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PROFESSOR: Hi everyone.
Spring has regressed.
So, we have--
we're going to have
a guest at the end of
lecture today, which should
kind of entertaining.
Just as a warning, if
you see someone come in.
So questions, before
we get started?
No questions about anything?
At all?
Math?
Nothing?
Yeah?
AUDIENCE: Can you explain
the physical significance
of the crystal momentum?
PROFESSOR: Yeah.
OK.
Let me go over that.
That's a good question.
So the question is what again is
the significance of the crystal
momentum?
So let me answer that in
a slightly backward way.
So this is a form
of the explanation
I haven't given you.
It's going to be a
slightly different one.
Let's step back and
think about the momentum,
and ask what the momentum is.
Now you guys showed on a
problem set, the following fact.
That if you have a wave
function, sine of x, such that,
the expectation value in the
state SI of x in the state SI
is equal to x naught,
and the expectation value
of p in the state
SI is p naught.
Hat, hat.
Then if you want to change the
momentum, increase momentum
by h bar k, the way to do
that is to take SI and build
a new wave function,
SI tilda, is
equal to e to the
i, k x, SI of x.
And then the
expectation value of x
is the same, SI tilda,
still equal to x naught,
because this phase goes
away from the two complex,
from the wave function
is complex content
we give the inner product.
But the expectation
value, the momentum
is shifted in state SI tilda,
is shifted by each h bar
k, p naught plus h bar k.
So all the intuition
you have about momentum,
you can translate into intuition
about the spatial variation
of the phase of
the wave function.
Yeah?
AUDIENCE: [INAUDIBLE]
PROFESSOR: OK, good.
OK, good, So we have
a sneaky [INAUDIBLE].
So, the information
about the momentum
can be encoded in
these spatial variation
of the phase of
the wave function.
So another way to answer the
question of what is momentum,
apart from it's the
thing that-- so what
are ways to answer the
question, what is momentum,
you could ask well
what is momentum?
It's the thing that commutes
with p or with x by i h bar.
That's one way to answer it.
Another way to answer is to
say that translations by l
can be expressed in terms of
momentum as e to the minus i
upon h bar p l.
So these are both
ways of describing
what the momentum is.
But another way of
talking about the momentum
is the momentum p governance
the spatial variation,
the x dependence of the
phase of the wave function.
So these are always talking
about what the momentum is.
So now let's turn
this around, and let's
ask about the crystal momentum.
Oh, and one last thing,
a last defining property
of the momentum,
a central property
from the Schrodinger equation is
at the time variation d dt of p
is equal to the expectation
value of minus d the potential
of x d x.
Also known as the force.
So this is the Ehrenfest Theorem
Statement that the classical
equation of motion, p dot, is
equal to the minus d v d x is
equal to the force, Ehrenfest's
Theorem tells us that the
classical equations of motion
are realized as
expectation values.
And equivantly, if
there's no potential,
the potential is
constant, this tells us
that the momentum expectation
value is time independent.
Right?
A familiar fact.
So these are all true lovely
and things about the momentum.
So let's turn all
these facts around
into the crystal momentum.
So let's talk about
crystal momentum.
Which was the question, what
is the crystal momentum?
So the crystal momentum
is defined from beginning,
from the following property.
If we have a potential
v of x, which
is invariant under shifting, by
one lattice spacing, by some l,
v of x, then this tells us
that the energy operator
is invariant if we shift by l.
If we translate
by l equals zero.
And from this fact, we deduced
via block or a la block,
that the wave functions
are really the energy
eigenfunctions, can
be written in the form
e cubed is equal to e to the
i q x, u of x, where u, we're
going to take to be
a periodic function.
So what is this
parameter q doing?
Q is governing the
spatial variation
of the phase of
the wave function.
Cool?
So in precisely this sense,
the momentum difference
is space of the wave function.
Here, in the case of
a periodic potential,
the crystal momentum q is
governing the spatial variation
of the phase of
the wave function.
So q is the thing the governs
the phase as a function of x.
Well what about-- another fact
about the crystal momentum
which you show in
your problems set,
is that if you impose an
external force d q d t,
and really d h bar q.
d t is equal to-- d
dt of the expectation
value of h bar q,
is equal to the expectation
value of the force.
I'll just write-- OK?
So again, this is a
quantity, and this
was assuming that we had a
sharply peaked wave packet.
So this is for a wave packet
sharply peaked at q naught.
And so let me just write
this as h bar q naught.
So the central value
of your wave packet--
so this is what you've
shown on the problem
set that the central
value of your wave
packet, the peak
of your wave packet
varies in time according
to the external force.
And so in particular,
if the force is zero,
we turn no external
driving force,
your wave packet maintains
its crystal momentum.
It's time independent.
So the crystal
momentum is something
that time independent, unless
an external force is applied,
just like the momentum.
And it's something that
governs the phase of the wave
function just like the momentum.
However, it's different
in a crucial way.
It is not the eigenvalue
p on five sub e q
is not equal to a constant
p naught times 5 sub e q.
Because when we take--
when we active p
or we active the derivative,
you pick up a term
from here, which
gives us a constant,
but we also have this
overall periodic piece.
And its spatial variation
is generically non-zero.
And if the potential
is nontrivial,
it's always non constant.
So when the momentum
operator hits this guy,
it will generically
not give us zero.
It'll get two terms and we will
not get an eigenvalue equation.
So q is not the eigenvalue h bar
q is not the eigenvalue of p.
And what's the last
important property
of q that's different
from the momentum?
It comes from the
commutator, which tells us
that the thing that's conserved
is the expectation value of p l
is really the precise statement.
And in particular,
what this tells
us is that the eigenfunction,
or the eigenvalue of our wave
function, under
translations by l,
is a quantity that can be
determined simultaneously
with knowing the energy.
However, the eigenvalue
of t sub l, on this state,
is equal to e to the i q l.
Which means that
q is only defined
for determining the
eigenvalue up to 2 pi over l.
If you have q, which is 0, and
you increase it to pi over l,
that value, pi over
l, is effectively
the same as the value
minus pi over l.
Because at least they're
the same eigenvalue.
But that's really
strange because that
means that q itself, it's
not strictly conserved.
It's conserved mod 2 pi over l.
When you have
momentum conservation,
momentum is strictly
conserved if there's no force.
And even if there is a force,
it's increasing control
by the force as you
turn on the force,
it just constantly increases.
For the crystal momentum,
that's not the case.
You turn on a
force, it increases
according to the
conservation law.
But it's not
increasing constantly.
It's periodic.
It's periodically defined.
So it increases then it
ends up at a smaller value.
It increases and ends
up at a smaller value.
OK?
So it carries many of
the same properties.
It governs the phase.
It's time independent
unless there's
an external force applied.
It's the eigenvalue.
Controls the eigenvalue
of an operator that
commutes with the energy when
you have a periodic potential,
in the same way that
the momentum commutes
with the energy when you
have no external force,
when you have a
constant potential.
Does that help?
Good.
OK.
So developing an intuition for
the crystal momentum, I think,
is best done by just
playing with examples.
And you'll do that
more in the course
on solids, which I
encourage you all to take.
Because it's really
beautiful stuff.
But for our purposes,
this is going
to be the full set of
ideas we'll need for 8.04.
Yeah?
AUDIENCE: [INAUDIBLE]
PROFESSOR: Ah.
So good.
So thank you.
So this involves
a slight subtlety,
which I've been glossing over
in the entire story here.
Which of the following.
So, is u of x a real function?
Well, so when we
started out asking
what are the eigenfunctions
of the transit by l operator,
all we showed was
that, and I'm going
to do this on a separate
board just to make it clearer.
Tell me if this turns off,
because it kept bumping.
OK.
So when we started
with translate by l,
and we constructed
it's eigenfunctions,
we said that translate
by l q Phi sub cubed
is equal to some phase,
and this is unitary,
so we're talking about they
must be an actual phase in the i
alpha of Phi sub q of x.
And let's just suppose
that this is true.
Then this tells us that Phi sub
q times e to the minus i q l
equals u.
So, I'm just going to use this
to define a new function, u sub
q.
Or just u.
I'll use sub q.
Fine. of x.
So this defines a
new function, sub q.
I take an eigenfunction, I
multiply it by some phase.
