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PROFESSOR: We have the
hydrogen atom Hamiltonian.

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Hamiltonian.

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And that was given by
the kinetic operator

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for the proton plus
the kinetic operator

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for the electron plus
the potential, which

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was a function of the
distance between the proton

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and the electron.

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And what we achieved last
time was the introduction

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of two new pairs of
canonical variables.

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We had the electron
position momentum, that's

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a pair of canonical variables.

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The proton position
and momentum,

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that's another pair of
canonical variables.

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They commute each
pair, the two operators

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commute to give IH bar, but
the two pairs are independent.

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So we search for another
two pairs of variables,

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and we found another two pairs.

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One was the P and X associated
with the center of mass motion,

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and then we had the small
p and small x associated

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with the relative function.

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And these four variables were
a function of the original four

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variables, the X and
P of the electron

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and the x and p of the proton.

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So we define these two pairs,
and they were canonical pairs.

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This x with this p gave
IH bar, this x with this p

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gave IH bar, these p's and x's
commute with any combination

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of p's and x's over there.

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But not only was
that pretty good,

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it simplified the Hamiltonian.

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So at the end of the day,
we had a Hamiltonian,

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which was as if the center of
mass moves like a free particle

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plus a kinetic energy
for the relative motion,

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with a mass called
the renews mass,

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and the potential for
the relative position.

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And here, the mass, capital M,
was the sum of the two masses.

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And the relative
mass was the product

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of the masses over
the sum, which

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has the property that
if one of the two masses

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is much bigger
than the other, it

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gives you a mass mu proportional
roughly equal to the lower

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mass.

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So this is the hydrogen
atom reformulated.

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And now, we want to write
the Schrodinger equation

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and just see effectively how the
central potential formulation

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of the relative motion arises.

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Although, it starts to be
a little somewhat clearer,

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I think, that that's
going to happen.

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Another thing to
notice of course,

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is that if you're already
thinking of a Schrodinger

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equation, in which
you will think

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of the momenta as the
derivative operators,

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the center of mass
momentum should

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be thought as the derivative
operator with respect

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to the center of mass position.

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So think of the center of
mass as three coordinates,

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and you differentiate
with respect to them.

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Similarly, for the
relative momentum,

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we'll think of it as a
gradient with respect

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to the relative
position, because that's

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the canonical coordinate
that goes along with it.

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That's how we should think of
this operator as derivative.

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And before, we didn't have
to put those subscripts

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in [INAUDIBLE],, because we
always had just one coordinate

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to work with.

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But now, you have
two coordinates.

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What you've learned here
in doing this analysis

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was that we have a wave function
that has coordinate dependence

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on both the electron
and the proton.

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So let's do the separation
of variables that shows

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how to deal with this system.

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So you want to write the wave
function for the whole system,

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so it depends on
the center of mass

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and on the relative coordinates.

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Now, I don't put
time, because we're

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discussing time independent
Schrodinger equation, where

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the time you can
put it later, if you

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wish, with the total
energy into the minus IET.

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So what we will consider
is a simple solution

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that is of the product type.

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So there will be a wave
function associated up

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factor, which is a wave function
associated with center of mass,

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and the wave function associated
to the relative motion.

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And we want to replace this
into the Schrodinger equation

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into H psi equals E psi.

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So let's see how it would go.

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Each term of the
Hamiltonian is going

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to act on this product of
functions that determines

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the whole wave function.

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The center of mass
momentum, being

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derivative, respect to
this x, will act just

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on the first term.

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So we'll have p squared over
2 m acting on psi c m of x.

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And the relative wave
function in that term

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just goes for the ride.

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So that's the first
term in the Hamiltonian.

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For the second term
in the Hamiltonian,

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we would have p
squared over 2 mu.

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We could add them psi
relative, and let's put

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even the potential
here, v of x relative--

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we don't put the relative
on the x, but they'll just--

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it's a small x
psi relative of x.

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So we've looked at the second
term and the third term.

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The third term is
grouped with the second,

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because it uses the
little x, not the big X.

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And then you have psi
cm of x multiplicativly,

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it doesn't do anything to it.

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All this is equal to
e times the psi cm

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of x psi relative of little x.

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OK, that's the
Schrodinger equation.

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And this should remind you,
it's very similar to what

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you did months ago of having
motion's in say, in two

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dimensions, and you wrote part
of the wave function dependant

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on x, part of a wave
function dependant on y,

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and you separated the
Schrodinger equation.

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So the next thing to do is
to divide by the total wave

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function, by the product.

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So divide 5 by the
total wave function.

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So what do you get?

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From the first
term, you will get 1

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over psi cm of capital X times
this p squared over 2 m psi

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cm of capital X. And that's all
what comes of the first term.

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From the second, you get
plus 1 over psi relative

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of x and this whole
bracket squared

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over 2 mu psi relative
plus V of x psi relative.

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And the whole thing being equal
to E. The two wave functions

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are divided there.

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So we have a situation
where a number

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is the sum of two functions.

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Now, what is funny of
course, is the argument

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you've heard several times.

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This first term depends just
on the capital X-coordinates.

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The second term depends just
on the lower x-coordinates,

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small x-coordinates.

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I didn't write them in some
places, but here they are.

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And therefore, the only way
these two things can always

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be true, is if the
first term is a number.

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And we'll call it Ecm.

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The whole thing must be a
number, we'll call it Ecm.

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And the second term
should be another number,

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and I'll call it E relative.

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That's E relative.

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So our conclusion is that
if this first term is

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a number equal to Ecm we
can multiply by psi cm

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and get p squared over
to 2m psi cm of x is

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equal to Ecm times psi cm of x.

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Which is a time independent
Schrodinger equation

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for a wave function psi
cm that is moving freely.

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The next equation is
the one within brackets,

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which is p squared over 2 mu
psi relative of x plus v of--

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now I can put r, when I
say that r is the magnitude

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of x psi relative of
x equals e relative--

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I should move this--

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psi relative of x.

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And we said that r was this.

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So the second equation
comes from this term

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identified with relative.

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And the last equation is to
say that the total energy is

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equal to Ecm plus e relative.

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So the whole two body
problem has been reduced

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to these three equations.

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This is what we aim to show.

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This is a gradient squared
on this wave function,

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the central potential term,
and the rest of the Schrodinger

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equation.

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So this is a
Schrodinger equation

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for a particle in the
central potential.

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That particle happens to
be the relative distance

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between these two particles,
but it obeys a central equation

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potential.

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Here is the center of mass.

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Solutions of this is a plane
wave, momentum plane waves,

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because this is like
a free particle,

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and that's our intuition.

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The hydrogen atom can
move like a free particle.

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That's an overall
quantum system and then

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there's the relative
degrees of freedom.

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The total energy of this system
must be the sum of the two.

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So that said for the
system, we are allowed now,

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to consider the hydrogen atoms.

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So that's what we'll do next.