Hello everyone, welcome back.
We're now going to look a little bit more
deeply into these mysterious gaps in the
solar spectrum.
So, just to remind you, when seen at high
resolution, the
light of the sun exhibit the, exhibits
these, these dim spots.
Which could be interpreted as absorption
features.
Indeed that's what they are.
There's a complimentary phenomenon with
which you're probably
familiar undoubtedly have seen these
vivid, gas signs.
Neon signs they're often called in the
United States, that exhibit very bright
colors.
And these are due to the very sharp
emission lines
that are observed in low-density gases.
Here are two pertinent examples.
One is a spectrum of atomic hydrogen.
It's a, a, a, discharge lamp, and what
you're seeing here is
the increase of the wavelength going from
left to right.
And you see a, a, very distinct isolated
lines.
red is sort of cyan, blue and violet.
actually rather near our reference laser
points.
this red line of hydrogen, 656 nanometers,
the Balmer alpha.
so the balmer beta and so on, so called.
And down below the spectrum of neon more
complicated.
But as you can see it cons, consists of
fairly well resolved, quite sharp lines,
rather than the sort of rainbow feature
that we see for dispersed sunlight.
Now as it happens, the emission lines that
are observed in the low density
gases, often correspond to the absorption
features of the sun.
there's this, in fact, this line at 656
nanometers is this large dark gap here.
this line is down here,
I believe.
And the other, the others have, have their
counterparts in the relevant portions of
the spectrum.
So, it became understood that the features
seen in the absorption
spectrum in the sun, were due to the the
same
transitions that were giving rise to
emission in, in low density gases.
But the, the presence of, of these sharp
and
distinct lines in atoms of, was a great
puzzle,
because in classical mechanics a, a
system of interacting particles can have
any set of positions and velocities.
And there's nothing to single out
a particular configuration for special
attention.
So atomic theory at the end of the 19th
century had, there were many
clever ideas deriving from things like
vortices in
fluids, where there was son, some type of
regular structure that could be inferred.
but none of these bore any fruit in
terms of explaining the specific
properties of atomic spectra.
I'll just mention in passing that the, the
vortex nought idea is coming back into
vogue in atomic physics due to the ability
to produce
vortex structures in gases and
Bose-Einstein condensates.
But, I mean, these, these are not, these
are not elementary atomic structures.
They're, they're rather a type of a super
fluid pattern, that can
be generated by appropriate excitation of
the condensate.
But the whole idea of the continuum
models, as atoms are some kind of fluid
confronted real difficulty in the
discovery that there were, that atoms
seemed to be made of particles interacting
with electric charges.
And thereby interacting, by electric
forces.
A, a notable development was Rutherford's
experiment in
1911, that showed that virtually all of
the mass of
the atom is concentrated in a minute
fraction of its volume, 1, whatever that,
quadrillionth.
So it's, ju, it, it was as if this, this
thing that
had a pretty well defined size as known
from a variety of measurements.
some as simple as looking at the thickness
of
a an oil film on, on the surface of water.
Taking the volume of the ordinary oil drop
and seeing how far it was
spread out give, gives it a pretty good
indication of the size of an atom.
And most of it, from Rutherford's
experiment had to be empty space.
with all the mass concentrated at the
center.
So this is actually a system somewhat
like the the, the planetary system.
the mass, the sun is very massive, it sits
near the center of the solar system, and
the earth has, a much smaller mass.
And stands off from the sun, and, and
orbits it.
We're going to see what classical
mechanics
has to say about the, about the planetary
atom.
That is one consisting of two particles
ordinarily well separated in an orbit.
Very much like the orbits of the planet
off the sun and solar system.
This year is the 100th anniversary of this
great discovery by Niels Bohr.
which is reported in the paper that's
contained in the
additional materials section for this
course.
And Bohr
developed the postulate that there were
certain orbits certain planetary
orbits of the electron
about the proton in a hydrogen atom that
would be stable.
And that would correspond, correspond to
the stationary states of atoms.
And he so he basically solved the
equations
of motion for the planetary system,
there's nothing
novel about that, those solutions are well
known.
And then stated a postulate that certain
of
the, certain orbit, certain of the orbits
in an
atom would, would consist of, would, would
provide
the stationary states that were seen in
optical spectra.
And then, the, that the, the radiation
observed
in atoms would be, would be associated as
in this
case here with an atom falling from a
higher energy state.
A lar, larger orbit, into a lower energy
state, and emitting a photon into the
radiation field.
So that the, the frequency of the light
that is seen, either an absorption or
emission,
corresponds to the energy transfer from
the atom to
the ra, between the atom and the radiation
field.