Sorry, minus i q x.
If we choose q l to be equal to
alpha, then acting on u sub q,
by translate by l,
on u sub q, of x,
is equal to-- well, if we act
on Phi sub q with translate
by l, what happens to Phi sub q
we pick up a phase e d i alpha.
What happens to e
to the minus i q x?
x goes to x plus l.
We pick up a phase e
to the minus i q l.
So if q l is equal to alpha,
those two phases cancel,
and we just get u
back. u sub q of x.
But translate by l, if
u sub q, by definition,
is equal to u sub q of x plus l.
So we've determined is that if
we take q l is equal to alpha,
then Phi sub q if eigenvalue
label by its eigenvalue, q,
can be written in
the form e to the i q
x u sub q of x, where
this is periodic.
Everybody agree with that?
OK.
So that's step one.
Step two is to say well
look, since the eigenvalue
of this guy, under t
sub l, e d i alpha is
equal to e to the i q l.
Since this is periodic under
shifts of q, by 2 pi upon l,
I can just choose to
define q up to 2 pi over l.
So 2 q, I will take
to be equivalent to q
plus 2 pi over l.
And the reason I'm
going to do that
is because it gives
the same eigenvalue,
and if I want to label
things by eigenvalues,
it's sort of redundant
to give multiple values
to the same eigenvalue.
Now there's a
subtlety, here though.
And this little
thing here is this.
Suppose we have a free particle.
Does a free particle
respect translation by l?
So if we have a free particle,
the potential is zero.
That constant function is also
periodic under shifts by l.
Right?
Because it's just zero.
So it's stupidly periodic,
but it's periodic nonetheless.
So now I'm going to ask
the following question.
What are the common
eigenfunctions
of the energy and translate
by l for the free particle?
We did this last time.
So the common eigenfunctions of
translate by l and the energy
are the wave functions
Phi sub q, comma e,
are equal to e to the i
q x times some function
u of x, on general grounds.
But we know what these
eigenfunctions are.
They're just e to the i k x.
Where k squared upon 2 m is e.
[INAUDIBLE] So we
know that these
are the correct eigenfunctions,
but we're writing them
in the form e v i q x u.
Now you say that's fine.
There's nothing wrong with this.
We just say u is constant
and q is equal to k.
These functions
are of this form,
but they're of this
form with e v i q x
being e d i k x and u
of x being constant.
Right?
There's nothing wrong with that.
Everyone agree?
Perfectly consistent.
However, I thought we said
that q is periodic by 2 pi?
If q is periodic
by 2 pi, then that
would seem to imply that
k is periodic by 2 pi,
and we know that's not
true because any k is
allowed for a free particle.
So if we want to think about
q is periodic by 2 pi upon l,
then we cannot require
that u is real.
Because it must be the
phase that makes this up.
It must be, so I can always
write this as e to the i q x
where q is less
than 2 pi upon l.
I'm sorry, where q is between
2 pi or pi upon l and minus pi
upon l.
So that it's defined only
after this periodically thing.
But times some additional
phase, e to the i k minus q
x This is trivially
equal to e to the i k x.
But now u is not
a real function.
On the other hand, if we
hadn't imposed the requirement
that q is periodic, we wouldn't
have needed to make u real.
We could just taken q to be
equal to k, for any value k,
and then u would be
constant. u would be real.
So this is important
for answering
the excellent question
that our fearless restation
instructor provoked
me to answer.
Which is that so
what-- we'll come back
to the question
in just a second.
But what I want
to emphasize this,
that if we're going to take
q to be not periodic, Sorry.
If we're going to take q to
be defined only up to shifts
by 2 pi over l, it's important
that we allow u to be not real.
It must be able to
be an overall phase.
But if we want u to be
always real, we can do that.
We just can't impose
this periodicity.
Different values of q mean
different wave functions.
And this is really what's going
on when you see those plots,
sometimes you see the
plots as parabolas.
The bands are represented
by parabolas with wiggles,
and sometimes they're folded up.
And that's the difference.
The difference is that
when you fold them up,
you're imposing this
periodicity and you're
labeling the
eigenfunctions by q,
and the overall amount of the
number effectively of k phases
that you're subtracting off.
Yeah?
AUDIENCE: So is this an
arbitrary choice? [INAUDIBLE]
PROFESSOR: Yeah.
I mean, how to say?
It's exactly akin to
a choice of variables.
In describing the
position of this particle,
should we use
Cartesian coordinates,
or should we use
Spherical coordinates?
Well it can't possibly matter.
And so you'd better make sure in
any description of your system,
that changing your coordinates
doesn't change your results.
And here, that's
exactly what's going on.
Do we want to define our
variable to be periodic by 2 pi
upon l?
Well, OK then.
But u can't be real.
Or we could take q to
be not periodic by 2 pi
l and impose that u is real.
It's just a choice of variables.
But it can't possibly
give different answers.
The point is, this is a
subtle little distinction it
we gloss over, and is glossed
over into my knowledge
every book on intro
to quantum mechanics
that even covers
periodic potentials.
It can be very confusing.
Anyway, the reason that I
had to go through all this,
is that in order to answer
the very, very good question
professor Evans posed, I'm going
to need to deal with this fact.
So for the moment, let me deal
with-- let's work with u real.
And q, q an unconstrained,
real number.
OK.
So not periodic.
Are we cool with
that for the moment?
So if we do that, then notice
the falling nice property
of our wave function.
Our wave function, Phi sub
q, is equal e to the i q
x times u of q, or u of x.
Which is real.
So when we can construct
the current-- remember
that j boils down to
the imaginary part,
h bar over 2 m i.
Well, h bar over m times
the imaginary part of SI
complex conjugate
derivative, with respect
to x, which is the current,
in the x direction of SI.
And we need this
to be imaginary,
or we will get no current.
You show this in a problem set,
if you have a pure, real wave
function, for example.
A single real
exponential, that's
decaying, as on the
wrong side of a barrier.
Then you get no current.
Nothing flows.
And that make sense.
It's exponentially decaying.
Nothing gets across.
So we need the wave
function to be real.
So if q were zero
we would get zero.
And what you can immediately do
from this, compute from this,
is that while the derivative,
if the derivative doesn't
hit e to the i q x, if it hits
u, than the phase e to the i q
x cancels.
And so the contribution
from that term vanishes.
So the only term that's
going to contribute in here
is when the derivative
hits the e to the i q x.
But then this is going
to be equal to h bar q.
And we want the
imaginary parts, that's
going to be e to the I over m.
And then we're left
with u squared of x.
So this is the
current, but we have
to do it-- we had take
advantage in order for this
to be sort of clean, we had to
take advantage of u being real.
Everybody cool with that?
Now there's one
last twist on this,
which is that if I
have k-- if I have q.
So this is a side note.
Going back up here,
to this logic.
If I have q, and I
want, I can always
write it as some q
naught plus n pi over l.
And so now what
I want to do is I
want to take sort of a
hybrid of these two pictures.
And I want to say
Phi sub q is going
to be equal to e to
the i q naught x.
Where this is the value
that's periodic by 2 pi.
e to the I n pi over l x u.
And so now really what's going
to happen, what I'm doing here
is I'm labeling q, not
by a single number.
I'm labeling my wave function
not by single number q,
but by q naught
and an integer n.
Comma n.
So q naught and n.
So now q naught is periodic.
It's defined up
to shifts by 2 pi.
n is an additional integer,
and what it's telling
you is how many times did
you have to shift back
to get into that fundamental
zone between pi and minus pi.
And this fits nicely
into this story,
because now all we're
going to get here
is q, which is q
naught plus n pi.
So the current depends on both
the part defined mod 2 pi over
l, and the integer,
which tells you
how many factors of 2 pi over
l did you have to subtract off
to get into that
fundamental domain.
So let's think back
to our band structure.
So what is this n quantity?
Let's think back to
our band structure.
In our band structure, we had
something that looks like this.
And here's the value of q.
But am I plotting q?
No.
I'm plotting here q naught.
I'm plotting the part
that's periodically
defined up to 2 pi over l.
So this is pi over l.
This is minus 2 pi over l.
Or minus pi over l.
OK.
And what we see is that
there isn't a single energy.
Because this is the energy the
vertical direction for the band
pictures.
There isn't a single energy
for a given value of q.
