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Hello everyone, welcome back.

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We're now going to look a little bit more

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deeply into these mysterious gaps in the
solar spectrum.

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So, just to remind you, when seen at high
resolution, the

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light of the sun exhibit the, exhibits
these, these dim spots.

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Which could be interpreted as absorption
features.

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Indeed that's what they are.

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There's a complimentary phenomenon with
which you're probably

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familiar undoubtedly have seen these
vivid, gas signs.

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Neon signs they're often called in the

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United States, that exhibit very bright
colors.

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And these are due to the very sharp
emission lines

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that are observed in low-density gases.

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Here are two pertinent examples.
One is a spectrum of atomic hydrogen.

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It's a, a, a, discharge lamp, and what
you're seeing here is

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the increase of the wavelength going from
left to right.

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And you see a, a, very distinct isolated
lines.

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red is sort of cyan, blue and violet.

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actually rather near our reference laser
points.

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this red line of hydrogen, 656 nanometers,
the Balmer alpha.

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so the balmer beta and so on, so called.

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And down below the spectrum of neon more
complicated.

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But as you can see it cons, consists of
fairly well resolved, quite sharp lines,

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rather than the sort of rainbow feature
that we see for dispersed sunlight.

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Now as it happens, the emission lines that
are observed in the low density

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gases, often correspond to the absorption
features of the sun.

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there's this, in fact, this line at 656
nanometers is this large dark gap here.

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this line is down here,

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I believe.

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And the other, the others have, have their

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counterparts in the relevant portions of
the spectrum.

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So, it became understood that the features
seen in the absorption

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spectrum in the sun, were due to the the
same

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transitions that were giving rise to
emission in, in low density gases.

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But the, the presence of, of these sharp
and

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distinct lines in atoms of, was a great
puzzle,

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because in classical mechanics a, a

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system of interacting particles can have
any set of positions and velocities.

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And there's nothing to single out

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a particular configuration for special
attention.

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So atomic theory at the end of the 19th
century had, there were many

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clever ideas deriving from things like
vortices in

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fluids, where there was son, some type of
regular structure that could be inferred.

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but none of these bore any fruit in

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terms of explaining the specific
properties of atomic spectra.

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I'll just mention in passing that the, the
vortex nought idea is coming back into

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vogue in atomic physics due to the ability
to produce

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vortex structures in gases and
Bose-Einstein condensates.

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But, I mean, these, these are not, these
are not elementary atomic structures.

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They're, they're rather a type of a super
fluid pattern, that can

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be generated by appropriate excitation of
the condensate.

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But the whole idea of the continuum
models, as atoms are some kind of fluid

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confronted real difficulty in the
discovery that there were, that atoms

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seemed to be made of particles interacting
with electric charges.

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And thereby interacting, by electric
forces.

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A, a notable development was Rutherford's
experiment in

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1911, that showed that virtually all of
the mass of

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the atom is concentrated in a minute

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fraction of its volume, 1, whatever that,
quadrillionth.

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So it's, ju, it, it was as if this, this
thing that

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had a pretty well defined size as known
from a variety of measurements.

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some as simple as looking at the thickness
of

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a an oil film on, on the surface of water.

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Taking the volume of the ordinary oil drop
and seeing how far it was

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spread out give, gives it a pretty good
indication of the size of an atom.

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And most of it, from Rutherford's
experiment had to be empty space.

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with all the mass concentrated at the
center.

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So this is actually a system somewhat

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like the the, the planetary system.
the mass, the sun is very massive, it sits

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near the center of the solar system, and
the earth has, a much smaller mass.

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And stands off from the sun, and, and
orbits it.

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We're going to see what classical
mechanics

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has to say about the, about the planetary

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atom.

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That is one consisting of two particles
ordinarily well separated in an orbit.

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Very much like the orbits of the planet
off the sun and solar system.

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This year is the 100th anniversary of this
great discovery by Niels Bohr.

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which is reported in the paper that's
contained in the

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additional materials section for this
course.

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And Bohr

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developed the postulate that there were
certain orbits certain planetary

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orbits of the electron

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about the proton in a hydrogen atom that
would be stable.

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And that would correspond, correspond to
the stationary states of atoms.

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And he so he basically solved the
equations

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of motion for the planetary system,
there's nothing

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novel about that, those solutions are well
known.

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And then stated a postulate that certain
of

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the, certain orbit, certain of the orbits
in an

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atom would, would consist of, would, would
provide

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the stationary states that were seen in
optical spectra.

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And then, the, that the, the radiation
observed

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in atoms would be, would be associated as
in this

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case here with an atom falling from a
higher energy state.

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A lar, larger orbit, into a lower energy

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state, and emitting a photon into the
radiation field.

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So that the, the frequency of the light

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that is seen, either an absorption or
emission,

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corresponds to the energy transfer from
the atom to

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the ra, between the atom and the radiation
field.

