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ROBERT FIELD: All right.
So last time, I said
that atomic sizes
are interesting or useful to
keep in mind, because you want
numbers for them, which are
somewhere between 0.1 and 100,
or something like that.
Because then you have a sense
for how big everything is,
and you're in the
right ballpark.
If you have to remember
a number and an exponent,
it's a little trickier.
And so I'm going to say
something about atomic sizes
in just a few minutes, but the
main things from the previous
lecture were this relationship
between the wavelength
and the momentum, which is true
for both waves and particles,
or things that we think
are wave-like, like light,
and things that we think are
particle-like, like electrons--
and so this unifying principle.
Then I have a little question.
And that is suppose the
wavelength for a particle
is known, and suppose
we have n particles.
And so if we say each particle
requires a volume of lambda
cubed, and there
are n particles,
and we stick them into a
volume smaller than n times
lambda cubed, what's
going to happen?
Yes?
AUDIENCE: They interfere
with the structure?
ROBERT FIELD: Their
identities get corrupted,
and the person who does
that for the first time
gets a Nobel Prize, and
that was Wolfgang Ketterle--
and others.
So when you have quantum
mechanical particles
that are too close
together, they
lose their individual identity.
And so this is a
very simple thing
that anybody who was beginning
to understand quantum
mechanics in the
1920s would say,
this is a puzzling thing, maybe
we should think about that.
And it's really hard,
so it took a long time.
Then a whole bunch of
experiments led to the idea
that we need to have a way for
atoms to fill space and satisfy
all of the other stuff, and that
was Rutherford's planetary atom
picture.
The problem with that
is it does fill space,
but there's no way
for the electron
to continue orbiting around
a nucleus, because it
will radiate its energy, and
fall into the nucleus, and game
over.
And so Bohr and De Broglie
both had ways of fixing this.
And Bohr's way was simply to
say the angular momentum is not
just conserved, but it
has certain values--
an integer times h bar,
h bar is h over 2 pi.
And De Broglie said in
order to keep the electron
from annihilating itself,
since it's going around
in a circular orbit of
known circumference,
there must be an integer
number of wavelengths
around that orbit.
Now, that's a much more
physical and reasonable ad
hoc explanation, but
it's still ad hoc.
It assumes that
the particles are
moving around circular orbits.
I hinted that the
way out of this
is going to be that the
particles aren't moving.
Then we could still
have angular momentum.
We can still have all
sorts of useful stuff,
but they're not moving,
and we don't radiate
the energy of the particles.
But that requires a
completely new way
of looking at particles
in quantum mechanics.
But the thing about both the
Bohr hypothesis and the De
Broglie hypothesis is
that any sophomore who
made such a proposal
would be laughed
at in the 1910-1920 period,
because it's just ridiculous.
But the reason these hypotheses
were taken seriously is that
this planetary model with those
corrections explain the spectra
of all one-electron systems--
hydrogen, helium plus,
lithium two plus,
uranium 91 plus, all
of them, to better
than measurement
accuracy at the time,
and better than measurement
accuracy for a long time
after this was proposed.
So getting a whole bunch--
now, a whole bunch is an
infinite number, actually--
of 10 digit numbers makes you
think there's something here,
but nobody knows where
here is except the numbers.
And the idea of the planetary
model with the fixes
doesn't explain anything other
than the spectral lines, which
is a lot, but it tells you
there's something really good
there.
I guess one thing I forgot
to mention in the summary
is that the energy levels that
you get from the Bohr model
are going to explain
spectra if you
say a spectrum are transitions
between these energy levels.
And that was also a
brilliant suggestion,
and it was suggested
by numerology.
But let us go back to work now.
We're going to talk about
the two-slit experiment.
And I have a personal
thing to report about that.
The first time I
gave a talk, it was
going to be a 15-minute talk
at a spectroscopy conference,
and so I did a
practice talk, and it
had to do with the two-slit
experiment in relationship
to spectroscopy.
But I did a practice talk,
and it took two hours.
So I have a thing about
the two-slit experiment.
And I think this lecture is
going to be not two hours.
It's going to be on time.