In fact, the set of
energy eigenvalue--
or the set of allowed
states or energy eigenvalues
for an allowed value of q
would say this particular value
of q naught, how many
of them are there.
Well, there are as many
as there are integers.
One, two, three, four, count.
So to specify a state, I don't
just have to specify q NAUGHT,
I also have to specify N.
Which one of these
guys I'm hitting.
And when you unfold this
into the parabola picture,
remember where these came from.
These came from these curves.
Came from shifting over.
And the higher up you go, the
more you had to shift over.
And that's exactly the
integer piece in n pi over l.
And so we can write
the current now,
in terms of h bar q naught
upon m, u squared-- I'm sorry.
h bar q naught upon m plus n
pi h bar upon m u squared of x.
So we get a contribution
from the crystal momentum
and from which we're in.
OK?
So sort of an elaborate story
to answer the phase question.
Yeah?
AUDIENCE: [INAUDIBLE]
PROFESSOR: Good.
So here we had SI--
so SI-- I'm sorry.
I should have done this for Phi.
But I meant this
wave function, right.
This is Phi, this is Phi q.
So from here we're going
to get the imaginary part.
So we get the imaginary
part of this wave function
which is u to the minus i
q x u of x derivative of e
to the i q x u of x.
Now the term that contributes
is when the derivative hits
the e to the i q.
x pulls down a factor of
i q, and the two phases
cancel from these guys,
leaving us with a u of x here,
and a u of x here.
AUDIENCE: [INAUDIBLE]
PROFESSOR: Oh sorry.
This is a potential.
Good.
That's the point.
So this is the potential.
So in this statement that what
we have this translation by x.
So this is just some function.
It has nothing to the potential.
It's defined in terms
of the wave function.
The eigenfunction
of translate by l.
So the logic here
goes, if we know
we have a function
of translate by l,
then I construct
a new function u.
Nothing to do with the
potential, just a new function.
Which is e to the
minus i q x times it.
You can't stop me.
You hand me a function, I will
hand you a different function.
And then we pick q felicitously,
to show that u is periodic.
So u is just some
periodic function
which is contained which is
defined from the wave function.
From the energy.
From eigenfunction of t l.
Did that answer your question?
OK.
So here, it just
came from the fact
that u is Phi is the e to
the i q x u, x and then
a factor of u for each of these.
Other questions.
Yeah.
AUDIENCE: [INAUDIBLE]
PROFESSOR: So this picture,
when it's unfolded,
first off, you know what
it is for a free particle.
So we want the energy
as a function of q.
So what is it for
a free particle?
Parabola.
Yeah, exactly.
And now let's add in-- let's
make this a function of q,
not q naught, but
so here's pi over l.
Here's 2 pi over l.
Here's 3 pi over l.
And I need to do this
carefully, because it's
incredibly difficult
to get the straight.
OK.
My artistic skills are not
exactly the thing of legend.
OK.
So here's the parabola
that would have been,
if we had not turned on
a periodic potential.
As we turn on the
periodic potential,
we know that the
energies change.
And so in the first
band it's easy to see,
because for minus pi
over l, it's pi over l.
We don't have to do anything.
So it look exactly the same
as the lowest band over here.
So in particular-- OK?
So what about this second band?
Well what I want to know what's
the allowed, the other allowed
energy that's say,
plus pi over l.
Plus pi over l, it's going
to be something greater
than this value.
But plus pi over l, we
already know the answer
from that diagram,
because plus pi over l is
the same as minus pi over l,
so what's the value over here?
Well, the value over
there for the second band
is slightly above, and then
it increases and decreases.
So slightly above,
and then it increases.
Shift by pi over l.
Whoops.
Did I shift by pi
over l for this guy?
That's one, two.
Yes.
I did.
Good.
And it goes the other way.
So just noting that it
goes away from the top.
I have a hard time
drawing these things.
So for every value of q,
there's an allowed energy.
But it's different
than it would have
been for the free particle.
And then we do the same
thing for the next state.
And it looks like this.
So now imagine what
happens when we take this,
and we it over one two.
What we get is a
band the looks--
that should look like this.
That's what the second
band should look like.
And indeed, when we put it
in the fundamental domain,
this is what we get.
This is what the first
band and the second band
together look like.
And then the third
band, we'll move this
over once, and then twice,
it's going to look like this.
And this guy, move it over
once, twice, looks like whoops.
Yeah?
AUDIENCE: If we wanted to plot
u with respect to k instead,
would that just be a
parabola dotted line?
If so, why do we
not have really--
PROFESSOR: If we
just wanted-- sorry.
Say it again?
AUDIENCE: E as a function
of k instead of q.
PROFESSOR: Oh.
Yeah.
E as a function of k is always
going to look like that.
But k is not a
well-- so what is k?
K is just defined as h bar
squared, k squared upon 2 m
is equal to e.
So this doesn't
tell you anything.
Right.
Because any allowed k.
Sure any allowed k is
some valid value of e.
But this didn't tell you
which values of e are allowed.
Only some values of
e are allowed, right?
There are no values of e--
there are no energy eigenstates
with energy in between
here and here, right?
And so that tells you they're
no allowed k's because k is just
defined, it's just
completely defined by e.
So this doesn't tell you
anything about which states
you're at.
It just that given an e,
there's some quantity that
could define k.
This is a definition
of k, in terms of e.
What this diagram is telling
you is which e's are allowed.
AUDIENCE: [INAUDIBLE]
PROFESSOR: Yes.
Yes.
There should be.
Let's see.
What's
AUDIENCE: [INAUDIBLE]
PROFESSOR: Oh, here.
Yes.
Yes, you're absolutely right.
Over out.
Thank you.
Excellent.
That's exactly right.
Yeah.
Oh man.
I made a dimensional mistake.
Thank you.
Jesus.
OK.
Good.
Yeah.
AUDIENCE: Could
you like re-explain
how imperfections and a lattice
leads to actual conduction?
PROFESSOR: Yeah.
I'm going to do that.
So that's an excellent question.
The question is could
you explain again
how imperfections and a lattice
leads to actual conduction.
As we talked about
last time, when
you have a perfect
lattice, there
is actually no current
flowing in response
to an applied
electromagnetic field.
If you put on a
capacitor, played
across your perfect lattice,
you don't get any current.
So the particle, the charged
particle in your lattice,
just oscillates back and forth
in a block oscillation, running
up the band, and down the
band, and up the band,
and down the band.
So, let me slightly
change your question,
and turn it into
two other questions.
The first question is given
that that's obviously not what
happens in real
materials, why don't we
just give up on
quantum mechanics
and say it totally failed?
And so this is a totally
reasonable question,
and I want to emphasize
something important to you.
Which is the following.
That model led to
a prediction, which
is that if you put a capacitor
plate across a perfect crystal,
then you would get no
current flowing across,
you would just see
that the electron wave
packets oscillate.
Or block oscillations as
we discussed last time.
And that is manifestly
what happens with copper.
But the experimentalist comes
back to you and says look dude.
That is a ridiculous model
because the copper isn't
in fact perfect, it's messy.
So how do you test the model?
Well there are
two ways to test--
to deal with the situation.
One is you improve the model
to incorporate properties
that copper actually has.
And see if you can actually
get the same conductivity
that you see.
But the other is you could
improve the material, instead
of improving the theory.
So let's make up what--
can we actually build
a perfect crystal?
This is actually something that
I'm doing research on right
now.
Not on the building side,
but on the theory side,
because I'm a theorist and you
should not let me in a lab.
But I collaborate
with experimentalists,
so they're nice people.
They're very good physicists.
So here's something you can do.
You can build a system that has
exactly a periodic potential.
It turns out it's
very difficult to do
this with quantum systems.
But what you can do is you
can do it with lattices
not of atoms, but
lattices of dielectric.
So the equation.
Here's a cool fact,
the equation for light
going through a dielectric,
where the dielectric has
different constants,
like wave guides.
You've got glass, you got air.
You've got glass, you got air.
That equation can be put
in exactly the same form
as the Schrodinger equation for
the time evolution of a wave
function.
They're both waves.
And so it's not so surprising
these two wave equations are
related to each
other a nice way.
Meanwhile, the index
of the dielectric
turns into the potential for
the quantum mechanical problem.
