The following content is
provided under a creative
commons license.
Your support will help MIT
OpenCourseWare continue to
offer high quality educational
resources for free.
To make a donation or view
additional materials from
hundreds of MIT courses, visit
MIT OpenCourseWare at
ocw.mit.edu.
PROFESSOR: All right, a couple
of announcements.
Next week two minor
celebrations, maybe receptions
we would call them.
Quiz 2 Tuesday based on homework
2, and periodic table
quiz Thursday.
I provide the numbers, you
provide the letters.
I guess that's how it works.
And the contest ends
Friday five PM.
Last day we looked at the Bohr
Model and we developed
equations for the radius of the
electronic and the orbit
of the one electron atom.
The energy of the electron and
the velocity of the electron.
And we found that for all
of these they were
a function of n.
Quantum number.
n takes on discrete values.
One, two, three, and so on.
We say that these
energies radii
velocities are quantized.
They take discrete values.
And then later in
the lecture we
started looking for evidence.
And we found ourselves in an
exercise of reconciliation
with data taken by Angstrom
about 50 years earlier and fit
to an equation by J.J.
Balmer.
And we were part way through
that and adjourned.
So I'd like to pick up the
discussion at that point.
I've done a different drawing of
what's going on inside the
gas discharge tube.
Last day I had the ballistic
electronic here, and this is
boiling off the cathodes.
The cathode is inside the gas
discharge tube and this
electron, if the voltage is high
enough, will leave the
cathode and shoot across this
low pressure gas, which
contains, among other things,
atomic hydrogen.
And I'm trying to depict the
atomic hydrogen atom here.
Here's the proton, which
is the nucleus--
that's the sum total of the
contents of the nucleus--
and here's the lone electron
that is orbiting the nucleus
at some initial value.
n sub i.
Could be ground state.
Doesn't necessarily
have to be.
With some thermal energy this
could be n greater than one.
Then we reason that if the
electron, ballistic electron,
and it's trajectory across the
gas discharge tube over to the
anode, which is charged
positively.
If it collided with this
electron it could impart some
of its energy, thereby promoting
the electron from ni
up to nf, the final level.
And the electron would
be up here.
And for this transition there
would be an energy cost. That
energy cost is delta e.
Delta e is the energy
to go from ni to nf.
And so the kinetic energy--
half mv squared of the
incident electron--
is diminished by this amount.
And the electronic continues
on it's merry
way at a slower speed.
We assume it's mass
doesn't change.
The only way we can change its
energy is to slow it down.
And there's a conservation of
energy, so the sum of the
energy of the scattered electron
and the transition
energy of the electron within
hydrogen must equal the
incident kinetic energy of
the ballistic electron.
But there's more.
This is a--
no pun intended--
a one shot deal.
This is ballistics, and
so the electron is
not sustainably promoted.
It falls back down.
And when it falls back down
we have the transition
energy now given off.
Here, to promote we had
to call for energy.
When the electron falls down
it gives off that energy.
And that energy is given
off in the form
of an emitted photon.
And it's that emitted photon and
it's that emitted photon
that ultimately gives
rise to the lines.
The lines in the spectrum are
generated by the emitted
photon here.
Everything else is preamble to
this event, and this event
gives rise to the
emitted photons.
And I think that's about where
we got last day with the
reconciliation.
So let's look carefully here.
We recognize that there
needs to be
conservation of energy again.
In other words, the energy of
the emitted photon, the energy
of the photon, which we know
from Planck is h times nu.
I'm trying to distinguish.
This, I'm making nu, the
Greek symbol nu.
And I put a little
descender on it.
It looks like a v, but I put
a little ascender here to
distinguish it.
This is lowercase v,
as in mv squared.
This is nu.
So h nu, or it could
be hc over lambda.
Or it could be hc nu bar.
Three ways of writing the
energy of the photon.
And that must equal delta
e of the transition.
So let's keep going.
We know that the delta e, the
transition is given by the
Bohr Model.
Delta e transition will equal e
final minus e initial, which
will be minus kz squared--
I'm writing this generally,
in this case with atomic
hydrogen z as 1--
with minus k times 1 over n
final squared minus one over n
initial squared.
What we can do then is equate
these and roll them around to
isolate nu bar.
nu bar, then, equals minus kz
squared over the product of
the Planck constant, the speed
of light, 1 over nf squared
minus 1 over ni squared.
Now for the Balmer series,
that is to say the Balmer
series of lines, that turns out
to be a series where all
of the transitions end
up on n equals 2.
We said nf equals 2.
z equals 1 because we're talking
about atomic hydrogen.
Then what we have is a set of
translations that go 1 over 2
squared minus 1 over
ni squared.
ni must be greater than nf, so
ni must come from the set 3,
4, 5, et cetera.
Furthermore, I'm going
to put z equals 1.
Let's evaluate k.