So we're going to talk about
the two-slit experiment,
and the important thing
about the two-slit experiment
is that it's mostly
ordinary wave interference.
There is no quantum mechanics,
and so most of the hard stuff
to analyze is classical physics.
And I'm going to do the
best I can with that.
But after we get to
understanding this problem,
there will be a surprise
at the end, which
is a quantum surprise.
And it's something that
absolutely requires
a postulate, the first
postulate of quantum mechanics,
some idea of what we
are talking about here.
What are we allowed to
know about a system?
And it tells you
there's something there.
And then I'm going to give you a
little description of something
which I think I have to
call a semi-classical optics
uncertainty principle.
Semi-classical or semi-anything
is usually quantum mechanics,
just a little bit,
mixed into something
that was well understood
before, or that
is very convenient
to use, and you only
bring in quantum mechanics
when you have to.
And it's the easiest
thing to understand.
And that introduces the
uncertainty principle,
and we get a taste of the first
of several quantum mechanical
postulates.
So let's start with the sizes.
I should have given you these
sizes in the previous lecture,
but I was being
mostly number-free.
So the radius of a Bohr
orbit is given by n
over z squared, where z is the
integer charge on the nucleus,
times 0.5292 angstroms.
So half an angstrom is the
radius of, basically, any atom.
And this charge on the
nucleus, it gets smaller.
And this is a very useful thing.
The wavelength is given by n
over z times 3.32 angstroms.
And that's the
situation where n times
the wavelength is
equal to 2 pi rn.
So we have n wavelengths
around an orbit,
and this is really the
De Broglie hypothesis.
But again, something which
is on the order of something
that you can remember.
And the energy levels--
these energy levels are z
squared times 13.6 electron
volts over n squared.
If we were talking about
energies in joules,
or in any units you want, it's
likely to have a big exponent.
But this one is
in electron volts,
and that's actually
what's happening.
An electron is being
attracted to a positive thing,
and there's, basically,
a voltage difference.
And so that's
another useful thing.
Now this z--
I left out the Rydberg.
This Rydberg constant--
there's a bunch
of fundamental constants, and
since I'm a spectroscopist,
I think in terms
of wave numbers,
reciprocal centimeters.
And for me, that's energy, it's
frequency, it's everything,
but anyway, for hydrogen,
it's 1097677.581.
For something with an
infinite mass nucleus,
it's 109737.3153
reciprocal centimeters.
And this is the number
that I have in my head,
and I use it in all
sorts of places,
and you can imagine where.
And to get to any
particular nucleus,
this is just r infinity
mu nucleus, or mu atom,
over the mass of the electron.
And it turns out that almost
everything except hydrogen
is very close to this number.
And so this hardly
matters, but it
does give you a little bit of
dependence on the nuclear mass.
So I said before that
this Rydberg equation,
or this equation,
tells you nothing.
It tells you where all
the energy levels are,
and anyone could tell
you where the rest are.
So it's a pattern,
which is nice,
but a pattern which says
if things are well-behaved,
like hydrogen atom, this is
what the energy levels will be.
But life is difficult.
Life is not with everything
well-behaved, and so
this is a pattern which
says I'm interested in
how the real life is
different from that pattern.
It's a way of thinking
about structure
and how we learn
about structure.
Information about the details
of a molecule or an atom
is encoded in the spectrum,
and this is the magic decoder,
or one of the magic
decoders, we use
to begin to assemble
the new insights.
This doesn't appear
in textbooks.
In textbooks, you get the
equations, you get the truth,
and you don't get any
strange interpretations.
That's what you're
getting from me.
You're getting strange
interpretations,
and you'll have them
throughout the course.
So now what we
want to do is talk
about the two-split experiment.
Let's just begin.
So here's a diagram.
And here we have
a source of light.
It's a light bulb in your notes.
It's a candle here.
And then we have two
slits, and the slits
are separated by distance d.
And so this is the first slit,
s1, and the second slit, s2.
And the distance between
them is much larger
than the width of each slit.
And now we go down
to the screen.
And the distance from the
slits to the screen is l,
and l is much,
much larger than d.
