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00:00:00,499 --> 00:00:02,729
PROFESSOR: We're talking
about angular momentum.

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00:00:02,729 --> 00:00:08,766
We've motivated angular
momentum as a set of operators

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00:00:08,766 --> 00:00:12,690
that provided observables,
things we can measure.

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00:00:12,690 --> 00:00:15,250
Therefore, they are important.

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00:00:15,250 --> 00:00:18,980
But they're particularly
important for systems in which

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00:00:18,980 --> 00:00:21,440
you have central potentials.

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00:00:21,440 --> 00:00:27,230
Potentials that depend
just on the magnitude

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of the radial variable.

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A v of r that depends just on
the magnitude of the vector

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00:00:34,160 --> 00:00:40,340
r relevant to cases where you
have two bodies interacting

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00:00:40,340 --> 00:00:42,740
through a potential
that just depends

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on the distance
between the particles.

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So what did we develop?

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Well, we discussed the
definition of the angular

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momentum operator.

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You saw they were permission.

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We found that they satisfy
a series of commutators

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in which lx with
ly gave ih bar lz,

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and cyclical versions
of that equation, which

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ensure that actually you
can measure simultaneously

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the three components
of angular momentum.

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You can measure,
in fact, just one.

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Happily we found there
was another object

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we could measure, which was
the square of the total angular

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momentum.

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Now, you should
understand this symbol.

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It's not a vector.

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It is just a single
operator. l squared

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is, by definition, lx
times lx, plus ly times ly,

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plus lz times lz.

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This is this operator.

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And we showed that any
component of angular momentum,

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be it lx, ly, or lz,
commutes with l squared.

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Given that they commute,
it's a general theorem

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that two permission
operators that commute,

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you can find simultaneous
eigenstates of those two

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operators.

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And therefore, we set up for the
search of those wave functions

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that are simultaneous
eigenstates of one of the three

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components of angular momentum.

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Everybody chooses
lz and l squared.

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lz being proportional to
angular momentum has an h bar m.

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We figured out by looking at
this differential equation

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that if we wanted single
valued wave functions--

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wave functions would
be the same at phi,

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and at phi plus 2 pi,
which is the same point.

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You must choose m
to be an integer.

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For the l squared
operator we also

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explained that the
eigenvalue of this operator

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should be positive.

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That is achieved when
l, whatever it is,

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is greater than 0,
greater or equal than 0.

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And the discussion that led
to the quantization of l

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was a little longer,
took a bit more work.

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Happily we have this
operator, and operator

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we can diagonalize, or we
can find eigenstates for it.

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Because the Laplacian,
as was written

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in the previous
lecture, Laplacian

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entering in the
Schrodinger equation

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has a radial part
and an angular part,

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where you have dd
thetas, and sine thetas,

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and the second defies square.

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All these things were
taken care of by l squared.

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And that's very useful.

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Well, the differential
equation for l

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squared-- this can be though as
a differential equation-- ended

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up being of this form,
which is of an equation

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for the so-called Associate
Legendre functions.

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For the case of m equals 0
it simplifies very much so

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that it becomes an equation
for what were eventually

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called Legenre polynomials.

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We looked at that differential
equation with m equals 0.

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We called it pl 0.

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So we don't write the zeros.

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Everybody writes pl
for those polynomials.

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And looking at the
differential equation one

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finds that they have
divergences at theta

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equals 0, and a theta
equal pi, north and south

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pole of this spherical
coordinate system.

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There aren't divergences unless
these differential equations

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has a polynomial solution that
this is serious the recursion

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relations terminate.

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And that gave for us
the quantization of l.

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And that's where we stopped.

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These are the
Legendre polynomials.

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Solve this equation
for m equals 0.

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Are there any questions?

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Anything about the
definitions or?

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Yes?

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AUDIENCE: Why do we care about
simultaneous eigenstates?

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PROFESSOR: Well, the
question is why do we care

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about simultaneous eigenstates.

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The answer is that
if you have a system

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you want to figure out what are
the properties of the states.

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And you could begin by saying
the only thing I can know

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about this state is its energy.

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OK, well, I know
the energy at least.

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But maybe thinking harder
you can figure out, oh, you

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can also know the momentum.

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That's progress.

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If you can also know
the angular momentum

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you learn more about the
physics of this state.

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So in general, you will be
led in any physical problem

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to look for the maximal
set of commuting operators.

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The most number of operators
that you could possibly

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measure.

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You know you have success
at the very least,

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if you can uniquely characterize
that states of the system

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by observables.

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Let's assume you have
a particle in a circle.

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Remember that the free
particle in a circle

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has degenerate
energy eigenstates.

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So you have two
energy eigenstates

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for every allowed energy,
except for 0 energy,

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but two energy eigenstates.

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And you would be baffled.

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You'd say, why do I have two?