So if you have a periodic
potential, what do you want?
You want a periodic
dielectric constant.
Yeah.
And so you can build a
system which incredibly,
cleanly, has a periodic
dielectric constant
and no disorder.
And then you can put
light into the system,
and you can ask what
happens to this system.
So here's the idea,
I take a system which
is a periodic-- I'm
going to draw the
potential here.
So I'm going to draw
the dielectric constant.
So small, large, small, large,
small, large, small, large, et
cetera.
But instead of having it be
a one dimensional lattice,
I'm going to make it a
two dimensional lattice.
So now, basically, I've
got a set of wave guides.
Let me draw this differently.
So does everyone get
the picture here?
So literally what
you have, is you
have glass, glass with
a different index,
glass, glass with a
different-- if you
can think of those as
a line of glass fibers.
Optical fibers.
And you shine your light that's
reasonably well localized,
in both position, and in phase
variation, or crystal momentum.
Because you can control
the phase of the light.
So you send this wave packet
in and you ask what happens.
Well not a whole lot happens.
It's a wave packet.
It's going through
a wave guide, but we
haven't implemented
an electric field.
To handle an electric field,
you need the potential
to be constantly varying.
Uh huh.
So it's at a linear
ramp into the potential.
Instead of making it
just perfectly periodic,
let's make the index
ramp just a little bit.
And this experiment
has been done.
In this experiment, so as
the wave packet moves along,
what's discovered is that
the position-- if I draw
the x as a function of
t, so now the role of t
is being played by
the distance it's
moved along the wave guide, what
you find is that it does this.
It exhibits beautiful
block oscillations.
And this has been proved
in a very small number
of real honest quantum
mechanical systems.
The most elegant
experiment that I know of
was done by Wolfgang
Ketterle, who's here at MIT.
And he got three data points
because it was preposterously
difficult and declared victory.
So I talked to him about
this in the hallway one day.
And he said yes,
this was ridiculous,
but we got three data points.
We got small, we got large.
Victory.
We declared victory.
But it really needs
to be done well.
So one of the
interesting questions
in this part of the field right
now is we know that it's true.
But we want to see it.
We want to feel it, so various
people around the world
are working on making a
truly beautiful demonstration
of this bit of physics.
Yeah.
AUDIENCE: [INAUDIBLE]
PROFESSOR: It's
totally impractical,
because any interference
is just going to kill you.
Unfortunately.
So, you have to
work ridiculously
hard to make systems clean.
So the question is
really a question
about quantum computation,
which we'll come to next week.
But, the basic question
is how robust is this.
And the answer is it's
not robust at all.
But which you can tell because
everything in the real world
has enough impurity
that it conducts.
Or as an insulator.
Yeah.
AUDIENCE: What place sort of
like the larger role in sort
of like the perfection of
a lattice like temperature
or impurities.
PROFESSOR: That's a
very good question.
So the question is what's
the most important property?
What's most
important disordering
property that leads
to conduction?
And there's temperature
fluctuations,
there are impurities
in the lattice.
There are decohereing
effects which
is a more complicated story.
And that's actually, it
depends on the situation,
it depends on the system.
And exactly how it
depends is something
that is an active
area of research.
Now there are many, many
ways to probe this physics.
So we know that these block
oscillations are true.
We see them in all sorts
of different systems
that are analogous.
So there's lots of
[INAUDIBLE], it's not
like this is an
ambiguous bit of physics.
But it's one that turns
out to be surprisingly
difficult to tease apart.
The reason I bring all this up
is to emphasize the following,
our model made a prediction
that disagreed explicitly
with the connectivity property
of copper and other materials.
So don't throw away the model.
Observe that you've
modeled the wrong system.
If you find a system
that fits your--
that is-- that shares
the assumptions
of your model, that's
when you ask did it work.
And it worked like a champ.
OK.
So now let's talk
about real materials.
This is going to close up our
discussion bands and solids.
And this is actually
what I wanted
to get to at the
beginning of the lecture.
But that's OK.
There are lots of questions
and they were good questions.
So this is an extremely brief.
But I want to ask you
the following question.
What happens in the
following three systems?
So first, imagine
we take why don't we
take a system with built
out of single wells,
which have some set
of energy eigenstates,
and then we build the
periodic array out of them.
What do we expect?
And let me draw this bigger.
What do we expect to see
when we build a lattice?
We expect that this is going
to-- that these states are
going to spread out
into bands a funny way
Yeah and let's just talk
about the 1 d potential.
So what we'll find is that this
band turns into-- I'm sorry.
This state, this
single state turns
into a band of allowed
energy eigenstates.
There's now a plot
of the energy.
And similarly,
this state is going
to lead to another
band with some width.
And this state is going to
lead to another band, which
is even wider.
Everyone cool with that?
Quick question?
In 1 d, do these
bands ever overlap?
No.
By the node theorem.
Right?
OK.
Now let's take a single
electron, and let's
put in-- let's take
a single electron,
and let's put it in the system.
What will happen?
Well if we put it in
the system, what state
will this single
electron fall into?
Yeah one event.
But which state?
AUDIENCE: [INAUDIBLE]
PROFESSOR: Yeah, if you
kick the system around,
you let it relax a little bit.
It's going to fall down
to the ground state.
You have to couple to
something else like hydrogen
has to be coupled with an
electromagnetic field to decay.
But couple it, kick
it, and let it decay.
It'll settle down
to its ground state.
So you get an electron down
here in the ground state,
and looking back
at that band, we
know that the band for that
ground state looks like this.
So, here it is.
There's our electron.
It's sitting in the
lowest energy eigenstate.
Is it moving?
Well, it's in a
stationary state.
Is the expectation value of
the position changing in time?
No.
The expectation values
don't change in time,
in the stationary state.
That's part of what it
is to be a stationery
state, to be an
energy eigenstate.
OK.
Great. it's not moving.
Now, in order to make it
move, what do you have to do?
What kind of state corresponds
to the position changing
in time?
Yes.
Superposition.
Right?
From the superpositions
we'll get interference terms.
So if we put in a superposition
of say, this state,
and this state,
which corresponds
to different energies.
If we put it in a
superposition of these guys,
then it's meaningfully moving.
It has some meaningful,
well defined time variation
of its position
expectation value.
So in order to induce
a current, in order
to induce a current
of this system
where the electron
wave packet carries
a little bit of momentum is
changing in time it's position,
what do I have to do to the
electron in the ground state?
I have to excite
it, so that it's
in a superposition
of the grounds state
and some excited state.
Or more generally, into a
superposition of other states.
Yes?
In order to induce
the current, I
must put the electron
into a higher energy state
and in a particular
superposition of higher energy
states.
Everyone down with that?
Here's why this is so important.
Imagine each one of
these wells is actually
not some square well,
but it's an atom.
And let's say the
atom is hydrogen,
just for-- this doesn't
actually happen,
but just imagine-- in
particular what it means
is it has the ion, the
nucleus is charge plus 1.
And so in order for the
system to be neutral,
I must have one
electron for every well.
So if I have n wells, I must
have n electrons in the system.
Everybody agree with that?
In order to be neutral.
Otherwise, the thing's
charged and all sorts
of terrible things--
electrons will
get ripped off from nearby cad.
So we must have an
electron per well.
How many states
are in this band?
For n wells?
n.
Right?
OK.
So if I put in the n electrons
I need to neutralize a system,
where do those n electrons go?
Yeah, they fill
up the first band.
And if we let the system
relax with lowest energy
configuration, every
state in this lowest band
will be filled, and none of
these states will be filled.
Everyone down with that?
So here's my question.
When I've got that ground
state configuration
of this lattice of atoms
with one electron per well,
in these distributed
wave functions,
filling out these bands,
is anything moving?
Wow, you guys are
so quiet today.
Is anything moving?
This system is in an
energy eigenstate.
In particular, it's in a
completely antisymmetrized
configuration, because
they're identical fermions.
So, nothing is moving.
If we want to induce a
current, what do we have to do?
Yeah.
We have put them
in a superposition.
But where's the next
allowed energy eigenstate?
Next band.
So it's in the next band.
The next allowed
energy eigenstate.
So the configuration
we have now is
that these guys are all filled,
these guys are all empty,
but in order to take
an electron from here
and put it into
this excited state,
we have to put in a minimum
amount of energy, which
is the gap between
those two bands.