This means, of
course, we're going
to be using small-angle
approximation
and simple solutions, because
everything is much larger
than something else.
And that's very convenient.
Now I want to just put on axes.
So this is the x-axis, and
this is 0, and this is l.
And now the screen--
we're going to see
something on the screen that
looks like this.
I'm giving it away.
And the distance here is
0, and the distance here
is the z-axis.
So the distance to this
slit is on the x-axis,
and the pattern, the diffraction
pattern, is on the z-axis.
And this 0 is right in
the middle of the pattern.
It would correspond to this
point, the midpoint here.
So what we want
to do is calculate
what's going to
appear on the screen,
and I've already given it away.
What you see is a bunch of
equally-spaced intensity maxima
where we have
constructive interference.
And in between, we have less
constructive interference--
or destructive interference,
and we want to understand that.
Now this is optics.
This is no quantum
mechanics at all.
Let's look at this
in more detail.
So we have the
z-axis, horizontal,
and we have a path
to the screen,
and another path to the screen.
So what we're interested
in is here's one slit,
here's the other slit, and
we have two parallel lines
that meet at infinity, which
is where the screen is--
and what we're going
to be interested in
is what is the path difference
between this one and this one.
So we have an angle which is
given by the perpendicular
to this right.
So this distance is d.
This distance is
L. And this angle
is theta, as is this angle.
And what we're
interested in is this--
the extra path traveled
by the lower slit.
So we use trigonometry,
and we can figure that out.
And so delta is the
path difference.
Delta is equal to d sine theta.
And in order for it to be
constructive interference,
we have to have this path
difference to be an integer
number of wavelengths.
Now this is optics,
and so that's something
that we don't need
quantum mechanics for.
We know we're going
to get interference.
So we can now solve for where
the constructive interference
occurs.
And so we have theta n is
equal to the inverse sine of n
lambda over d.
But this is a small angle,
even though I drew it
not as a small angle.
And so we can replace sine x
by x or inverse of sine y by y.
And anyway, we can
say that the angles
for constructive interference
are given by n lambda over d.
So we've derived the
diffraction equation.
We've solved.
And now what we want to know
is where do the spots occur.
They occur at z equals 0,
z equals plus or minus l
sine theta, which is
approximately equal
to l over d n lambda.
So we have l times lambda
over d times an integer.
So what we're going
to see on the screen
is a series of bright lines
for constructive interference--
and they're not lines.
It's a curve.
But we can measure the
maximum of the intensity,
and we can say
they're like this,
and they're equally spaced.
And they tell us things we knew.
We knew l.
We knew d.
We knew lambda.
So there's nothing surprising.
This is just optics.
So now suppose we go in
and we cover one slit.
What happens?
Yes?
AUDIENCE: The
interference stops?
ROBERT FIELD: The
interference goes away.
And you could imagine that
there would be a little sign.
If you covered the
top slit, the pattern
would be skewed a little
bit in the direction
of the bottom slit.
And so there'll be a
little bit of asymmetry,
but you could
actually know which
slit your colleague covered.
So if both slits are open,
you have interference.
If one slit is covered,
you have no interference.
We're getting into the
realm of quantum mechanics.
In quantum mechanics, one of
the things we do is we say,
suppose we did a
perfect experiment.
Maybe it's an experiment
that's beyond what you're
capable of doing with
the present technology,
but you can say, I could
measure positions in time
as accurately as I
want, and one could also
say that I could do this an
infinite number of times.
I could do the same experiment
an infinite number of times.
Without quantum mechanics, if
you did the same experiment
an infinite number
of times, you'd
get the same answer an
infinite number of times.
But with quantum mechanics,
you're going to discover you
don't.
It's probabilistic,
not deterministic.
And under certain conditions,
the range over which you have
a finite probability
is very small,
and it looks deterministic,
but it isn't.
The perfect experiment business
is an interesting hypothesis,
but you can imagine
defining what
is perfect in terms of
what is intrinsically
possible to achieve, even if
it's not currently possible.
So what we want to do is
decrease the intensity
of the light that's
going into the apparatus,
so that there is never more than
one photon in the apparatus.
Never is a strong word,
and we could never do that,
and so that's not legal.
