And this system
is our electrons.

193
00:14:52,060 --> 00:14:59,240
So it's minus the electron
times the potential, electric

194
00:14:59,240 --> 00:15:02,120
potential, scalar potential.

195
00:15:13,770 --> 00:15:19,890
Now, needless to say,
actually, the electron

196
00:15:19,890 --> 00:15:21,720
that is going to
be kicked out it's

197
00:15:21,720 --> 00:15:24,450
going to be non-relativistic.

198
00:15:24,450 --> 00:15:27,360
That's also kind
of obvious here.

199
00:15:27,360 --> 00:15:31,020
You see h omega is 2.3 keV.

200
00:15:31,020 --> 00:15:33,750
You subtract 13.6 eV.

201
00:15:33,750 --> 00:15:35,590
Doesn't make any difference.

202
00:15:35,590 --> 00:15:40,560
So that's the energy of the
emitted electron roughly,

203
00:15:40,560 --> 00:15:48,120
and for that energy, that's much
smaller than 511 keV, which is

204
00:15:48,120 --> 00:15:49,830
the rest mass of an electron.

205
00:15:49,830 --> 00:15:53,170
So that electron is going
to be non-relativistic,

206
00:15:53,170 --> 00:15:56,610
which is important, too,
because we're not trying

207
00:15:56,610 --> 00:15:59,490
to do Dirac equation now.

208
00:15:59,490 --> 00:16:03,780
So here is our
potential, and then we'll

209
00:16:03,780 --> 00:16:07,300
write the electric field.

210
00:16:07,300 --> 00:16:07,800
Let's see.

211
00:16:07,800 --> 00:16:10,440
The electric field.

212
00:16:10,440 --> 00:16:13,940
We will align it to the
z-axis to begin with.

213
00:16:16,990 --> 00:16:23,190
2e0 cosine omega t times z hat.

214
00:16:23,190 --> 00:16:24,610
These are conventions.

215
00:16:24,610 --> 00:16:27,540
You see, we align
it along the z-axis,

216
00:16:27,540 --> 00:16:30,690
and we say it has a
harmonic dependence.

217
00:16:30,690 --> 00:16:32,190
That's the frequency.

218
00:16:32,190 --> 00:16:35,310
That's the frequency of the
photons that we've been talking

219
00:16:35,310 --> 00:16:41,800
about, and the intensity, again,
for these convention preference

220
00:16:41,800 --> 00:16:45,660
will put 2e0 times
cosine omega t.

221
00:16:45,660 --> 00:16:56,040
So some people say ep, which
is the peak e field is 2e0.

222
00:16:56,040 --> 00:16:58,230
That's our convention.

223
00:17:01,020 --> 00:17:06,220
Well, when you have an
electric field like that,

224
00:17:06,220 --> 00:17:11,910
the electric field is
minus the gradient of phi,

225
00:17:11,910 --> 00:17:16,500
and therefore, we
can take phi to be

226
00:17:16,500 --> 00:17:22,089
equal to minus the electric
field as a function of time,

227
00:17:22,089 --> 00:17:23,520
times z.

228
00:17:28,650 --> 00:17:33,540
So if you take the gradient
minus the gradient,

229
00:17:33,540 --> 00:17:37,500
you get the electric field.

230
00:17:37,500 --> 00:17:41,910
And therefore, we have
to plug it all in here.

231
00:17:41,910 --> 00:17:53,516
So this is plus e, e
of t, z, and this is e.

232
00:17:53,516 --> 00:18:00,570
e of t has been given
2e0 cosine omega t.

233
00:18:00,570 --> 00:18:05,910
And z, we can write
this r cosine theta.

234
00:18:05,910 --> 00:18:08,730
In the usual description,
we have the z-axis

235
00:18:08,730 --> 00:18:12,990
here, r, theta, and
the electric field

236
00:18:12,990 --> 00:18:14,830
is going in the z direction.

237
00:18:14,830 --> 00:18:19,480
So z's are cosine theta.

238
00:18:19,480 --> 00:18:23,760
So I think I have all
my constants there.

239
00:18:23,760 --> 00:18:33,580
So let's put it 2 e, e0,
r cos theta cos omega t,

240
00:18:33,580 --> 00:18:36,670
and this is our
perturbation that we said

241
00:18:36,670 --> 00:18:41,980
it's 2h prime cosine omega t.

242
00:18:41,980 --> 00:18:45,400
That's harmonic perturbations
were defined that way.

243
00:18:45,400 --> 00:18:51,010
So we read the value
of h prime as this one.

244
00:18:53,760 --> 00:18:59,080
e e0 r cosine omega t.

245
00:18:59,080 --> 00:19:01,060
No, r cosine theta.

246
00:19:06,890 --> 00:19:07,950
OK.

247
00:19:07,950 --> 00:19:11,440
We have our h prime.

248
00:19:11,440 --> 00:19:14,490
So we have the
conditions of validity.

249
00:19:14,490 --> 00:19:17,580
We have our h prime.

250
00:19:17,580 --> 00:19:23,650
Two more things so that
we can really get started.

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00:19:23,650 --> 00:19:26,110
What is our initial state?

252
00:19:26,110 --> 00:19:28,290
The initial state
is the wave function

253
00:19:28,290 --> 00:19:34,590
of the electron, which is 1
over square root of pi a0 cubed

254
00:19:34,590 --> 00:19:37,300
e to the minus r over a0.

255
00:19:37,300 --> 00:19:41,070
That's our initial state.

256
00:19:41,070 --> 00:19:45,690
What is our final state,
our momentum eigenstates

257
00:19:45,690 --> 00:19:47,050
of the electron?

258
00:19:47,050 --> 00:19:58,410
So you could call them psi,
or u sub k of the electron.

259
00:19:58,410 --> 00:20:04,230
And it would be 1
over l to the 3/2, e

260
00:20:04,230 --> 00:20:07,470
to the ik of the
electron times x.

261
00:20:10,670 --> 00:20:15,390
These are our initial
and final states.

262
00:20:15,390 --> 00:20:20,870
Remember, the plane waves had
to be normalized in a box.

263
00:20:20,870 --> 00:20:23,060
So the box is back.

264
00:20:23,060 --> 00:20:27,440
u is the wave function of
the plane wave electron.

265
00:20:27,440 --> 00:20:30,650
And we could use a plane
wave, because the electron

266
00:20:30,650 --> 00:20:33,080
is energetic enough.

267
00:20:33,080 --> 00:20:35,480
And it has the box thing.

268
00:20:35,480 --> 00:20:37,790
This is perfectly
well normalized.

269
00:20:37,790 --> 00:20:40,460
If you square it,
the exponential

270
00:20:40,460 --> 00:20:44,330
vanishes, because
it's a pure phase,

271
00:20:44,330 --> 00:20:47,090
and you get 1 over l cubed.

272
00:20:47,090 --> 00:20:48,980
The box has volume l cubed.

273
00:20:48,980 --> 00:20:51,080
It's perfectly normalized.

274
00:20:51,080 --> 00:20:55,570
This is all ready now
for our computation.