Right?
So now think about it this way.
Suppose I take light and I
send my light at this crystal.
In order for the light to
scatter off the crystal,
you must have electrons
in superposition states
so that they can have a
dipole and absorb and radiate
that energy.
Yeah.
But in order for that
to happen, the light
has to excite an
electron across the gap.
It has to give it this
macroscopic amount of energy.
Well, it's not macroscopic.
it's large.
It's not infinitesimally small.
That means that there's a
minimum amount of energy
that that incident
light must have in order
to excite the electron
in the first place.
So very long wavelength
light will never do that.
Light along wavelength
will not have enough energy
to excite an electron across
this gap into the next band
to allow there to
be a current, which
could oppose the electric field.
So the only for light to
scatter off of this crystal,
is if the energy, h
bar omega, of the light
is greater than or
equal to, let's say
greater than approximately,
the band gap delta e.
That cool?
We've just discovered something.
Crystals are
transparent unless you
look at sufficiently
high frequencies.
That's cool.
Right?
A crystal is
transparent unless you
look at sufficiently
high frequencies.
If you look at low
frequencies, your crystal
should be transparent.
Well that's really interesting.
In particular, we immediately
learn something cool
about two different materials.
Consider diamond and copper.
These are both crystals.
They're solids made
out of a regular array,
perhaps not perfect, but
extraordinarily good,
regular array of atoms
of the same time.
Array in a particular structure.
Diamond, anything and I
think face inner cubic.
I don't remember.
I really should know that.
Anyway, copper.
It's a lattice.
That's embarrassing.
I really should know that.
So we have these
two materials which
one has the larger band gap?
Diamond, because
it's transparent.
At in the visible.
So the band gap,
delta e of diamond
is much larger than the
band gap for copper.
But in fact, this is
a little more subtle,
because copper in fact,
doesn't even have a band gap.
We made an important
assumption here.
So I want to think about--
we're going to come back
to copper in a
second, but I want
to point out the
nice thing here.
Which is that diamond
has to have a band gap.
It's transparent.
It must have band gap.
It must be such that when you
fill up all the electrons you
need for it to be neutral, there
is a gap to the next energy
states.
And that gap must be larger than
visible wavelengths of light.
Yeah.
That's cool.
And that must be true of
all the transparent crystals
that you see.
Otherwise, they
wouldn't be transparent.
They would respond by
having free electrons that
could respond like a metal.
Yeah.
AUDIENCE: So, [INAUDIBLE]
PROFESSOR: Yeah.
Are diamonds good conductors?
No.
They're terrible conductors.
In fact, there
preposterously-- if you
compare the number of--
I'll get into this later.
But yes, they're terrible.
AUDIENCE: [INAUDIBLE]
PROFESSOR: Uh, that's a slightly
more complicated story, which
let me come back to.
Hold on to that, and if
I don't answer today,
ask me after in office hours
because it's a little more--
what?
Really?
Wow.
Well, MIT.
It's all about the intellect.
And everything else has to-- OK.
So, this is pretty good,
but here's the thing.
In one dimensional crystals,
the only thing that can happen
is, look if you have each
band come from allowed energy
state and each energy
state, each well
comes with one electron, or two
electrons, or three electrons,
you will always have filled
bands, and then a gap
and filled bands and then a gap.
Does everybody agree with that?
You can't have a
partially filled band
if each band comes from a
bouncy, in a single well,
and each well comes with an
integer number of electrons.
You just-- you're stuck.
Yeah.
AUDIENCE: [INAUDIBLE]
PROFESSOR: Oh.
I'm lying about spin.
But spin in one dimension is
little-- I'm lying about spin.
But do you really want
me to get in spin?
Man.
OK.
So if we include spin,
and we have splitting,
then it becomes a
more subtle story.
If we include spin, then
there are two states
for every allowed energy
eigenstate of the potential.
However, there are
generically going
to be interactions between
the-- there are generically
going to be magnetic
interactions which
split the energy of
those two spin states.
Electrons spin up, and
electrons spin down,
will generically have
different energies.
Now in 3D, this isn't
such a big deal,
because those
splittings are tiny,
and so the states
can sort of overlap.
But in 1D they can't.
So I mean, that's
also not exactly true,
but it depends on
exactly the details.
It depends on the
details of the system,
is what I wanted to get to.
Curse you.
So let me talk about
the same phenomena
in an easier context,
where we don't
have to worry about spin, which
we haven't discussed in detail,
in the class.
Which is in three dimensions.
Where the story changes
in a dramatic way.
Yeah.
AUDIENCE: [INAUDIBLE]
PROFESSOR: Oh.
It's not.
Generically, no, it's not.
It will depend.
AUDIENCE: [INAUDIBLE]
PROFESSOR: You say it happens
to be the salient one.
Yeah, exactly.
That is exactly right.
There the gaps are not the same.
That they do not
remain constant.
OK.
So let's talk more
about this system,
but let's talk about
it in three dimensions.
So in three dimensions, you
guys did an interesting thing,
when you studied,
you didn't know
this was about the structure
of solids, but it really was.
When you studied
the rigid rotor.
And when you studied
the rigid rotor,
you found that you
had energy eigenstates
and they were degenerate
with degeneracy 2 l plus 1.
The various different
l z eigenstates.
Yeah.
And then we turned
on an interaction
which was the energy costs,
the energy penalty for having
angle momentum in z direction.
Which added an l z
term to the energy.
And what you found is that as
a function of the coefficient,
which I think we called
epsilon, of that perturbation
of the energies of the energy
was equal to l squared over 2
i plus epsilon l z.
What you found is
that these guys split.
So this remained constant.
And this split into, so this
is the [INAUDIBLE] l equals 1.
So l equals zero.
So this is one, this
is three, this is five.
So the l equals zero
state, nothing happens.
l equals 1.
There's one that changes,
one that doesn't.
And then this guy has five.
One, two, three, four, five.
OK.
And what we found here is
that these guys could cross.
States from different
multiplates,
with different values
of l, had energies
that could cross as a
function of the strength
of the deformation
of your system.
Right?
The deformation is
where you have a sphere
and you stick out your arm.
So it's no longer symmetric top.
So here we can have
states crossing.
There's no nodes here
in three dimensions.
So as a consequence, when
you have a three dimensional
material built out of atoms.
So here's my sort of
pictorial description
of three dimensional
system built out of atoms.
You have a potential well,
potential, potential well.
Now, if the energy in one
particular potential well,
is like this, and like
this, and like this,
then when we add in a
lattice we get bands again.
The structure's a
little more intricate
because it depends
on the momentum.
But these bands now can overlap.
OK.
Everybody see that?
Because there's
nothing preventing
states from different--
in different multiplates
from having the same
energy in three dimensions.
There's no nodes
here that tells you
have to keep the
ordering constant
as you turn on the potential.
Now we turn on the multiple
particle potential,
and they can interact,
they can overlap.
As a consequence,
when we fill up,
let's say we two
electrons per potential,
or per well, when we filled
those first two bands, well,
there is the first-- so the
first band is now filled.
The second band and part of the
first-- part of the third band
and most of the second band
are going to be filled.
But part of the second
band is now available,
and much of the third
band is now available.
We filled in 2n electrons, but
we haven't filled up this band,
because it's really two
bands jammed together.
Or really bands from
two different orbitals
jammed together.
They happened to overlap.
So as a consequence
here, if this
is the length of the
energy of the last electron
that you put in, how much energy
do you have to give the system,
do you have to add the
system, to excite the energy--
or to excite the electrons
into excited states,
in particular into
superpositions
so that the electrons can move?
AUDIENCE: [INAUDIBLE]
PROFESSOR: Yeah.
Preposterously small amount.
An amount that goes like one
over the number of particles.
So in the continuum
limit, it's zero.
There's an arbitrarily
nearby energy.
So how much energy does it
take to excite an electron
and cause a current that opposes
the induced electric field?
Nothing.
Any electric field
that you send in
will be opposed by
an induced current.
So this behaves like
a classical conductor.
You turn on an electric
field, and the charges
will flow to oppose
that externally
imposed electric field.
You get charges then building up
on the walls of your capacitor
plates.
So, this is where
we have a conductor.
Because there's
an unfilled band.