But we can say, suppose
we decrease the intensity
so that for the time
it takes for the photon
to go from the slit to
the screen, which we know
because we know
the speed of light,
and for the intensity of the
light, which we can measure
with an energy meter, we can say
the probability of there being
more than one photon at a time
in the apparatus is small, as
small as we want, but not zero.
And so then we do
the experiment.
And what we discover when we
do the experiment is instead
of having a uniform intensity
or some kind of continuously
varying intensity on
the detector screen,
we get a series of dots--
events.
The photon went
in, and the photon
was a wave when it went in.
There was interference,
maybe, and the photon
died on this detector screen.
This is an example of
destructive detection, which
is something that is very
important in quantum mechanics,
because in quantum mechanics,
most measurements destroy
the system, or destroy the
state that the system was
in during the experiment.
So this business of what
is the state of the system
is a really important
quantum mechanical concept,
which you don't normally
encounter in classic mechanics.
We send photons one at a
time through the apparatus,
and we get something like this.
And we get something like this
whether both slits are opened
or one slit is covered.
So we do this, and we
do this for a long time,
and what we see is we
see a lot of events,
and they're starting
to arrange themselves
where the interference
fringes were supposed to be.
So this pattern only
emerges after you'd
allow a large number of photons
to go into the apparatus.
There is no way classical
optics gives you that.
And so if one slit
is covered, you
get a uniform
distribution of dots.
If both slits are open, you get
this kind of a distribution.
I'm going to ask
you to vote on this.
So we do the experiment--
and I've sort of
given away the answer,
but I still want
you to vote on this.
What are the possible things?
So we know there's
only one photon
in the apparatus at a time.
Our concept of interference is
light interference with itself.
And this is a possibility
that says, I think I know.
And here is another,
weak interference
on top of constant background.
This would reflect.
Even though we
decided that we would
have one photon
in the apparatus,
occasionally there are two.
And when there's two, there
could be interference,
and so we'd have some weak
interference superimposed
on the constant background.
Now we get this 100% modulated
interference structure.
And the last thing
is something else.
You have to transport yourself
back in time to around 1910.
You haven't heard
this lecture, but you
do know what the experiment is.
And so what would you expect?
First, no interference,
raise your hand.
I've got one-- two--
I've got a few votes
for no interference.
In 1910, that's what
you would have said.
Weak interference on top
of constant background--
that would be when you're
hedging your bets and saying,
the experiment,
it isn't perfect,
and this is really
the right answer,
but if someone was a little
sloppy, I'd get this.
Raise your hands for this one.
What would you
have said in 1910?
I got nobody with the
courage to say this.
You would have
gotten a Nobel Prize
if you could have defended it.
Something else-- maybe something
else-- maybe the photons
come in pairs or
something ridiculous.
So the correct answer
is 100% modulated.
What people would have
said in 1910 is this.
Some curmudgeons, like the
climate change deniers,
would have said there is a
little bit of this or maybe
that, but I can't
possibly accept
that, which is the truth.
We won't belabor that anymore.
This means that one photon
can interfere with itself.
It's a very disturbing
idea, but it
leads to a critical idea
in quantum mechanics.
In quantum mechanics,
the state of the system
is described by some
state function, which is
a function of position in time.
And so what happens
is you prepare
the system in some state.
You do something
to it, like force
it to go through two slits.
Then we get some
new state function.
And then we detect it,
and we get something else.
So the actual experiment is
a click, the preparation,
and the click, detection.
And somehow, what is the
nature of the experiment
is expressed on this
initial state of the system.
This is all very abstract,
but it's about interference.
So this guy had
better have phase.
So we have a wave that can
constructively or destructively
interfere with itself.
And so we start talking
about things like amplitude,
but the crucial
word is amplitude.
And mostly, when
we detect things,
we're detecting probability.
This is always positive.
These guys can be
positive and negative.
This is essential for
quantum mechanics.
It's essential for understanding
the two-slit experiment,
but we have to do an awful lot
more to make all this concrete.
Why don't we look at the
classical wave equation?
