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PROFESSOR: For a
couple of lectures,

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we're going to be discussing
now the hydrogen atom.

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So you've seen the
hydrogen atom before.

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You've seen it in 804.

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You've seen it in 805.

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Why do we see it again in 806?

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Well, the hydrogen atom is a
fairly sophisticated example.

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It's the harmonic oscillator
of atomic physics.

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If some of you are going
to be doing atomic physics,

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eventually, you have to
understand the atom perfectly

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well, the energy levels, the
degeneracies, why are they,

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how do you separate
the levels, how

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do you split the degeneracies.

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It's a fantastically
good example

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of perturbation theory,
our degenerate perturbation

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theory, non-degenerate
perturbation

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theory in some cases.

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But overall, it's a very
important physical system

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that affects many
areas of physics,

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astrophysics with
hyper fine splitting,

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all kind of processes in
atomic physics and lasers,

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and things like that.

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So it's something we really
want to understand well.

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And this time in 806, it's
the right time for you

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to understand the fine
structure of this atom.

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Bits and pieces of this
have been done before.

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But now, we're going
to do it in detail.

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So let's start with hydrogen
atom fine structure.

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Hydrogen atom fine structure.

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That is the main topic
for a couple of lectures.

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So in terms of
Hamiltonians, we've

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been talking of an unperturbed
Hamiltonian, an H0.

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This Hamiltonian is known.

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It's the momentum squared over
2m minus e squared over r.

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Here, this is the momentum
in three dimensions.

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It's a three-dimensional
system, of course.

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This m is the mass of the
electron times the mass

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of the proton, divided by
the mass of the electron

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plus the mass of the proton.

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And that's the reduced
mass of the system,

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but it's approximately equal
to the mass of the electron.

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So we will--
whenever we write m,

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it will be the mass
of the electron.

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Now, if you happen to have a
nucleus with z protons, what

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you have to do is replace
this e squared by z e squared.

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And this is because
the factor of e

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squared comes from the
product of the charge

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of the electron times the
charge of the nucleus.

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So if the charge of the
nucleus becomes z times bigger,

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this e squared is
replaced by this quantity.

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What important length
scales exist here

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is the bore radius is the
most important length scale,

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and it's constructed with
the quantities that appear

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in this Hamiltonian and H bar.

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So in particular, doesn't
involve the speed of light.

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There's nothing in this
system that is relativistic.

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So this quantity is h
squared over m e squared.

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And that this has
units of length

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is something that you
should be able to derive

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from here in a few instance.

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Please make sure you
know how to derive it

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without having to count mass,
length, time units here.

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This is something you should
be able to do very quickly.

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One piece of
intuition, of course,

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is that the e appears in the
denominator, which is intuition

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that as the strength
of electromagnetism

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is made smaller and smaller
by letting e goes to 0,

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the orbit of the electron
would be bigger and bigger.

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So this is about 53
picometers, where a picometer

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is 10 to the minus 12 meters.

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A nice unit.

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It's half an angstrom,
roughly, but picometers

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is the right unit for
people that think of atoms.

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Then the energy levels are given
by minus e squared over 2a0 1

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over n squared.

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This is a beautiful formula
in terms of quantities

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that you understand.

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First, the energy just depends
on n, and this 1 over n

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squared.

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And n is called the
principle quantum

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number, principle
quantum number,

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and it goes from
1 up to infinity.

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This has units of
energy, and that's

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why this is nice,
because e squared over

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r is the units of potential
energy in electromagnetism.

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So when you see this,
you know that this

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has the right units of energy.

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And for the ground
state of hydrogen,

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n equal 1, that energy
is about minus 13.6 eV.

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Now, in terms of thinking
ahead for perturbation theory,

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some of the
perturbation theory here

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will reflect the fact that
electromagnetism is weak,

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and the other fact will reflect
that the electron is actually

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moving with slow velocities.

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So let's see these two things.

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First, we can write
the energy, or in terms

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of defined structure constant.