And back here , we had an
insulator because we had filled
bands separated by gap.
The gap between the filled band
and the next available band.
This is actually called
a band insulator.
Because there are other
ways of being an insulator.
So from this so far, just from
the basic quantum mechanics
of a particle and a
periodic potential,
we now understand why some
crystals are transparent.
Why some materials conduct.
Why the materials that
are transparency are also
insulators.
And the things that conduct
are not transparent, generally.
Yeah.
AUDIENCE: [INAUDIBLE]
PROFESSOR: Excellent.
Excellent question.
So what's so special about
diamond and differ from copper?
And so the answer
goes like this.
So what determined the
exact band structure
in for a 1D periodic potential?
Two properties.
One was l, the periodicity.
And that came in
the q l and k l.
And the second is the detailed
shape of the potential.
Now in three
dimensions, the story's
going be a little
more complicated.
In three dimensions,
the things that
are going to determine
the potential
are not just the
distance between atoms,
but you have a three
dimensional lattice.
And the three
dimensional lattice
could have different shapes.
It could be cubic, it
could be hexagonal,
could be complicating in
all sorts of different ways.
Right?
It could be bent, it
could be rhomboidal,
and it could have all sorts
of different crystallographic
structures.
So that's going to go into it,
in the same way that l went
into it, which is the only
parameter in one dimension.
In the same way
that l goes into it.
So the crystal structure,
the shape of the lattice,
is going to determine it.
Secondly, the structure of
the orbitals is different.
Different atoms are
different wells,
so they'll give you
different band structure.
So different materials for
example, diamond versus copper,
are going to give you different
bands allowed energies,
because the potential
is different.
It has different shape.
And so when you
solve the problem
for the energy eigenvalues
is a function of now
the three different components
of the crystal momentum,
you'll just get a
different set of equations.
And working those out
is not terribly hard.
But it's a computation that must
be done, and it is not trivial.
And so one of the sort of, I
don't know if I'd say exciting,
but one of the
things that one does
when one takes a
course in solids,
is you go through a
bunch of materials.
And you understand
the relationship
between the potential, at
the atomic orbital structure
of the individual atom,
the crystal structure,
and the resulting
band structure.
And there's some sort
of nice mnemonics,
and there are calculations
you do to get the answer.
AUDIENCE: [INAUDIBLE]
PROFESSOR: You
will almost always
find overlapping bands
in three dimensions
in sufficiently high energy.
I can't off the top of my head
give you a theorem about that,
but yeah, it's generic.
Yeah.
AUDIENCE: --analog to
conductor in one dimension?
You have these like,
non-zero band depths?
PROFESSOR: Yeah.
And this is why Matt
was barfing at me.
So the answer to that is yeah.
There aren't [INAUDIBLE].
But what would we need?
What we need is
one of two things.
We need either the
band gap coincidentally
is ridiculously small.
What's a good example of that?
A free particle.
In the case of a free particle,
these band gaps go to 0.
Right?
And so that's a conductor.
Just an electron.
Right.
It conducts, right?
OK.
So that can certainly happen.
But that's sort of stupid.
I mean, it's not totally stupid.
But it's sort of stupid.
But a better answer
would be, well,
can you have a system
where there are bands
but you didn't have one
electron per potential well?
And yeah.
You could orchestrate
that in lots of ways.
Now it involves orchestration.
So it's not the
generic system that we
were talking about here.
But you can't orchestrate it.
So spin is a useful thing that
gives you an extra handle.
If you have twice as
many states per well then
you can have half a band filled.
So that's one way to do it.
Then it becomes dependent on
details of the system, which
is what I didn't
want to get into.
But yeah, you can
orchestrate it.
It's just not a generic
thing from what we've done.
And it's really not
for spin-less systems.
On the other hand,
accidental small gaps.
Easy.
That happens.
That certainly happens.
So that brings me
to the last thing
I wanted to talk about before
getting to entanglement,
which is accidental small gaps.
So what happens
to a system which
is-- so there are
some systems that
are neither conductors
nor insulators.
They are reasonably
good conductors
and reasonably bad insulators.
But they're not perfect.
And these materials are
called semiconductors.
I want to talk about why
they're called semiconductors
and what that means.
So this is going
to be very brief.
Then I'm going to
give you-- we're
going to get into entanglement.
So consider a
system exactly using
the same logic we've
used so far which
has the following property.
We have two bands.
And the bottom band is
filled because we've
got just the right number
of charged particles.
Bottom band is filled.
And this guy is empty,
but the gap is tiny.
OK.
Delta e is very small.
Now delta e has dimensions.
It has units, right?
So when I say small, that
doesn't mean anything.
I need to tell you
small compared to what.
So what's a salient
thing that controls
an energy scale for
a real material?
Well the temperature.
If you have a hot
piece of copper,
then the lattice
is wiggling around.
And every once in
a while, an ion
can hit one of the
electrons and excite it,
give it some momentum.
And so there's an
available reservoir
of energy for exciting
individual electrons.
You have it really
hot, what happens
is every once in a
while an electron will
get nailed by a little thermal
fluctuation in the system
and get excited above the gap.
And now it's in a super--
and generically, it's
going to be in a superposition
state of one of these excited
states.
So it's in general
going to be moving.
It can radiate.
It will eventually
fall back down.
But you're constantly
being buffeted.
The sea of electrons
is constantly
being buffeted by this
thermal fluctuation.
And as a result,
you constantly have
electrons being excited
up, cruising around,
falling back down.
So you end up with some
population of electrons.
And they can ask--
and both when asked,
although not quite
in this language,
how likely are you to
get an electron up here?
How likely is an electron
to be excited up thermally?
And those of you taking 8.04
will know the answer to this.
The probability goes as e to
the minus delta e over kt.
So let's think of this where
this is the Boltzmann constant.
So what does this mean?
At very low temperatures, if
the gap isn't 0, then this is 0.
It doesn't happen.
But at large temperatures,
the denominator here is large.
If the temperature
is large compared
to the width of the gap,
then this is a small number.
And e to the minus of a
small number is close to 1.
So at high temperature,
you're very
likely to excite
electrons up here.
And now if you have
electrons up here,
you have a bunch
of available states
down here-- also
known as holes--
and you have a bunch
of available electrons
up here with lots
of available states.
So at a high temperature, a
material with a small gap--
or at least at
temperatures high compared
to the size of the gap--
it's basically a conductor.
And at low temperatures,
it's basically an insulator.
This is called a semiconductor.
And there are notes on
the Stellar web page
that discuss in a little more
detail what I just went through
and show you how you
build a transistor out
of a semiconductor.
And the important bit
of physics is just this.
OK.
So that finishes us
up for the band gap
systems for periodic potentials.
We've done something
kind of cool.
We've explained why
diamonds are transparent.
We've explained why
they don't conduct.
We've explained why copper
does and it's opaque.
And that's pretty good
for 15 minutes of work.
It's not bad.
But along the way, we also
talked about the analogous
system of what are
called photonic crystals.
Systems of periodic
arrays of dielectrics.
Like wave guides.
And those have the
same structure.
They have bands of
allowed energy and gaps
of disallowed energies where
no waves propagate through.
So you might think
that's a little bit
of a ridiculous example.
So just to close
this off, you've
all seen a good example of
a photonic crystal flying
past you.
You know that highly reflective
at very specific frequency
structure on the
surface of a butterfly
wing that makes
it shiny and blue?
It looks metallic.
It looks like it's
a crystal reflecting
in a specific frequency.
At some sharp blue.
And the reason is, it's
a photonic crystal.
It is exactly this form.
If you look at it
under a microscope,
you see little rays
of protein which
have different
dielectric than air.
And they form exact crystals--
or not exact, but very good
crystals-- that reflect at
very specific wavelengths.
And as a consequence, they
have a metallic sheen.
So why would a butterfly
put a photonic crystal
on its surface?
Well it's extremely light.
It's fairly rigid.
It looks shiny and
metallic without actually
being shiny and metallic.
And it's not a pigment, so it
doesn't absorb light and decay
over time.
It's like the best
thing you could ever
do if you wanted to be a
shiny, fluttery, flying thing.
Anyway.
So there's an incredible
amount of physics
in this story of the band gaps.
And consider this an
introduction to the topic.
OK.