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There must be some difference
between these two states.

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If there are two states,
there must be some property

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that distinguishes them.

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If there is no property
that distinguishes them,

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they should be the same state.

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So you're left to search
for another thing.

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And in that case the
answer was simple.

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It was the momentum.

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You have a particle with some
momentum in one direction,

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or in the reverse direction.

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So in general, it's a
most important question

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to try to enlarge the set
of commuting observables.

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Leading finally to
what is initially

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called a complete set of
commuting observables.

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So what do we have to do today?

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We want to complete
this analysis.

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We'll work back
to this equation.

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And then work back to
the Schrodinger equation

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to finally obtain the
relevant differential

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equation we have to
solve if you have

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a spherical symmetric potential.

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So the equation will be
there in a little while.

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Then we'll look at
the hydrogen atom.

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We'll begin the hydrogen
atom and this task why?

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Having a proton
and an electron we

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can reduce this system to
as if we had one particle

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in a central potential.

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So that will be also very
important physically.

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So let's move ahead.

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And here there is a simple
observation that one can make.

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Is that the differential
equation for p l m depends on m

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squared.

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We expect to need values of m
that are positive and negative.

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You have wave functions
here, of this form.

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The complex conjugate
ones should be

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thought as having m negative.

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So we expect positive and
negative m's to be allowed.

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So how did people
figure this out?

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They, in fact, figured out that
if you have these polynomials

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you can create automatically
the solutions for this equation.

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There's a rule, a simple
rule that leads to solutions.

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You put p l m of x is
equal to 1 minus x squared,

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to the absolute
value of m over 2.

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So there are square
roots here, possibly.

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An absolute value of m
means that this is always

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in the numerator, whether
m is positive or negative.

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And d, dx acting exactly
absolute value of m times on p

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l x.

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The fact is that this definition
solves the differential

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equation star.

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This takes a little
work to check.

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I will not check it, nor
the notes will check it.

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It's probably
something you can find

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00:11:27,850 --> 00:11:30,690
the calculation in some books.

172
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But it's not all that important.

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The important thing to
note here is the following.

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That this provides solutions.

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00:11:40,310 --> 00:11:47,956
Since this polynomial is like
x to the l plus x to the l

176
00:11:47,956 --> 00:11:53,510
minus 2 plus
coefficients like this.

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You can think that most
m equal l derivatives--

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if you take more than l
derivatives you get 0.

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And there's no great
honor in finding

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00:12:05,680 --> 00:12:08,010
zero solution of this equation.

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00:12:08,010 --> 00:12:10,270
These are no solutions.

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So this produces solutions
for an, absolute value of m,

183
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less than l.

184
00:12:20,390 --> 00:12:30,242
So produces solutions
for absolute value

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of m less or equal to l.

186
00:12:35,080 --> 00:12:42,516
And therefore m in
between l and minus l.

187
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But that's not all that happens.

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There's a little more that
takes mathematicians some skill

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00:12:56,090 --> 00:12:56,940
to do.

190
00:12:56,940 --> 00:13:00,720
It's to show that there
are no more solutions.

191
00:13:00,720 --> 00:13:03,360
You might seem that
you were very clever

192
00:13:03,360 --> 00:13:07,840
and you found some
solutions, but it's a theorem

193
00:13:07,840 --> 00:13:12,460
that there are no
more solutions.

194
00:13:12,460 --> 00:13:22,836
No additional regular solutions.

195
00:13:22,836 --> 00:13:24,405
I mean solutions
that don't diverge.

196
00:13:27,100 --> 00:13:30,710
So this is very important.

197
00:13:30,710 --> 00:13:36,030
It shows that there is one
more constraint on your quantum

198
00:13:36,030 --> 00:13:38,800
numbers.

199
00:13:38,800 --> 00:13:43,000
This formula you may
forget, but you should never

200
00:13:43,000 --> 00:13:45,246
forget this one.

201
00:13:45,246 --> 00:13:52,830
This one says that if you
choose some l which corresponds

202
00:13:52,830 --> 00:13:57,600
to choosing the magnitude
of the angular momentum,

203
00:13:57,600 --> 00:14:00,880
l is the eigenvalue
that tells you

204
00:14:00,880 --> 00:14:03,940
about the magnitude of
the angular momentum.

205
00:14:03,940 --> 00:14:07,870
You will have several
possibilities for m.

206
00:14:07,870 --> 00:14:13,140
There will be several
states that have the same l,

207
00:14:13,140 --> 00:14:15,910
but m different.

208
00:14:15,910 --> 00:14:19,960
So for example you'll
have l equal 0, in which

209
00:14:19,960 --> 00:14:23,220
case m must be equal to 0.