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So alpha, which is defined
structure constant.

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This e squared over [? hc, ?]
the units we're working with.

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And it's about 137.

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So whenever you see
e squared over a0,

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which is a unit of energy,
you should realize that--

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if you wanted to change
that, for example,

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for the case that you have
a nucleus with z protons,

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you cannot just do e
squared going to ze squared,

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because in fact, a0,
itself, has an e.

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So don't think that the
energy depends like e squared.

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It actually depends more
like e to the fourth times

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other constants.

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So let's do that.

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So we have here m e
to the fourth over h

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squared, using the value of a0.

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And then we can use e to the
fourth from this equation.

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So it would be alpha m
alpha squared h bar squared

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c squared over h bar squared.

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And it's equal to alpha
squared mc squared.

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A nice way of thinking
of the ground state

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energy of the hydrogen
as a small quantity alpha

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squared times the rest
energy of the electron.

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After all, the system begins
just with an electron,

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and it has some rest
energy, and there

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should be a natural
way of expressing this.

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Of course, you start
seeing this c squared here.

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But the c squared
really is nowhere there.

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We've put it for convenience.

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But it allows us
to think of scales,

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and in particular, the
rest mass of the electron

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is alpha is much bigger,
because alpha squared is about 1

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over 19,000.

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1/137 squared is about that.

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So the energy, en, can
be written in a way

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that we will use often as
minus 1/2 alpha squared

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mc squared 1 over n squared.

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All right.

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And another observation, the
momentum of the electron, we

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could estimate it
to be h bar over a0.

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So this is me
squared over h bar.

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And we use the e
squared for what it is.

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So this is alpha.

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Well, I'll put m,
alpha hc over h bar.

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So this is alpha mc.

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It's a nice thing, alpha mc.

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It says that the momentum
that you could construct

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relativistically, the
mass of the electron

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times the velocity, you
still have to divide by 137.

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But it's clearer if you write
it like m alpha c, and then

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momentum, which is mass
times velocity, at least

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for slow velocities.

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Now is mass times alpha times c.

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So the approximate velocity
that we estimate on the electron

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is c over 137.

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So that's very nice.

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It's kind of non-relativistic.

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OK.

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So this is the basic things.

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But the most important
stuff is really

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getting the table of
how the atom looks.

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So this is still review.

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Half of this lecture,
in a sense, is review.

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We put here, not in scale,
n equals 1, n equals 2,

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n equals 3, and n equals 4.

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In reality, the hydrogen
atom, if this is 0 energy,

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and this is the ground state,
the ground state that's here,

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the second excited state
is here, the third is here,

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the fourth is here, fifth,
they all accumulate here.

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But that I've
written it this way.

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Now, we'll have l
equals 0 here, l

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equals 1, l equals
2, l equals 3.

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And there is another
notation for this state.

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I don't know why, but there is.

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A capital l that
is a function of l.

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So capital l of l
equals 0 is called s.

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So these states are
called s states.

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Capital l of l
equals 1 is called p.

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So states with l equal
1 will be called p.

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Then d and f.

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These are names.

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We'll use those names.

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They're OK.

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And here are the states.

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For n equal 1, there's
just one state here.

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And here there's
one state as well.

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But for n equal 2, there
is a state with l equal 0,

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and there is a state
with l equal 1.

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For n equal 3,
there's 0, 1, and 2.

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For n equal 4, there's
0, 1, 2, and 3.

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So these are the states in the
spectrum of hydrogen, something

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that we should know very well.

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And what are the patterns here.

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They're degeneracies.

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Why?

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Because when we talk l
equal 2, for example,

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that means a multiplet
of angular momentum

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with total angular momentum 2.

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And that comes with as
azimuthal angular momentum

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that goes from minus 2 m to
plus 2, that is five states.

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So in principle, each bar here
is five states, five states.

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l equal 1 has m equal
1, 0, and minus 1.

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So three states, three
states, three states.

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And here, really zero states.

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So there's lots of degeneracy.