So that's it for band gaps.
And I want to move
on to the remainder,
the last topic of our course.
Which is going to be
entanglement and quantum
computation.
And here I need to give
you one quick observation
and then move on to
the punchline of today.
The one quick
observation is this.
We've talked about
identical particles before.
And we've talked about identical
particles in funny states.
So for example, imagine I
have two particles described
by a wave function where
the first particle could
be in the state a and the
second particles in the state b.
And I can build a wave function
for the first particle being
in state a and the
second particle in state
b in the following way.
So let's say position
a and position b.
I could take a single particle
wave function, chi of a,
and a single particle
wave functions phi of b.
And we've talked about
what this tells us.
And you've studied this
on your problem set.
What this tells you is that
the probability of finding
the particle at point A
is given by chi a squared.
And this is normalized, so
when we integrate against it,
we get 1.
And similarly, the
probability that we
find the second particle at
b is this thing norm squared.
And it's independent
of what a is.
But we also
studied-- and so this
was called the distinguishable.
We also studied the symmetric
configuration, which
was equal to 1 over
root phi, root 2.
Chi of a.
Phi of b.
Symmetric, plus
chi of b phi of a.
And this tells us
something totally awesome.
What's the probability that I
find the first particle at a?
It's the norm squared of
chi of a phi of b, right?
If we integrate
over all phi b, this
is the norm squared
integrates to 1.
So it's fine.
So there's a factor of one half.
We either find it at
chi of a or chi of b.
However if I tell you that I've
measured the first particle
and I find it in
the state chi, what
can you say about
the second particle?
It's in the state phi.
If you know the first
particle's in the state chi,
the second part is
in the state phi.
Because we measured it and
it's not in the state--
the first particle's
not in the state phi.
So measuring one
particle tells you
something about the
second particle.
And this is deeply
disconcerting,
because I could've
taken these particles,
put them in this
entangled state,
and sent one particle
off to a distant planet
and the second particle
to my sister in DC.
And my sister measures
this second particle
and determines what state
it's in and is immediately
determined what state
the first particle is
in over in this distant
planet Zorg, right?
So that's deeply disconcerting.
And to those of us who have
studied quantum mechanics up
to this point-- which
we all in this room
have-- to those of us
who have studied quantum
mechanics to this
level of development
and understand that it
is a correct description
of many experiments, this
should be yet another moment
of serious discomfort.
We've run into a bunch of
these over the semester.
But this one should
be troubling to you.
Because look.
How can something
here dramatically
change the state,
the configuration,
the initial configuration, of a
particle arbitrarily far away?
Isn't that deeply concerning?
And if you think
about relativity,
this should be all the
more deeply disconcerting.
Because how does relativistic
causality fit into this?
So there was a person
that roughly this time,
a little earlier, who was
troubled by this problem.
And his name was Einstein.
And so one of the things
that's kind of amazing
is that he created a thought
experiment which we're
going to study in detail next
week called the EPR experiment.
And there's a beautiful
historical story
about the setting
and the meaning
and the particular person.
And unfortunately,
I'm not a historian
so I can't tell you that story.
It sure would be nice
if we had someone
who wrote a
biography of Einstein
to tell you a little
bit about that story.
Oh look, it's Tom
Levenson who wrote
a biography about Einstein.
So Tom is--
TOM LEVENSON: Oh, I
need a microphone.
Those of who have taken
courses in [INAUDIBLE]--
and I'm sure that's all of
you because of the GIRs--
know this is larger than
the usual [INAUDIBLE] class.
So I'm very used to microphones,
but not in this context.
OK.
Is this-- yeah, it's on.
Can you hear me?
All right.
So there are lots of ways to
slice the story of Einstein
by the time he reaches the EPR
experiment, which is Einstein,
Podolsky, and Rosen for the
three people who actually wrote
the paper.
Just to dot the I's and cross
the T's on the paper itself,
Rosen is apparently for person
who first came to Einstein.
Podolsky and Rosen were two
young physicists in Princeton
after Einstein
moved to Princeton.
Einstein moved to
Princeton in 1933.
About three weeks
before-- I'm sorry,
he moved to Princeton '33.
He left Germany
in 1932 December,
about three weeks before
Hitler took power.
And he did so with decisiveness
and dispatch and a head
of almost all of
his-- in fact, I
think all of his German-Jewish
physicist colleagues
and those German physicists
for whom the Hitler
regime was unacceptable.
Which shows that Einstein
really was smarter
than most of his peers.
That's one of many different
ways you can ascertain that.
And so he came to
Princeton in '33.
He actually went to Caltech
before we went to Princeton.
As part of an ongoing
visitor-ship he had there.
Came back to Europe, hung
with the queen of Belgium who
was a friend of his.
Went to England.
And then headed
across the Atlantic
and took up residency
in Princeton
at the Institute
for Advanced Studies
where he stayed for
the rest of his life.
And over the course of the--
that was '33, he died in '55,
I think.
I should know that, but
I think that's right.
22 years.
He worked with a lot
of different, mostly
younger physicists.
And Podolsky and Rosen were
early members of that chain.
So Rosen was talking
with him some day
and starts to frame
this experiment.
Einstein develops it.
The three of them talk about it.
They write the paper
and they put it out.
And I want to share
with you, actually,
a really lovely description
of the way the problem was
represented in a way
by-- this is from a book
that I recommend to all of you.
It's actually
really hard to find.
It's really sweet.
Jeremy Bernstein,
who is a physicist.
He's sort of been around.
A physicist and writer.
He's in his eighties now.
He lives in Aspen.
He worked with CERN
for a number of years.
He's always been independent.
He wrote for the New Yorker.
Anyway.
So you've all heard of the
physicist Bell, I assume?
Bell's inequality?
OK.
So Bell had a lovely way to
describe-- I'm trying to find.
I had this marked and then
I lost my piece of paper.
I have already lost it.
That's terrible.
So Bell has this wonderful
way of describing
this problem of entanglement.
And it's based on
his description
of an actual person.
I was going to read
you his actual quote.
Now I'm just going to
paraphrase it for you.
He had a friend named,
I think, Bartelstein.
Or at least someone
known to him.
Who had two quirks.
An unusual color sense and a
taste for mismatched socks.
And so Bell used to say, if you
saw Bartelstein and you could
only see one leg and
that sock was pink,
you knew to a certainty that
the other sock was not pink.
He comes up I think--
I'm trying to remember
who this is originally
attributed to.
Same thing.
If you have a coin and you
cut it in half down the-- so
you've got two coin shape disks.
You cut the disk in
half, not-- and you
have one side that's the head
and the other side that's
the tail.
And they're separated.
They get handed to two
different gamblers.
And one gambler tries to cheat
the gambling establishment
by tossing in his half coin.
And you see the
head that you know
somebody-- somebody
at some other casino
is cheating by tossing
in the half coin that
only has a tail on it.
So there are lots of
ways to represent this.
And many physicists
being very witty indeed
have come up with
different metaphors for it.
So Allan just described for
you the basic claim in EPR.
Its weirdness.
That you have two
particles that are
entangled in some way and
then go their separate ways.
And thus you have--
if you have knowledge
of what's the state of one,
you have certain knowledge
of the state of the
other, violating
relativistic ideas of locality.
And just kind of
making you queasy
if you're sort of
approaching it naively.
What Einstein,
Podolsky, and Rosen
argued was actually something
a little bit-- in fact,
the paper comes to an end
on that note of queasiness.
But what they argue is a
little bit more subtle.
Because what they said is, OK.
You perform this
thought experiment.
You send the two particles off.
You measure position of one,
you know absolutely the position
of the other.
You've conferred--
and the paper turns
on a discussion
of the connection
between a measurement--
a physical measurement--
and a property of
physical reality.
And they have definition
for what reality is.
And that is something whose-- if
you can perform a measurement,
you know that
quantity absolutely.
I don't have the mathematics
to express that properly.
But that'll do for
this hand waving.
You can then do
another experiment
and measure a
complementary property.
And you know that
piece of reality.
But you can't do the--
so on the one hand,
quantum mechanics says you
can't know physical reality
to this level of precision.
And on the other hand,
the fact that you
can do that measurement violates
the relativistic picture
of reality.
So you have what they
claimed was a paradox.
And this paper was published.