We know that's 2.18 times 10
to the minus 18 joules, or
13.6 electron volts.
And we know the Planck constant,
6.6 times 10
to the minus 36.
And this is 3 times 18 of
the 8th all in SI units.
So this gives me 1.1
times 10 to the 7th
in reciprocal meters.
And if I put all this together
I end up with exactly they
equation that was published
by Balmer.
Exactly Balmer's equation in
1885, rewritten to express it
in SI units.
1 over 2 squared
minus 1 over--
I'm just going to
put ni squared--
or ni equals 3, 4, 5, 6.
This is Balmer exactly.
Balmer exactly.
So the assumption of this
planetary model, with all of
the restrictions that Bohr
placed on it in order to get
this set of equations,
reconciled
with laboratory data.
Very, very significant.
So here we are.
Those four lines all can be
derived from the Bohr model.
And here's another cartoon from
your book and showing
what I'm trying to depict here,
namely it's the falling
down, the return to the state
from which the electron was
promoted that generates
the photon.
And the set of those lines
is what gives you this.
There so there's the validation
of the sixth piece.
So Bohr Model agrees with
Angstrom's data, but it also
suggests other experiments.
Let's think about this
for a second.
OK, here's another cartoons
from your thing.
You know, I told you that
this thing is unstable.
And in the Balmer series it
goes from n equals 2 up.
But there's a ground
state, n equals 1.
What was wrong with those
electronics in Sweden in 1853
than Angstrom could never find
any electron that would fall
all the way down to
the ground state?
What's wrong with it?
Well, here's the answer.
It has to do with
instrumentation.
So this is an example where
science goes further thanks
for the advent of new
instrumentation that allows
this to make measurements that
previous people couldn't make,
even though they were very
competent experimentalists.
Angstrom could have found n
equals 1 series, but couldn't
see them because he was using
a photographic plate.
This shows you the range
of sensitivity for
photographic plates.
Here's the electromagnetic
spectrum.
Out here you have low energy
radio waves and up here you
have x-rays and gamma
rays and so on.
And the visible spectrum is
parked right here in the
middle, and here it is
unpacked for you.
And it roughly runs from
400 to 700 nanometers.
That's the invisible spectrum.
So wavelength increasing from
left to right, which means
energy frequency in wave
number increase
from right to left.
They're complimentary, right?
e nu, nu bar on the top, lambda
is on the bottom.
Some spectra are plotted in
lambda, some are plotted in
wave number, whatever.
And by the way, I want to
show you the power of
knowing a few things.
I don't expect you to know a lot
of facts, but I expect you
to know a few things.
Every educated person ought to
know that the visible spectrum
runs round numbers 400
to 700 nanometers.
But look, I can take 700
nanometers and use this
formula and convert
it to energy.
And I'm going to get something
like 3 times 10 to
the minus 19 joules.
Yuck.
Instead, I go in
electron volts.
1.8 ev.
Over here, 400 nanometers
is 3.1 ev.
So round numbers, the visible
spectrum spans 2
to 3 electron volts.
Our eyes are photo detectors
that operate on the band width
2 to 3 electron volts. that's
easy to remember.
Those are good numbers.
So where does that leave us?
It leaves us here.
We go back and we see these
numbers, 656, 486, 434,
they're all in the
visible spectrum.
So I went and I did a
little calculation.
I said, well what would I have
if we'd gotten the wave length
for the transition
from 2 down to 1?
This is n equals 2 down
to ground state.
If you plug in the numbers to
the Bohr model, you'd find
that that would give
you 122 nanometers.
122 nanometers?
Well, 122 nanometers is going
to put you way over.
It's too high energy, right?
122 nanometers is going to put
you off to the left there into
the ultraviolet, where
the photographic
film was not sensitive.
So he couldn't measure
those lines.
So now I'm going to end by
putting the master equation
that captures all of this.
And the master equation
that captures of
all of this is here.
It's that nu bar goes as r times
z squared, 1 over nf
squared minus 1 over
ni squared.
So this is the most general form
for all 1 electron atoms.
That's why I've got z squared
in there for all 1 electron
atoms.
And this is called the
Rydberg equation.
Named after another Swedish
spectroscopist at the
University of Lund.
I think a Swede would probably
pronounce this something
Rydberg, but you don't
have to say that.
You can just say Rydberg
and it'll be fine.
And in honor of Rydberg, the
constant here is given the
symbol, capital R.
The capital R as the Rydberg
constant and it has a value of
1.1 times 10 to the
7th reciprocal
meters in good ST units.
Well, there was more
evidence for the
support of Bohr's Model.
More evidence for the support
of Bohr's Model.
By the way, as the detectors got
better and better we could
get more and more lines.
You see these, as you get
higher and higher series
ending on higher and higher end
numbers, you move off into
the infrared.
Because this is not to scale.
These n equal 4, n equal 5 are
closer and closer to closer
together in terms of energy.