210
00:14:23,220 --> 00:14:27,841
But if you choose
state with l equals 1,

211
00:14:27,841 --> 00:14:31,040
or eigenfunctions
with l equal 1,

212
00:14:31,040 --> 00:14:37,150
there is the possibility of
having m equals minus 1, 0,

213
00:14:37,150 --> 00:14:38,660
or 1.

214
00:14:38,660 --> 00:14:43,810
So are three waves
functions in that case.

215
00:14:43,810 --> 00:14:51,700
Psi 1, minus 1, psi
1, 0, and psi 1, 1.

216
00:14:51,700 --> 00:14:56,230
So in general when we
choose a general l,

217
00:14:56,230 --> 00:14:58,420
if you choose an
arbitrary l, then

218
00:14:58,420 --> 00:15:05,780
m goes from minus l, minus l
plus 1 all the way up to l.

219
00:15:05,780 --> 00:15:15,858
These are all the values
which are 2l plus 1 values.

220
00:15:15,858 --> 00:15:19,140
2l and the 0 value in between.

221
00:15:19,140 --> 00:15:20,430
So it's 2l plus 1 values.

222
00:15:26,500 --> 00:15:30,860
The quantization in
some sense is done now.

223
00:15:30,860 --> 00:15:35,080
And let me recap about
these functions now.

224
00:15:35,080 --> 00:15:40,270
We mentioned up there that
the y l m's are the objects.

225
00:15:40,270 --> 00:15:42,380
The spherical
harmonicas are going

226
00:15:42,380 --> 00:15:44,786
to be those wave functions.

227
00:15:44,786 --> 00:15:50,230
And they have a normalization,
n l m, an exponention,

228
00:15:50,230 --> 00:15:51,090
and all that.

229
00:15:51,090 --> 00:15:54,600
So let me write,
just for the record,

230
00:15:54,600 --> 00:16:00,440
what a y l m looks like
with all the constants.

231
00:16:00,440 --> 00:16:06,230
Well, the normalization
constant is complicated.

232
00:16:06,230 --> 00:16:10,925
And it's kind of a thing you
can never remember by heart.

233
00:16:10,925 --> 00:16:12,750
It would be pointless.

234
00:16:16,330 --> 00:16:16,830
OK.

235
00:16:16,830 --> 00:16:17,930
All of that.

236
00:16:17,930 --> 00:16:22,806
Then a minus 1 to
the m seems useful.

237
00:16:22,806 --> 00:16:32,720
e to the i m phi, p
l m of cosine theta.

238
00:16:32,720 --> 00:16:41,120
And this is all valid for
0 less 0 m positive m.

239
00:16:41,120 --> 00:16:44,950
When you have negative m you
must do a little variation

240
00:16:44,950 --> 00:16:55,140
for m less than 0 y l m of
theta and phi is minus 1

241
00:16:55,140 --> 00:17:07,359
to the m y l minus m of theta
and phi complex conjugated.

242
00:17:07,359 --> 00:17:12,569
Well, if m is negative,
minus m is positive.

243
00:17:12,569 --> 00:17:14,150
So you know what that is.

244
00:17:14,150 --> 00:17:16,750
So you could plug
this whole mess here.

245
00:17:16,750 --> 00:17:18,504
I don't advise it.

246
00:17:18,504 --> 00:17:20,750
It's just for the record.

247
00:17:20,750 --> 00:17:24,260
These polynomials
are complicated,

248
00:17:24,260 --> 00:17:27,369
but they are normalized nicely.

249
00:17:27,369 --> 00:17:30,070
And we just need to
understand what it

250
00:17:30,070 --> 00:17:32,580
means to be normalized nicely.

251
00:17:32,580 --> 00:17:36,370
That is important for us.

252
00:17:36,370 --> 00:17:42,040
The specific forms of these
polynomials we can find them.

253
00:17:45,370 --> 00:17:50,680
The only one I really remember
is that y 0 0 is a constant.

254
00:17:50,680 --> 00:17:53,490
It's 1 over 4 pi.

255
00:17:53,490 --> 00:17:55,230
That's simple enough.

256
00:17:55,230 --> 00:17:58,230
No dependents. l
equals 0, m equals 0.

257
00:18:01,541 --> 00:18:03,600
Here is another one.

258
00:18:03,600 --> 00:18:11,954
y1 plus minus 1 is minus plus
square root of 3 over 8 pi

259
00:18:11,954 --> 00:18:20,690
e to the plus minus
i phi sine theta.

260
00:18:20,690 --> 00:18:26,170
And the last one,
so we're giving

261
00:18:26,170 --> 00:18:30,300
all the spherical
harmonics with l equals 1.

262
00:18:35,670 --> 00:18:39,230
So with l equals 1
remember we mentioned

263
00:18:39,230 --> 00:18:43,670
that you would have
three values of m.

264
00:18:43,670 --> 00:18:46,760
Here they are.

265
00:18:46,760 --> 00:18:50,496
Plus or minus 1 and 0.