Actually, we'll look at
the classical wave equation
next time, but I will say
that the solution to the wave
equation is some
function of x and t,
which has the form a
sine kx minus omega t.
So I'm doing this
to introduce you
to the crucial actors in this
game, which is amplitude,
wave number, and frequency.
And this is a
probability amplitude.
It can have either sine, because
you can see the sine function.
So this is a wave of
frequency omega propagating
in the plus x direction.
Now let's just identify
the crucial quantity.
Wavelength is the
repeat distance.
So that if we said
we have u of x and t,
it has to be equal to u
of x plus lambda and t.
So that's how we
define the wavelength.
And we discover
that the wavelength
has to be related to k times
lambda is equal to 2 pi.
Because this part has to
change by 2 pi in order
for there to be an exact
replica of what we had before.
So we know that the wavelength
is the repeat distance,
and k is 2 pi over lambda.
It's called wave
number, and it's
the number of waves,
complete waves, that occur
in 2 pi times the unit length.
Now, in 3D, we have a vector
as opposed to a number.
And that points in the direction
of propagation of the wave.
We know from quantum mechanics--
or from the experiments--
that the wavelength is
related to the momentum.
There's several reasons
for this for waves.
This could still
be optics, but it
could have been
relativistic optics,
because Einstein proposed
that the momentum is e over c.
So we put that together,
and we get the relationship
between k and momentum.
Now h bar is h over 2 pi.
And so the wave number is
large if the momentum is large,
and the wavelength is small
if the momentum is large.
Now what about the velocity?
So we have a wave.
Let's sit on this
wave here and say,
we're now sitting on some point
where the phase is constant.
I like to call this the
stationary phase point,
but that has other
meanings, so you just
have to be careful with this.
So we want to know how
fast this moves in space.
And so we say, the phase is the
phase of this function here,
and so it's k x--
I'll put a little phi on this--
minus omega t.
And we want to
find how this moves
in time, this stationary phase.
We could choose this to be zero.
We could choose a phase.
This is zero, this
is some maximum,
but we can choose
anything we want.
So let's make it zero.
And so then we can solve for
x phi is a function of t,
and that's omega t over k.
We want the phase velocity,
the velocity of this wave.
So we take the derivative
with respect to t.
And so the velocity,
which we'll call c,
is equal to omega over k.
That's true for light
traveling in a vacuum.
This is called the
dispersion relation.
Whenever you do a calculation
of waves in material,
the goal is to get the
dispersion relation.
This is the simplest possible
one, because it says everybody
at the same frequency--
I'm sorry-- that there is a
relationship between omega
and k so that everybody
travels with the same speed.
If that didn't happen,
the waves would get out
of phase with each other.
That's what's dispersion is.
So we have from simple optics,
basically, everything we need.
And now, the last thing is
that the intensity of this wave
is given by kkix
minus omega i t.
So this is the wave function.
It's something that has sines.
And this is the
intensity, which is
proportional to this quantity.
We sum the amplitudes
and then square.
We do this in quantum
mechanics all the time.
And this is like quantum
mechanics in the sense
that we have our fundamental
building block, which
is something with phase.
And the relationship
of the thing
with phase to the thing
which is probability
is sum of the square.
Yes?
AUDIENCE: What's the first
character after the capital
sigma?
ROBERT FIELD: I'm sorry?
AUDIENCE: The first character
after the sum there?
ROBERT FIELD: This?
No, this.
That's a constant.
That's the amplitude of that
particular frequency and wave
vector.
I'm sorry about the
mess on the board.
We have enough time.
Oh, good.
So this is sort of a
taste of what you're
going to be doing,
but now let's produce
a form of the
uncertainty principle.
You've heard about the
uncertainty principle,
and it's a very important
part of quantum mechanics,
but it's also something
that you can have in optics.
And so again, we resort
to this simple idea
of sending a particle
through a slit
and looking at what
happens over here.
Instead of having two
slits, we just have one.
But there are two
important things.
The two edges of this slit are
special, because they're edges.
And so we can ask, what
about the interference
between particles
that was diffracted
by this edge versus that edge.
And the same thing goes.
You get constructive
interference
when the difference
in path length
is an integer number of waves,
and destructive interference
when it's an odd integer
number of half waves.