They're farther and farther
apart in terms of spacing, but
they're closer and closer
together in terms of energy.
Because they're farther
from the nucleus.
You say, gee, shouldn't
it cost more
energy to go farther?
Uh-uh.
Because you're farther from
the positive nucleus.
Be careful.
Don't let your intuition send
you in the wrong direction.
It's all about Coulombics.
Anyway, so the Lyman series
ends at n equals 1.
And these are different
scientists.
Paschen, Bracket, Pfund,
Humphreys, and so on.
So maybe, I don't know, if
somebody hasn't claimed n
equals 214, all the lines that
end there, you know, maybe
that could be your name
on the series.
As if anybody cares.
Looks like this quantum
condition is validated.
See this is really important
because this was the big break
away from classical theory.
That the motion of a body,
something with mass, could be
quantized in its behavior shook
this physics community.
But this reconciliation
of the data says that
assumption is valid.
There's more that happens.
So in 1913 in Berlin--
remember 1913 is when Bohr
published the paper--
1913 in Berlin there was James
Franck and Gustav Hertz.
James Franck and Gustav Hertz.
And they were conducting
experiments on
gas discharge tubes.
Only they filled a gas discharge
tube, instead of
with hydrogen, they filled
it with mercury vapor.
So gas discharge tube--
GDT--
gas discharge tube containing
mercury vapor.
The same thing.
Put the electrodes, connect on
a power supply, and started
varying the potential.
So I'm going to show you
what they found.
This is the Planck voltage,
and this is the current
between electrodes, or if you
like, across the tube.
Between the electrodes,
or through the tube.
Or if you like, tube current.
Meaning from one electrode
to the other.
The tube current.
Well, low voltage,
low current.
High voltage, high currently.
They get up to a certain value
of voltage, all of sudden the
tube starts glowing blindly and
the current falls to 0.
Then they continued to
raise the voltage.
More voltage, more current,
up, up, up, up, up.
And then they get to another
critical value of voltage,
even more intensity.
And then the current
falls to 0.
So you look at those data and
say, well, what's that got to
do with the Bohr Model?
Because mercury is not
a 1 electron atom.
It's got a boat load
of electrons.
This is not a 1 electron atom.
So you say, I know what it is.
It's ionization energy.
Must be ionizing the mercury.
So you go to the periodic table
and you look up the
ionization energy of mercury and
you discover that that's
10.4 volts.
10.4 electron volts is the
ionization energy.
And this first null
is at 4.9 volts.
Well, 4.9 is a long
way from 10.4.
And this second null occurs
at 6.7 volts.
6.7 volts.
So what's this telling us?
What this is telling us is that
when you get to a value
of 4.9 volts, you've hit a
certain value that allows you
to promote electrons within
mercury between one level and
the next level.
And those electrons are
being promoted and
then cascading down.
And they're cascading down and
they're emitting in the
visible and it's blinding you.
Say, OK, so what
does that mean?
Well, it means that the Bohr
Model, which is for a 1
electron atom, assumes
that energy levels
within it are quantized.
These data indicate that on the
basis the behavior of this
gas discharge tube, there must
be quantized energy levels
inside of mercury, which means
all atoms have quantized
energy levels.
You understand?
Everything is quantized.
That's really powerful.
It starts off with this nerdy
little 1 electron atom, and
now he's applying it
across matter.
And this is gas, this is
more elaborate gas.
Heaven forbid, it might exist
in liquids and solids.
So that's the Franck,
Hertz experiment.
So his stock goes way, way
up as a result of that.
And they win a Nobel Prize.
Here's James Franck.
Here's Gustav Hertz.
You know what the Hertz is.
200 kilohertz, so on.
That's Hertz.
James Franck was at Gottingen
when he won this, but he
ultimately came to the United
States when the political
changes started occurring in
the thirties in Germany.
Franck decided to seek safer
surroundings and ended up at
the University of Chicago, where
there is to this day the
James Franck Institute
of Physics.
Very, very high-end physics
institution.
So this is good.
But all good things
come to an end.
So 1913 was a bittersweet
year for Bohr.
Because he got some good news,
but he also got some bad news.
So now I want to move over to
limitations of the Bohr Model.
Limitations of the Bohr Model.
So I know what you're going to
say-- well, it only talks
about 1 electron atoms, so
that's a limitation.
No, there's more to
it than that.
Even the 1 electron atom model
doesn't capture everything.
I'm going to summarize
the limitations.
I'm going to show you three,
and they all fall under the
general umbrella of
fine structure.
Fine structure.
In other words, the Bohr Model
is good, give us the big
lines, but when you start
looking more carefully it
fails to capture some
of the physics.
So first of all, let's go back
to some earlier data.
1887.
1887, there were already data
out there that were going to
give heartburn to
the Bohr Model.
And those data were taken
by Michelson and Morley.