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And in fact the the the balmer or alpha
line, the red line that we

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see in, in the absorption spectrum is
around

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656 nanometers, rather close to our
reference laser.

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Okay, so let's see what classical
mechanics tells

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us about these hydrogen like systems.

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So we have a single nucleus with a
position denoted by r sub n,

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the mass, capital M, a charge of plus ze.

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Z is the atomic number so it would be 1

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for hydrogen 2 for helium, 3 for lithium
and so on.

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And then e is the, what we call the
specific charge.

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That's the the absolute value of the
charge of the electron.

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Then there's the electron, with a position
vec, described by position vector r

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sub e, little mass little m, and a
negative charge of minus e.

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We're going to keep, we're going to keep
the masses in play.

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represented explicitly in this treatment
because there's not such

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a big simplification from changing it to
something else.

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And one of the exercises that we want to
do, is

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to go through the process that led to the
discovery of

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[UNKNOWN].

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It's a very important event, both from the
standpoint of scientific development of

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nuclear physics, and also the development
of nuclear energy and nuclear weapons.

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And it was a discovery that was made by
looking at the balmer series of hydrogen.

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Okay, now here are the Newton's equation.

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It's ma, mass times the second

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derivative, the acceleration of the
nuclear, coordinate.

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And then these are the, this is Coulomb's
Law.

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So there's two classical mechanical,
classical, there's an

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equation of motion for the nuclear
coordinate, which

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is just inversely propor, the force on it

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is in the direction of the electron
nuclear distance.

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And it's inversely proportional to the
square of that distance.

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You see there're a factor of r from the
vector, and a factor of

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r cubed below.
So, that's one over r squared force law.

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And then according to Newton's, what is
it, third law?

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There's an equal but opposite force on the
electron.

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So you have the same thing there.

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Now, I th, the I'm using this, this
coordinate

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r here, which is the separation between
the nucleus and the electron coordinates.

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Because that is in fact,

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the relevant length for, that describes
the electrical force law.

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What you can do now, is you can get, you
divide this, you divide.

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Take this equation and divide it by 1 over
m, multiply

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by 1 over m, and you multiply this
equation by 1 over little, 1 over little

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m.

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And, you subtract them, and so you get an
equation of motion, for

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the the electron nucleus separation alone.

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So you see on this side you have the
second derivative of that, is just a

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function of itself.
So we've now managed to get an equation

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that can be solved directly for the the
separation between the two particles.

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So from, from this equation, we go
directly here.

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You see we're just

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taking the mass, this is a, an in, this
term here has, is a dimension.

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This term here as the dimensions of 1 over
mass.

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So we just invert it to get what's called
the reduced mass, and now we have a simple

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Newtonian equation for the motion of the
of the

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relative distance between the, the
electron and the nucleus.

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Now we define the momentum p associated
with this coordinate.

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It's just the mass times the velocity.

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Usual, the usual definition.

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And this then allows us to rewrite the
equations of motion as a, a pair of

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coupled first ordered equations in time.
So here, here,

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here's that equation we just made up.
It's the r dot is 1 over mu times

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p.
And then p dot is just taken from,

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from this equation.

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because p dot is r double dot.

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And so it's just minus zed e squared r
over r cubed.

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Now, why do

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we reduce this to first order equations,
you might ask?

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So I emphasized the development of
equation, equations that are first order

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in time, because the whole idea of solving
the

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equations of motion for a system, means
that if you,

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if you have a specification of the state
at time t.

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You can, you can find it's the state at
some later time.

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And that, for a small time, that means you
need to have an

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equation of motion that, that is first
order in the time for that state.

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That's why Schrodinger's equation must be
first order in time.

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So the whole idea of quantum mechanics is
that the wave function

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of psi, defines a state.
And if we know psi at time t,

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and we know the Hamiltonian operator, then
we can, then we

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can, then we can determine psi at some
slightly later time.

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So this is something that seems rather
different from some of

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the classical equations of motion with
which you might be familiar.

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Here's an example to consider in an inline
quiz.

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So I hope you appreciate

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it, from that previous example, that the
familiar second order wave equation, that

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one often deals with it's, it's a

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convenient way of solving the wave
equation.

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But it is it's sort of a secondary

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equation, that's based on primary, first
order equations.

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So, there's no, there's no real difference
between

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quantum mechanics and classical mechanics
in that respect.

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In classical mechanics in order to solve
the equations of

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motion, say for electrodynamics to solve
the equations of motion

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in time, you need to know the value of the

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electric field and its time, and its first
derivatives in time.

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Let's say as an initial value problem.

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Whereas in quantum mechanics, we just have
this one

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wave function that provides all the
information needed to solve

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the equations of motion, once the
Hamiltonian's known.