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That's why we spend
lots of time studying

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degenerate perturbation
theory, because this gigantic

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degeneracies here.

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So to determine the
origin or the way

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we parametrize the
degeneracy, there's

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just one formula
that says it all.

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And that formula
is degeneracies.

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It's the formula
that explains it all,

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and it says that the
principle quantum

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number is equal to capital
n plus l plus 1, where

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n is the degree of a polynomial
in the wave function.

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Wave function.

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l is the orbital
angular momentum.

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So this says, for example, that
this degree of the polynomial n

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can be 0, 1, 2.

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It cannot be negative.

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l can also be 0, 1, and go on.

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But here, for example,
you see the main rule

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that for a fixed n, you
can have l equals 0 1, 2,

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up to n minus 1,
because by the time

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you take l equals to n
minus 1, this whole thing

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is equal to little n,
and capital n is 0.

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And that's as far as you can go.

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So the angular momentum cannot
exceed the principle quantum

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number minus 1.

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That is what we see
here for n equal 3.

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You can have up to l equal 2.

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For n equal 4, you can
have up to l equal 3.

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So this is kind of well known.

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Now, for each l,
each l, you have

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m from minus l all
the way to plus l.

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And that's 2l plus 1 values.

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So the degeneracy at n, at
the principle quantum number

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equal n is the sum from l
equals 0 up to n minus 1.

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Those are all the
possible values

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of l of the number of
states in each l multiplet.

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And you've done this one
before, and that actually

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turns out to be equal
to n squared, which

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says there should be
four states at n equal 2,

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indeed one state for S equals
0, three states here is 4, 1, 3,

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5.

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It's that property that the
sum of consecutive odd numbers

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comes up equal to the
square, a perfect square.

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It's a very nice
thing geometrically.

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So the wave function--

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we'll write it here-- the
wave function psi mlm.

248
00:18:06,050 --> 00:18:11,790
These are our quantum numbers
and principal quantum number.

249
00:18:11,790 --> 00:18:15,560
Once you know n, you know
the sum of capital N plus l.

250
00:18:15,560 --> 00:18:19,850
But l is a little
more physical than n.

251
00:18:19,850 --> 00:18:24,290
So we'll use l, and
once you know l,

252
00:18:24,290 --> 00:18:27,740
you still need to know where a
given state, which value of m

253
00:18:27,740 --> 00:18:28,380
you have.

254
00:18:28,380 --> 00:18:31,400
So those are your
three quantum numbers,

255
00:18:31,400 --> 00:18:39,260
and this wave function goes
like a constant times r over a0

256
00:18:39,260 --> 00:18:47,690
to the l times that polynomial
we spoke about, 1 plus beta r

257
00:18:47,690 --> 00:18:58,590
to the r over a0 all the way
up to a number times r over a0

258
00:18:58,590 --> 00:19:12,370
to the capital N times e to
the minus r over n a0 times

259
00:19:12,370 --> 00:19:16,370
ylm of theta and phi.

260
00:19:20,670 --> 00:19:21,975
So that's your wave function.

261
00:19:24,780 --> 00:19:27,090
Those numbers I
have not determined.

262
00:19:27,090 --> 00:19:29,550
I have not determined
the normalization.

263
00:19:29,550 --> 00:19:32,940
But if you're looking
at the wave function,

264
00:19:32,940 --> 00:19:36,450
the easiest place to see
the principal quantum number

265
00:19:36,450 --> 00:19:37,200
is here.

266
00:19:41,020 --> 00:19:45,730
The easiest place to see
the orbital angular momentum

267
00:19:45,730 --> 00:19:46,710
is here.

268
00:19:46,710 --> 00:19:48,400
You identify it from here.

269
00:19:48,400 --> 00:19:50,590
It must be multiplying
a polynomial that

270
00:19:50,590 --> 00:19:54,790
begins with 1, because the
leading power in the solution

271
00:19:54,790 --> 00:20:01,330
must be r to the l,
and has degree n here.