And in fact the the the balmer or alpha
line, the red line that we
see in, in the absorption spectrum is
around
656 nanometers, rather close to our
reference laser.
Okay, so let's see what classical
mechanics tells
us about these hydrogen like systems.
So we have a single nucleus with a
position denoted by r sub n,
the mass, capital M, a charge of plus ze.
Z is the atomic number so it would be 1
for hydrogen 2 for helium, 3 for lithium
and so on.
And then e is the, what we call the
specific charge.
That's the the absolute value of the
charge of the electron.
Then there's the electron, with a position
vec, described by position vector r
sub e, little mass little m, and a
negative charge of minus e.
We're going to keep, we're going to keep
the masses in play.
represented explicitly in this treatment
because there's not such
a big simplification from changing it to
something else.
And one of the exercises that we want to
do, is
to go through the process that led to the
discovery of
[UNKNOWN].
It's a very important event, both from the
standpoint of scientific development of
nuclear physics, and also the development
of nuclear energy and nuclear weapons.
And it was a discovery that was made by
looking at the balmer series of hydrogen.
Okay, now here are the Newton's equation.
It's ma, mass times the second
derivative, the acceleration of the
nuclear, coordinate.
And then these are the, this is Coulomb's
Law.
So there's two classical mechanical,
classical, there's an
equation of motion for the nuclear
coordinate, which
is just inversely propor, the force on it
is in the direction of the electron
nuclear distance.
And it's inversely proportional to the
square of that distance.
You see there're a factor of r from the
vector, and a factor of
r cubed below.
So, that's one over r squared force law.
And then according to Newton's, what is
it, third law?
There's an equal but opposite force on the
electron.
So you have the same thing there.
Now, I th, the I'm using this, this
coordinate
r here, which is the separation between
the nucleus and the electron coordinates.
Because that is in fact,
the relevant length for, that describes
the electrical force law.
What you can do now, is you can get, you
divide this, you divide.
Take this equation and divide it by 1 over
m, multiply
by 1 over m, and you multiply this
equation by 1 over little, 1 over little
m.
And, you subtract them, and so you get an
equation of motion, for
the the electron nucleus separation alone.
So you see on this side you have the
second derivative of that, is just a
function of itself.
So we've now managed to get an equation
that can be solved directly for the the
separation between the two particles.
So from, from this equation, we go
directly here.
You see we're just
taking the mass, this is a, an in, this
term here has, is a dimension.
This term here as the dimensions of 1 over
mass.
So we just invert it to get what's called
the reduced mass, and now we have a simple
Newtonian equation for the motion of the
of the
relative distance between the, the
electron and the nucleus.
Now we define the momentum p associated
with this coordinate.
It's just the mass times the velocity.
Usual, the usual definition.
And this then allows us to rewrite the
equations of motion as a, a pair of
coupled first ordered equations in time.
So here, here,
here's that equation we just made up.
It's the r dot is 1 over mu times
p.
And then p dot is just taken from,
from this equation.
because p dot is r double dot.
And so it's just minus zed e squared r
over r cubed.
Now, why do
we reduce this to first order equations,
you might ask?
So I emphasized the development of
equation, equations that are first order
in time, because the whole idea of solving
the
equations of motion for a system, means
that if you,
if you have a specification of the state
at time t.
You can, you can find it's the state at
some later time.
And that, for a small time, that means you
need to have an
equation of motion that, that is first
order in the time for that state.
That's why Schrodinger's equation must be
first order in time.
So the whole idea of quantum mechanics is
that the wave function
of psi, defines a state.
And if we know psi at time t,
and we know the Hamiltonian operator, then
we can, then we
can, then we can determine psi at some
slightly later time.
So this is something that seems rather
different from some of
the classical equations of motion with
which you might be familiar.
Here's an example to consider in an inline
quiz.
So I hope you appreciate
it, from that previous example, that the
familiar second order wave equation, that
one often deals with it's, it's a
convenient way of solving the wave
equation.
But it is it's sort of a secondary
equation, that's based on primary, first
order equations.
So, there's no, there's no real difference
between
quantum mechanics and classical mechanics
in that respect.
In classical mechanics in order to solve
the equations of
motion, say for electrodynamics to solve
the equations of motion
in time, you need to know the value of the
electric field and its time, and its first
derivatives in time.
Let's say as an initial value problem.
Whereas in quantum mechanics, we just have
this one
wave function that provides all the
information needed to solve
the equations of motion, once the
Hamiltonian's known.