Michelson and Morley.
Everything I've taught you so
far, with one exception, has
been European science.
Americans were not active in
science because this was a
young country.
We were really good engineers
because we were blockaded by
the rest of the world.
We had to live by our wits--
that's where you get the term
"yankee ingenuity." Science
was hifalutin stuff.
We didn't have time for it.
But towards the latter half of
the 19th century, we started
moving into fundamental
science.
The first American to win the
Nobel Prize was Michelson.
Michelson was doing work at
Case in Cleveland, which
eventually became Case Western
Reserve University.
So he was at Case in Cleveland
and he was studying optics.
And he was a brilliant
experimentalist. In fact, he
made the first reliable measure
of the speed of light.
Back before 1900.
What they were doing is they
were looking at Angstrom's
lines and they noticed
something peculiar.
If you take a look at even this
drawing, you notice the
red line is a little bit
fatter than the others.
Now you might just say, well,
that's just the artist taking
liberties and somebody didn't
catch it in proof reading.
But in point of fact, what he
found was that if you look at
that line, which is really the
line for the 3-2 transition--
the 3-2 transition in
the Balmer series--
what you find is that if you
look at the photographic plate
more carefully, you find that
this thing in fact is a pair
of lines, but very, very
closely spaced.
This is known as a doublet.
Two lines very closely spaced,
centered at 656 nanometers.
And with his interferometer he
gets super, super good data.
And he could split
the doublet.
Well, what's that
mean for Bohr?
Bohr has no way of
explaining this.
If you look at the Bohr Model,
you've got n equals 2, you've
got n equals 3, alright?
So this is energy 2,
energy 3, right?
And so when the electron falls
from 3 to 2, we get a photon
of a certain value.
It's going to be nu 3 to 2.
That's the frequency or wave
number, what have you.
Now, the fact that you've got
a doublet here means that
there must be two transitions,
but darn close.
There's either a 3 and a 3
primed, or there's a 2 and a 2
primed, but it's not
simply 3 and 2.
So that piece of information
runs
counter to the Bohr Model.
Bohr Model is silent about it.
It gets the big picture, but if
you look more carefully it
can't capture the doublet.
And Michelson ultimately
gets the Nobel Price.
And I think I've got him here.
There he is.
The Nobel Prize.
By the time he got the Nobel
Prize he was at the University
of Chicago, but he did the work
that won the Nobel Prize
for him at Case.
So sometimes when you see even
Millikan, Millikan did his
work at University of Chicago,
but eventually took a position
at Caltech.
So the Nobel Prize says, Robert
Millikan, Caltech.
But he didn't do that
work at Caltech.
He did it at Chicago.
Anyways, you can go to
the Nobel website.
You can read about
these people.
And what's really cool is when
you win the Nobel Prize-- you
notice I didn't say if--
I say, when you win the Nobel
Prize, what you do is you get
on an airplane, you go to
Stockholm, and then you go and
you have dinner in this
beautiful hall.
I've been there and it's
gorgeous, gilded and so on.
Very nice kitchen, excellent
wine list. And--
yes-- and you can go there
and they serve meals.
the menu is taken from previous
Nobel Prize dinners.
So you can sit and--
whatever it is, it could be the
Nobel Prizes of 1927 and
that's what's going to be
on the menu today--
and after the dinner they have
a presentation ceremony with
the King of Sweden.
You get your Nobel Prize,
and then people
listen to your lecture.
And those Nobel lectures are
really, really expository.
So if you want to go and read
the Nobel lecture that
Michelson gave on the occasion
of winning the Nobel Prize,
you'll probably learn all
of about this and more.
It's really, really good,
so go there and read.
Now back to the story.
Second problem with
the Bohr Model.
1896--
see, all this data had
been accumulating--
1896, there was a postdoc
by the name of Zeeman.
Piet Zeeman.
He was a postdoc at Leiden.
Leiden in Holland
under Lorentz.
You'll learn about the Lorentz
force when you study 802.
What he was doing--
again, gas discharge tube.
So this was gas discharge tube,
and what Zeeman was
doing on his postdoc was
in a magnetic field.
These people were doing all
sorts of experiments.
They were trying to block out
the whole experimental space.
So one guy, his specialty
is high energy.
One guy's specialty
is low pressure.
These people are taking a gas
discharge tube and putting in
the jaws of a powerful,
permanent magnet and then
measuring the spectrum.
And what he found was that
for certain lines,
this was the rest--
b, I'm going to use as
magnetic field--
in the absence of
applied magnetic
field you have a line.
And this is not a doublet,
triplet--
it's just a plain old line.
Well behaved line.
But when they take that gas
discharge tube and put it into
a magnetic field, they see
a plurality of lines.
And furthermore, the spacing--
I'm going to use c here--
the spacing in the lines is
proportional to the intensity
of the magnetic field.