272
00:20:01,330 --> 00:20:06,220
And the m quantum number you see
it from the spherical harmonic.

273
00:20:10,010 --> 00:20:13,340
So in particular, here
you have a state with l

274
00:20:13,340 --> 00:20:15,500
equals 0, n equal 1.

275
00:20:15,500 --> 00:20:19,100
So this must have n equals 0.

276
00:20:19,100 --> 00:20:23,570
This must have n
equals 1, because you

277
00:20:23,570 --> 00:20:29,310
must get to 2 with a capital N
and l equals 0 and a 1 there.

278
00:20:29,310 --> 00:20:30,950
So this is n equal 1.

279
00:20:30,950 --> 00:20:34,430
This would be n
equal 2, n equal 3.

280
00:20:34,430 --> 00:20:40,685
Similarly here, n equals
0, n equals 1, n equals 2,

281
00:20:40,685 --> 00:20:46,970
n equals 0, n equals
1, n equals 0 here.

282
00:20:46,970 --> 00:20:51,650
The n's decrease in this
direction as l increases,

283
00:20:51,650 --> 00:20:56,930
keeping the sum of capital
N and l constant and equal

284
00:20:56,930 --> 00:20:58,790
to little m minus 1.

285
00:21:02,178 --> 00:21:06,120
Interestingly, this
makes sense as well.

286
00:21:06,120 --> 00:21:12,300
If you may remember, the
quantum number N, capital N,

287
00:21:12,300 --> 00:21:14,940
tells you the number of
nodes of the wave function,

288
00:21:14,940 --> 00:21:20,250
because a polynomial of
degree n can have n zeros,

289
00:21:20,250 --> 00:21:22,260
and therefore,
this wave function

290
00:21:22,260 --> 00:21:25,560
has no nodes, one node,
two nodes, three nodes.

291
00:21:25,560 --> 00:21:28,170
The number of nodes increase.

292
00:21:28,170 --> 00:21:30,990
That's another thing that
wave functions should have.

293
00:21:33,720 --> 00:21:38,220
So a couple more
comments and this.

294
00:21:38,220 --> 00:21:44,620
We'll write-- I'll write the
ground state wave function.

295
00:21:44,620 --> 00:21:45,540
I'll put it here.

296
00:21:45,540 --> 00:21:47,540
I'm cluttering
things a little bit.

297
00:21:47,540 --> 00:21:49,670
But the ground
state wave function

298
00:21:49,670 --> 00:21:55,190
is one that we can have
and normalize easily.

299
00:21:55,190 --> 00:22:03,150
Square root of pi over a0
cubed e to the minus r over a0.

300
00:22:03,150 --> 00:22:04,890
That's the ground
state wave function.

301
00:22:07,445 --> 00:22:07,945
OK.

302
00:22:10,450 --> 00:22:13,390
Comments on this thing.

303
00:22:13,390 --> 00:22:19,710
First, there is a very
large degeneracy here.

304
00:22:19,710 --> 00:22:23,700
And it has a nice
interpretation.

305
00:22:23,700 --> 00:22:28,200
This correspond to states with
different values of the angular

306
00:22:28,200 --> 00:22:30,330
momentum.

307
00:22:30,330 --> 00:22:35,520
Semi-classically, if you
were doing Keplerian emotion,

308
00:22:35,520 --> 00:22:39,480
and think semi-classically
of the electron orbit,

309
00:22:39,480 --> 00:22:44,370
all the orbits here
corresponds of orbits

310
00:22:44,370 --> 00:22:49,780
of the electron with the
same semi-major axis,

311
00:22:49,780 --> 00:22:53,370
but with different eccentricity.

312
00:22:53,370 --> 00:22:59,130
So as it turns out,
the orbit with least

313
00:22:59,130 --> 00:23:03,840
l is the most eccentric
of all orbits.

314
00:23:03,840 --> 00:23:06,210
And as you go in this
direction, the orbit

315
00:23:06,210 --> 00:23:08,760
becomes more and more circular.