And it received a range of
reactions from indifference
by younger physicists who said,
we don't care that it's weird.
We're going to keep on
doing quantum mechanics
and performing experiments
and making measurements.
And just see where
this leads us.
Remember, this is
happening in the mid '30s.
1935.
One of these three books will
tell me precisely in a moment.
And the quantum theory, as it
turned into quantum mechanics,
developed in its first
period between '23 and '27.
And by '35, you have enormous
numbers of productive results
and unexpected things
and the prediction
of the positron and
then its observation.
And I mean, the theory is
enormously, dramatically,
excitingly productive.
So those who are really
heads down doing the work
are, for the most part,
saying, this is fine.
We'll get back to it when we're
old and retired and bored.
But that wasn't
the uniform case.
And most notably
Niels Bohr found
this paper really troubling.
And spent about six
weeks, apparently,
discussing this and trying to
come up with a response to it.
And what he responded
was essentially
that-- in some ways, it
was the same reaction
as his younger colleagues.
Get over it.
But more precisely,
it was he said,
there's no
description of reality
that excludes the measuring
apparatus anymore.
You can't make statements
about physical reality
unless you include a description
of the measuring apparatus.
And you've said that we can
measure this one quantity
with precision and
know the other thing.
And then we can subsequently,
in a separate observation,
measure a complimentary
quality and know the other one.
You still can't know much
them at the same time.
It's still true that the
complementarity in essence
means that once you know
one part of the picture,
you know some other
part the picture.
And that's just the nature
of the quantum world.
Einstein had argued that
the EPR paradox suggested
that quantum mechanics
was incomplete.
And Bohr essentially
responded in effect
that Einstein's description of
quantum mechanical explanation
was inadequate.
The important
thing to remember--
and I want to just spend a
couple minutes going back
into the pre-history
of all this,
and then a couple
minutes speculating
on why Einstein reached
the position he did.
And what that might tell you
about the practice of science
as a lived experience
as opposed to one
reflected in your textbooks.
But the thing to remember
is that there's nothing
logically wrong
with the EPR paper.
Right?
You know.
It does what it says it does and
there's no overt error in it.
And there's nothing wrong
with Bohr's response.
And in fact, when the
experiments were--
Bell formalized the-- what
Bell's inequality really does
is it formalized
the two arguments.
It says, if Bohr
is right, you will
observe this in the experiment.
And if Einstein is
right, you would
observe something different.
The experiments were
done, and I imagine
are still being done, as
sort of demonstrations.
And they showed that Bohr's
interpretation was correct
and that yes, quantum
mechanics produces results
that are non-local just
as Allan described to you.
And that the world really
is as strange as people
first glimpsed in
1925, '26, and '27.
And the question of whether
or not that strangeness
is adequately explained
without the explanations
that you're going to
learn in this class
and subsequent ones are
quote "complete" or not.
And completeness is a very
funny, very, very tricky
concept.
But the question whether or
not the framework of quantum
mechanics is somehow
unsatisfactory in any kind
of formal a technical
sense is one
that's at least partly dependent
on your scientific temperament,
I think.
So that's the cartoon version
of what happened in '35.
Einstein with his
two young colleagues
proposes-- really you
should understand the EPR
paper as a description in detail
of a consequence of quantum
theory as it was then
expressed with the conclusion--
and I just want to
read you this thing--
no reasonable
definition of reality
could be expected to
permit such a result.
In fact, it's called a
paradox, but it isn't.
It's a complaint.
You know, it's a memo to
the Flying Spaghetti Monster
that the universe
shouldn't be this way
if, in fact, experiments
turn out to show that it is
and they have.
The oddity here for a
biographer of Einstein
opposed to a physicist
is given what
you know about Einstein between
1879 when he's born and say,
1925 or so when he
completes the last
of his really great physics.
How could-- I mean,
I actually keep--
I've been working on Einstein
off and on for years and years.
I keep finding out new ways in
which he's just inconceivably
bright and on target and with
a nose for the right problem
and insightful.
And yet by the 1935,
10 years later,
he's still a
relatively young man.
He's in his '50s.
Which being in my '50s I think
is an extremely young man.
Just 10 years after
doing work that's
right on the edge of
modern quantum mechanics
that is essential
to its foundation.
That's really extraordinary.
10 years after
that, he's saying,
no reasonable
definition of reality
should be permitted
to behave this way.
Where does that come from?
Well the first thing I
want to tell you-- again,
this is all going to be
a really cartoon version.
Because there's not
much time, I understand.
Is that Einstein-- I
mean, how much are you
aware of Einstein's role in the
creation of the quantum theory?
A lot.
I mean, a lot?
None?
OK.
I mean, I'm going
to make a claim
that except for Heisenberg,
Schrodinger, maybe Bohr.
Maybe Born.
Maybe a couple of others.
There's no one more
important to the quantum
theory than Einstein.
And you could maybe
even argue that
from a sort of
foundational point of view
that without Einstein,
rigorous thinking about quantum
mechanics would have
taken much, much longer.
I mean, he's really
central to it.
Planck in 1900 publishes
as an ad hoc solution
to the black body problem
the first quantum theory.
In 1905, Einstein says
it's not an ad hoc thing.
If you look at the
photoelectric effect
is the particular problem he's
dealing with in explanation.
But that's really-- the behavior
of the photoelectric effect
is really presented as the
confirmation of this idea
that light exists as quanta with
particular kinds of behavior.
And from 1905 on, he
spends probably more time
on quantum problems than he did
on any other physics problems.
Certainly more
than on relativity.
Though he spent
enormous energy on
special and general relativity.
One of most amazing things
about Einstein, in fact,
is that despite the fact
that he's seen and appears
by 1935 to be this hidebound
old guy who can't accommodate
himself to the new world is he
had an extraordinary capacity
to do what the Red Queen
did in Alice in Wonderland.
and believe two impossible
things before breakfast.
Just think in 1905.
April, he publishes The
Quantum Theory Of Light.
June he publishes
Special Relativity,
which treats light as a wave.
And makes no mention of
his revolutionary-- I mean,
he called it revolutionary
in a private letter in 1905.
So he knew what he had in
the quantum theory of light.
But in special relativity, go
read the special relativity
paper.
It's actually lovely reading.
And you'll see he doesn't
even nod in that direction.
He doesn't say, you know this
is a hero-- he says nothing.
So he's capable of
doing excellent--
and there's a
reason that year is
called the annus mirabilis,
the year of miracles.
And in part it's
because Einstein
is able to actually really
focus on these things.
And I realize class
is almost over.
So there's several more steps
in Einstein's quantum journey.
But you know, what
you should take away
is that Einstein's
ability to deal
with the problems
of quantum pictures
extends to the point--
he's the first person
to suggest there might be
a problem with causality
in quantum mechanics.
He does this in
1917, eight years
before quantum
mechanics is invented.
When he starts looking at what
the quote "classical" quantum
theory tells you
about the emission
of radiation from
an excited atom.
He realized you can't
predict it precisely.
Radioactive decay
has the same problem.
He says, well-- he
writes in a letter.
I don't want to give up
causality, but we may have to.
So he's aware of these things.
So I see class is
over now at 12:30.
OK, sorry.
So juts to finish
off, the question
here is why does
Einstein give up on this.
And the answer, I think, is
because in addition to his-- as
he started at the
beginning of his career,
he says with the
quantum theory of light
and with special relativity,
ignore your physical pictures.
Try and look at the
phenomena and explain those.
And by 1935, that becomes
very difficult for him.
Because the phenomenology
becomes too strange.
One of the things that
quantum mechanics does
is it takes away the
immediate ability
to visualize physical systems.
There.
English is my first
language sometimes.
And that's an aesthetic
failure on Einstein's part.
He had the intellectual
capacity and explicitly said,
quantum mechanics is a
logically consistent theory
that incredibly powerfully
describes lots of problems.
He said that in print.
He nominated Schrodinger and
Heisenberg for Nobel Prizes
twice.
I mean, he wasn't stupid.
He was Albert Einstein.
But he was aesthetically
incapable of pursuing
this new physics
in ways that were
possible under the research
possibilities of the time.
And that is what I
would leave you with.
Physics is an aesthetic as well
as an intellectual pursuit.
So thank you all.
[APPLAUSE]