We're interested in the
destructive interference.
So we analyze this in
exactly the same way
we did the two-slit experiment.
And now this is the z direction.
And this is the x direction.
And what you end up finding
is that the uncertainty
in the z direction
is going to be
related 2 times lambda
over delta s over l.
Delta s is the width
of the slit, so width
of the image along the z-axis.
So we have a maximum here, and
we have minima here and here,
and so this delta z is the
distance between minima.
So we can say, between minima,
we have the particle localized.
Its position in space--
or the photon localized--
its position in space
is uncertain by this quantity.
What about its momentum?
And so now we just have to draw
a little bit of conservation
of momentum.
So here we have the magnitude
of the momentum starting
from the middle of the
slit, and this magnitude
is constant along a circle--
the magnitude.
So if we do draw a line
here, this has got p.
And if we ask the length
here, that's also p.
But now what we
want to know is what
is the uncertainty of the
momentum in the z direction--
delta pz.
This is the same sort of
calculation we did before,
and what you find is
delta pz is approximately
equal to magnitude of
p lambda or delta s.
And now we put in the
Bohr relationship here,
and we get that
this is equal to h
over lambda, lambda over the
ds, which is equal to h over ps.
Or ds in the z direction--
no, ds is the slit width.
I got something wrong
here in my notes.
The final result
of this calculation
is dz is dpz
approximately equal to h.
Sorry about the glitch here.
I don't know how to
fix it right now,
but if this were
done correctly, we
would have gotten this result.
This is an
uncertainty principle.
If you tried to measure
z and pz simultaneously
with classical optics, you still
would get something like this.
Using the relationship
for momentum
that lambda is equal h over
p, we've got to use that.
But because of
that relationship,
we get this result.
If you do a perfect experiment,
and you make the slit smaller
and smaller, you make the
uncertainty in the momentum
larger and larger.
The best you can do is this.
Now, this is actually a
reasonable and rigorous
derivation if I had
done it a little better.
But I've never liked this
introduction to the uncertainty
principle, because it says,
for the kind of experiment
we thought about, you
can't do better than this.
Maybe there's a different
kind of experiment.
It's sort of an
artifactual as opposed
to a physical derivation.
We care.
We will do a physical derivation
of the uncertainty principle.
And it will have to do
with the ability of two
operates to commute
with each other.
That's a purely
mathematical definition,
but this is the first sign
of the uncertainty principle.
At the end of your
notes, there is
a set of postulates from
which, essentially, all quantum
mechanics can be derived.
Now, there are different
sets of postulates proposed
by different people,
but these are
things that can't be proven.
They are things that you
think is going to be true,
and then you look
at the consequences.
The first postulate
says, the state
of a quantum mechanical
system is completely
specified by a function,
psi of r and t,
that depends on the coordinates
of the particle and on time.
This function, called the wave
function or the state function,
has the important property
that this quantity
times its complex conjugate
integrated over the volume
element is the probability
that the particle lies
in the volume element
centered at r at time t.
So we are saying there
is a way, a complete way,
of describing the
state of the system.
It's a function.
It's not a bunch of
discrete quantities,
like velocity and position.
And things are smeared out.
And what we want to do when
we want to calculate anything,
we're going to be
using this function.
And this function comes from
the Schrodinger equation.
And we're going to get to
the Schrodinger equation--
not in the next lecture.
I'm going to spend the next
lecture really laboring
the wave equation, because the
Schrodinger equation is just
a tiny step beyond
the wave equation.
I've given you the
five postulates.
You have not a clue
what any of them
mean, except maybe a little
bit about the first one.
But what I don't want
to do is give a lecture
on the postulates.
I want to bring them into
action when you need them,
because they'll mean much more.
And so you won't be asked
to memorize the postulates.
You'll know when
they are applicable
and how to apply them.
So it's a little
premature to say
you understand the
first postulate,
but that's what's at play here
in the two-slit experiment
and in this hand waving
derivation of the uncertainty
principle.
So next time, we will
look at the wave equation.
Thank you.
Hey, I finished
on time this time.
That's a bad sign.