316
00:23:08,760 --> 00:23:12,420
So if you want an
immediate intuition,

317
00:23:12,420 --> 00:23:16,590
as to why are all these
states distinguishable,

318
00:23:16,590 --> 00:23:20,800
it's because they are orbits
with different eccentricity.

319
00:23:20,800 --> 00:23:23,790
There are ellipses with
different eccentricity,

320
00:23:23,790 --> 00:23:26,200
all with the same
semi-major axis.

321
00:23:26,200 --> 00:23:28,380
So it could be a
circle like that,

322
00:23:28,380 --> 00:23:32,920
or it could be just
an ellipse like that.

323
00:23:32,920 --> 00:23:34,015
So those are it.

324
00:23:36,550 --> 00:23:41,290
Then, most important
complication we've

325
00:23:41,290 --> 00:23:46,480
ignored here, it's the
spin of the electron.

326
00:23:46,480 --> 00:23:47,800
Electron has spin.

327
00:23:47,800 --> 00:23:50,650
So we know the spin
of the electron

328
00:23:50,650 --> 00:23:53,680
represents a degree
of freedom described

329
00:23:53,680 --> 00:23:56,990
by a two-dimensional
vector space.

330
00:23:56,990 --> 00:24:00,730
So there's two states
in every one of this.

331
00:24:00,730 --> 00:24:04,730
So we will have
to consider that.

332
00:24:04,730 --> 00:24:10,390
And if I want to put the
number of states on each one,

333
00:24:10,390 --> 00:24:12,340
here there was
one state we said,

334
00:24:12,340 --> 00:24:14,485
but now we know there
are really two states.

335
00:24:17,710 --> 00:24:21,310
And here, there's two
states is l equals 0, two

336
00:24:21,310 --> 00:24:24,220
states, two states.

337
00:24:24,220 --> 00:24:25,900
Here is l equal 1.

338
00:24:25,900 --> 00:24:30,440
That's three configurations
of orbital angular momentum.

339
00:24:30,440 --> 00:24:35,590
But the electron has, again,
up or down possibilities.

340
00:24:35,590 --> 00:24:41,380
So these are six states,
six states, six states.

341
00:24:41,380 --> 00:24:46,510
l equals 2 is five states,
but with the electron,

342
00:24:46,510 --> 00:24:49,600
there is 10 states
and 10 states here.

343
00:24:53,220 --> 00:24:59,400
In this one, l equals 3
is 7 states, 2l plus 1,

344
00:24:59,400 --> 00:25:03,490
but with the electron degrees
of freedom is 14 states.

345
00:25:03,490 --> 00:25:05,815
So these are the right
number of states.

346
00:25:08,370 --> 00:25:09,960
When we'll do the
fine structure,

347
00:25:09,960 --> 00:25:14,430
we'll have corrections due
to the spin and corrections

348
00:25:14,430 --> 00:25:17,100
due to relativity.

349
00:25:17,100 --> 00:25:19,710
Both things will
make our corrections.

350
00:25:19,710 --> 00:25:23,910
And by the time we
do that, we may also

351
00:25:23,910 --> 00:25:27,300
want to explore the
atom by putting it

352
00:25:27,300 --> 00:25:28,680
in an electric field.

353
00:25:28,680 --> 00:25:30,270
That's the stark effect.

354
00:25:30,270 --> 00:25:33,330
It will change energy
levels, and you learn more

355
00:25:33,330 --> 00:25:36,240
about the energy
levels of a system.

356
00:25:36,240 --> 00:25:38,920
You can put it in
a magnetic field,

357
00:25:38,920 --> 00:25:41,130
and that's a Zeeman effect.

358
00:25:41,130 --> 00:25:44,580
And the magnetic field can
be weak, or it can be strong,

359
00:25:44,580 --> 00:25:46,410
and it's a different
approximation,

360
00:25:46,410 --> 00:25:51,080
and there's several things
we have to do with it.