No magnetic field,
single line.
Modest magnetic field,
modest amount of what
is called line splitting.
So a modest amount of
applied magnetic
field, modest splitting.
Intense magnetic field,
intense splitting.
Bohr Model is silent
about that.
Because you know, if you've got
different lines, it means
you must have different
energy levels.
It's as though the energy level
diagrams opens up in a
magnetic field.
The Bohr Model can't
account for that.
And parenthetically, they got
the Nobel Prize, too.
So there's Piet Zeeman.
Got his PhD in 1896.
He's got his Noble
Prize, 1902.
He's off to a good
start, I'd say.
And there's Lorentz.
Two of them.
We'll get to him in a second.
So third piece of bad news
for the Bohr Model.
And that comes, again,
in 1913 in November.
In November of 1913 there
was a man by the
name of Stark in Germany.
And Stark was doing analogous
experiments.
He was studying gas discharge
tube in electric fields.
Obviously, you've got an
electric field across the
electrodes to excite
the electrons.
But he's taking a whole gas
discharge tube and putting it
between flights and then
applying an electric field.
And what did he find?
He found the same
sort of thing.
He got line splitting in an E
field, and furthermore that
extent of splitting, extent
dependent upon the intensity.
E intensity.
So no field, no splitting.
Modest field, modest
splitting.
Intense field, intense
splitting.
Well, again, that's a headache
for the Bohr Model.
So this is all three problems,
and it's all under aegis of
fine structure.
So we know the Bohr Model
has its limitations.
OK, Stark.
I know he's got his
Nobel Prize.
There he is.
So 1913 ends on a sour note.
But people don't give up.
1916, Arnold Sommerfeld
in Munich.
He was a professor of physics
and he proposed modifications.
Modifications to Bohr Model.
It's a patch, we would
call it a patch.
going?
To put a patch on
the Bohr Model.
And what's he going to do?
What's the gist of his idea?
Well, he retains the planetary
structure.
He liked that idea--
nice orbits, so on.
But he took a page out
of Kepler's book.
The planets in the Kepler
model, when they revolve
around the sun their orbit
is not circular.
It's elliptical.
So Sommerfeld said, why don't
we give that a try?
What if we said the electronic
orbit can be
elliptical or circular?
And he was quite specific.
He said, suppose--
and this, again, is not to
scale, but to emphasize this
is going to be elliptical or
circular, but very, very mild
eccentricity.
What I'm going to draw for you
is extreme eccentricity to
make a point.
But suppose we had the circular
orbit as I'm drawing
it now, and then we had an
elliptical orbit that is
centered on that circle.
So it's mild eccentricity.
We might have another one--
let's do one more.
This is good enough.
The gist here is that we have
a circular orbit and an
elliptical orbit, but
the bandwidth here
is very, very narrow.
So this is very, very thin.
And it's sort of like
an egg shell.
So if I asked you, what's
the dimension of an egg?
You'd say, well, it's
dimension of the
surface of the egg.
Then I'd say, but the
egg shell has
some thickness, right?
But that thickness is relatively
small in comparison
to the total dimension
of the egg.
So an analogy.
He said that the range of
distance from the nucleus,
whether it's circular
or elliptical,
is very, very narrow.
So we can say the set of
circular and elliptical orbits
lie within a shell,
as in egg shell.
So this is a shell model.
It's a shell model.
So now how do you designate
the different orbits?
You've got some that are
circular, some that are
elliptical.
He needs to distinguish them
and he needs to be able to
label them.
So he introduces new
quantum numbers to
allow us to name them.
So let's go and take a look at
the quantum numbers that
Sommerfeld gave us.
So he starts off with n.
He retains that from the Bohr
Model and he calls that the
principal quantum number.
And it's primary attribute
is size.
It captures the distance, the
principal r from the nucleus.
And it takes values 1, 2, 3,
all the way up to infinity.
So n equals 1, small radius.
n equals 10, large radius.
Oh, by the way, there's another
numbering system.
This is what we use, but the
spectroscopists use letters.
The spectroscopists
use letters-- why?
Because remember the
Balmer series?
Everybody was hooked on the
Balmer series and it ended up
being n equals 2.
And then later with better
detectors we find
there's an n equals 1.
So the spectroscopists
said, we're going
to get fooled again.
So we're going use letters.
And we're going to start
with the letter k.
It's in the middle
of the alphabet.
That way if we discover even
lower energies, we've got some
head room here, we
can label those.
But we never found any.
So if you go over to Building
Thirteen and you do some x-ray
refraction and you use the
line that emanates from a
copper target, n equals
1-- it's called the k
alpha line of copper.
To this day.
So k, l, m, and so on.
You can't get to infinity,
obviously.
You know, I didn't think
this thing through.
Now the l.
l is, what's his name?
Sommerfeld.
And it's called the orbital
quantum number.
Why?
Because he said that the
electron is in an orbital
instead of an orbit.
Orbit is Bohr, orbital
is Bohr-Sommerfeld.
And it speaks to the shape.
Somehow, I've got to distinguish
between elliptical
and circular.
And it takes values 0,
1, up to n minus 1.
So the n number controls
the range of l.
And again, the spectroscopists,
they're real
number weenies, they're afraid,
so they use s, lowercase.
See this is uppercase,
this is lowercase.
s, p, d, f.
For sharp, this is the sharpest
line from the l
equals 0-- then the principal
because as you go to z they
all seem to converge and
look like hydrogen--
d is diffuse, f is fine, and
then after that they ran out
of ideas so g and h.
So you'll talk about the one
s-orbital, meaning n equals 1,
l equals 0.
And there are some values
here for shapes.
I'm going to put that
right above it.
When l equals 0, l equals
0 means you
have a circular orbit.
And when l equals 1
it's elliptical.
And when l equals 2 it's much
more complex, and we'll just
leave it at that.
1, 2, and 3.
So there's l values.
And then m is the magnetic
quantum number.
And it talks about
orientation.
I'll show you what I mean
by that in a second.
The values are governed by l,
which is governed by n.
Starts at l, l minus 1, goes
down through 0, goes to minus
values and ends at minus l.
So for example, we could do
something like this--
when n equals 1, then l most
equal 0, so therefore
m must equals 0.
So this means for n equals 1,
it's only a circular orbit and
this thing is going to be immune
to line splitting in a
magnetic field.
When n equals 2, l can equals
0 or l can equal 1.
When l equals 0, m equals 0.
That's boring, that's
circular.
But here's another
possibility.
And that is, when l equals
1 then m can equal 1,
0, and minus 1.
Now I said it has something
to do with orientation.
Most of quantum mechanics
doesn't translate into the
Cartesian world, but this one
does, mercifully, and I think
it's a cute analogy.
If I were to tell you that
I've got three different
quantum numbers and I've got an
elliptical thing-- and one
way to think, see, the number 0
looks like a circle and the
number 1 has some asperity
associated with it, so you can
think of that as the ellipse--
so I know that I can, with no
prior knowledge of where the
true origin of the universe it
is, I can arbitrarily define a
set of rectangular coordinates,
orthogonal
coordinates, x, y, and z.
And that means I could put one
orbital here, one orbital
here, and one orbital here.
So those are three orthogonal
orientations, which I think is
consistent with the fact that
m takes on three values.
OK, that's cute.
So that's as far as
Sommerfeld went.
I'm going to go and do something
as a retronym.
I want to get the fourth quantum
number up here now,
but we're going to
pause the story.
We're going to fast forward to
1925 so I can get the last
quantum number up here.
And that's called the
spin quantum number.
And it takes values plus
or minus a half.
Where did that come from?
Well, in 1922--
oh, you know everybody's getting
Nobel Prizes and I
didn't give Niels Bohr his
proper recognition.
He gets the Noble
Prize, as well.
Oh, when Sommerfeld turned 80,
they had a symposium in his
honor and they published
a book.
And the book had papers and
well-wishes, papers that were
given at the symposium.
And in the front they had
Sommerfeld's picture and they
also had this twin picture,
this diptych.
So on the right is Sommerfeld,
and on the left is the same
picture but they've
morphed it.
Now remember, there's
no Photoshop.
Horrors, there's no Photoshop.
Can you imagine?
So how could they do this?
They had to take the negative,
which was a photographic
plate, and when they were
printing the negative using a
light box they had to hold the
negative on an angle to get
the distortion.
And in holding it on an angle
to get the distortion, they
turned this image into something
that was a little
more spread out.
And the caption that went with
this, "To Arnold Sommerfeld,
who taught us that the circle is
the degenerate form of the
ellipse." Now that's
geek humor.
I mean, they laughed.
They thought that was
so funny, hahaha.
You know.
They were having a great time.
It was Germany, and
nineteen twenties.
And there he is.
OK, so now let's go to 1922.
This is the Stern-Gerlach
experiment.
Very interesting experiment.
This is really physical
vapor deposition.
Over here I've got a
crucible and it's
full of molten silver.
So Stern and Gerlach were
studying the magnetic behavior
of liquid metals.
So what they were doing
is they had this--
over here you see it's red
even though it's silver,
because this is that about
1,000 centigrade.
Silver melts at about 960.
Everything, I don't care what
it's color is at room
temperature, at 1,000
degrees it's red.
It's called red hot.
All right.
So this is red hot silver and
there's a vapor here and
there's a slit and the silver
atoms come out of the slit.
And they go across over here to
a substrate and then they
pause it on the substrate.
So making little band of silver
on the substrate.
And furthermore, he
sometimes put them
through a magnetic field.
So he's got a slit here that
narrows the beam, and then he
sends it through a magnetic
field that is asymmetric.
It's divergent.
Can you see here?
Look at the end.
The south pole as a tip
and the north pole is
this arc, this cup.
So the field lines don't go just
directly from tip to tip,
they go from tip off
to the side.
So you can see the divergence
of the magnetic field.
And so he looked at what kind
of deposits he got as a
function of the magnetic
field.
Here's what the observed.
Very puzzling.
The whole thing was about
Maxwell's equation.
So he's got a silver beam.
And when it went directly from
the furnace to the substrate
he just got the shadow
of the slit.
And they have the split
crosswise with respect to the
divergent magnetic field.
So if you look at the substrate
you just see a band.
So this is a band of silver and
you can imagine there was
a slit out here and it just cast
a shadow and that's the
band of silver.
This is PVD, physical vapor
deposition of silver.
Now when b is not equal 0,
what would you expect?
You'd think the beam
would bend, right?
So what do you think happens?
The beam bends up?
The beam bends down?
Or beam bends to the right
or to the left?
Think about it.
I don't want to hear
your answer.
Think about it.
What do they observe?
What they observe is, if this is
where the original one is.
Two.
The beam splits in two.
And it gets two deposits,
one above, one
below, of equal intensity.
That's a problem.
Beam splitting.
But now it's a beam of matter.
Beam splitting.
Boy, they had them scratching
their heads on that one.
No way to explain that.
So along come a couple
of graduate students.
1925, couple of graduates.
So this is 1992 in Frankfurt.
1925, two graduate students
in Leiden, again.
Gaudsmit and Uhlenbeck.
They're just like my TA's.
Grad students.
And they looked at this thing,
I don't know, maybe sitting
around over a beer one night,
and they said, you know, so
far what we've been saying is
the electron revolves around
the nucleus.
And sometimes it revolves in a
circular orbit, and sometimes
it revolves in an elliptical
orbit, but here's
the electron revolving.
And they said, what if in
addition to revolve, the
electron rotated so that
it's going like this?
[GESTURES]
But there's two choices.
It can be going like this, or
can be going like this.
[GESTURES]
Now, it's a charged species and
it's rotating, which means
that it's going to have
a magnetic moment
depending on rotation.
And now I'm going to
send it through a
divergent magnetic field.
Doesn't it follow to reason
that if I put it through a
magnetic field and I've got some
of them doing this and
some of them doing that,
they're going to go in
different directions?
Opposite directions?
And what do you think
the numbers are?
If I give you Avogadro's number
of silvers, you think
I'm going to get a dominant
clockwise and a minority
anti-clockwise?
No.
We're going to get
equal numbers.
Some are going to spin like
this, some are going
to spin like that.
And you're going say, but
electrons don't spin, they're
not doing this.
But if you model them as though
they are doing this,
you get those results.
Those results make sense.
And so they introduced the
spin quantum numbers.
And I think these are the ones
that have been erased.
Such is education.
OK.
So you know it.
S plus or minus a half.
By the way, Gaudsmit and
Uhlenbeck were here during
World War Two.
They worked in Building Four.
You go down the corridor,
Building Four just off the
Infinite Corridor, there's a
plaque there for the Radiation
Laboratory.
That's where they worked,
in the Rad Lab.
That's where radar was
first engineered.
There was work in the
UK, there was
work in other places.
But this is the Radiation
Laboratory, started here and
they were both here
at the time.
OK, well, I think that's a--
So this is a plate
from the paper.
No magnetic field with
the magnetic field.
And by the way, why did
they choose silver?
They chose silver because it's
atomic number is 47--
it has an odd number
of electrons.
You're going to learn later that
you get two electrons in
an orbital, and if you have
two electrons, one will be
spin up, one will
be spin down.
There's no magnetic moment.
So they were clever about
choosing an element that had
an odd number of electrons so
that there would, at the end,
be an unpaired electron.
And there's Otto Stern
with his Nobel Prize.
And he came to the United
States, as well.
And you're going to see the
ascendancy of American Science
as people flee Europe up in
the nineteen thirties.
And America is the beneficiary,
and then you see
American science rise.
But for now it's European
science.
OK, so I'm going to talk a
little bit about hydrogen and
transportation.
And we're going to talk about
the Hindenburg because it was
full of hydrogen.
And to give you a sense of
scale, this is what a 747
would look like and is what the
Titanic would look like.
It was almost as long
as the Titanic.
It was built in Germany by
the Zeppelin company.
And the Titanic, the Hindenburg
rather, was LZ129
serial number.
That's Luftschiff Zeppelin.
Airship Zeppelin.
135 feet in diameter.
804 feet long.
How long is a football field?
So that's a big boat.
Seven million cubic feet of gas,
giving you 112 tons of
useful lift.
You ever have to lift something
very heavy, there's
your sky crane.
So why are they using
hydrogen?
Well, when the Nazis came to
power in Germany, Congress
passed the Helium Control Act.
The dominant supplier of helium
to the world was the
United States.
Helium comes from helium
wells in the earth.
And so as of 1933 the United
States refused to sell Helium
to Germany, so the engineers
were forced to use hydrogen.
Next best thing.
Here are some posters.
"Only 2 1/2 half days to
Europe." And here one, a
German one.
"And now over the North
Atlantic." That's Manhattan.
That's the lower tip
of Manhattan.
There's the Chrysler Building,
look at that.
Now look at that picture,
isn't that magnificent?
10 transatlantic
flights, 1936.
1002 passengers.
Cruising speed, 78
miles an hour.
Took two and a half days.
By the way, 100 feet
in diameter--
when people traveled, they
traveled in style.
They had a ballroom there
and a grand piano.
People didn't sit like this.
[CROUCHES]
With a plastic knife and fork.
That's progress, right?
Two and a half dancing,
tux, tails, champagne.
Now, like this.
[CROUCHES]
May sixth, 1937.
Arrival of first flight to
the U.S. while docking in
Lakehurst, New Jersey.
Why were they docking in
Lakehurst, New Jersey?
If you go to the top of the
Empire State Building, look
and you will see at the
corners moorings.
Moorings sticking out.
The plan was to dock airships at
the Empire State Building.
So you'd come in from Europe,
you'd dock at Fifth Avenue,
get on the elevator,
and there you were.
When they tried to dock the air
currents were so violent
that they couldn't safely
dock the ship.
So then they moved across to the
fair grounds at Lakehurst,
New Jersey, where obviously this
mooring is much closer to
the ground than the top of the
Empire State Building.
The wild currents,
they're bad, but
they're manageably bad.
At the top of the Empire State
Building, impossible.
There's another image.
So what happened?
It did not explode.
It did not explode.
It couldn't explode.
Seven million cubic feet of
hydrogen to explode requires
seven million cubic feet of
oxygen instantaneously.
And air is 20% oxygen.
So it was a very violent fire,
roman candle from the point of
egress of the hydrogen.
Most of the people on board
walked off the Hindenburg.
Most of the people walked off
the Hindenburg uninjured.
They think it was electrical
discharge in the vicinity of a
hydrogen leak.
Recent research has indicated
the skin was made of resin
finished with a lacquer dope.
And then to make it shiny they
put aluminum powder.
And why they put iron oxide on
the inside I don't know, but
this is what NASA uses for solid
rocket motor grains.
So when this thing catches
fire, this is a thermite
reaction and could
be very violent.
And this spelled the end
of rigid error ships in
commercial air transportation.
Now this a U.S. Navy airship
filled with helium.
And there was a small gasoline
fire, look what happened.
Again, it was the skin.
That's a blow up of that one.
So I looked at that and I
thought, geez, that looks
Lichtenstein, doesn't it?
You know this one?
This one.
Look at that.
Look at that.
So you know, I can be an
artist, too, right?
OK, I'm going to tell
you one more story.
Another Niels Bohr story.
So in 1896 there was a guy,
and astronomer at Harvard
called Pickering.
Pickering at Harvard, 1896, and
he was studying the lines
in star light.
And he attributed to some of
the spectra that he was
getting, he said he was seeing
atomic hydrogen in star light.
And then there was a fellow
in London called Fowler.
And Fowler, in 1912, reproduced
the experiments in
the laboratory.
He put gas in a tube and got
the same thing in the lab.
So this guy's at Harvard and
the other guy is at London.
Well, Bohr looks at this
stuff and he says,
you guys are wrong.
You guys are wrong--
your lines are off
by a factor of 4x.
You've got the right series, but
you got the wrong element.
What you guys are looking
at is helium plus.
You're not looking at hydrogen,
and you know, from--
it goes to z squared.
So the lines are going to be
shifted by factor of four
because this z is equal to 2.
So Fowler was a pompous ass
and he didn't like being
called on his bad science.
So he does a calculation and he
looks more carefully and he
says, Bohr, you're wrong.
In fact, our lines are
off by 4.0016.
Now don't laugh.
The reason is the spectroscopy
was so precise that they could
go to five significant
figures.
So Bohr says, hmm.
And he goes back and he says,
you know, we've been doing all
these calculations with a one
electron atom just neglecting
the center.
So he redoes the calculations
for the entire Bohr model,
including considerations of the
mass of the nucleus and
the mass of the electron in the
form of the reduced mass.
The reduced mass is-- you're
learning this in--
the reciprocal of the sum
of the reciprocals.
And when he does that, he gets
that the value of the line
shift should be 4.00163.
So he says, you guys
are wrong.
It should be 4.0016.
You got 4.0016, you idiots.
You're looking at helium plus.
That was Bohr.
Did not want to get into
an argument with Bohr.
All right.
Have a nice weekend.
