>> Welcome back to the fourth
lecture of Chem, Chemistry 131A.
Today we're going to talk
about complementarity,
quantum encryption, and
the Schrodinger equation.
First of all, suppose that
we actually know something
about a particle
but not everything.
So we know that a particle may
be localized or in the vicinity
of some region of
space, and we'll stick
with one-dimensional
problems for simplicity here.
So we have a variable x,
and we know that the
particle is probably
at a position around x naught.
The question is what should
the wave function look
like for a particle like that.
And if we assume a real wave
function, we can write something
that creates a peak near this
particular point, x naught,
in terms of a Gaussian function,
a function that's small
and comes up smoothly and down,
and that's a very nice function,
has a very simple analytical
form which I've written here,
and it's characterized by a
position where the peak is,
which in this case is x naught,
and by a standard deviation
which has to do with how
wide the distribution is
around the position x naught.
In other words, how peaked
the wave function is,
and how closely we know the
position of the particle.
So sigma in this formula
measures the width
of the peak distribution, and
a small value of sigma means
that the wave function
is peaked more strongly.
It's really very tall, and
a large value of sigma means
that the wave function, while
it has the same average value,
where the peak is, it's
much wider, in fact.
I've plotted here the wave
functions for sigma equals 2
and sigma equals 1, and
both the wave functions
when you square them
and integrate them
have unit probability
that the particle is somewhere
in the universe, but you can see
in terms of this graph
that it's very likely
that the particle is
quite near x naught.
That's the most likely
place, and if sigma is 2,
there's some likelihood that
the particle could be one
or two units away, and then by
4, it dies off quite quickly,
but for sigma equals 1,
there's 99 percent chance
that the particle is going
to be within two units
of the position x naught that we
think the particle is located.
And so the particle is much more
localized for sigma equals 1,
and this lets us have
a tunable parameter.
We can use the same formula.
We can put in different
values of sigma, get a family
of wave functions, and then
we can analyze how they behave
as we move forward and
try to discern things
about the momentum
or the uncertainty
and position of the particle.
Now we previously worked
out that the momentum operator
was minus ih bar d by dx,
and we found the eigen functions
for the momentum operator.
They're the complex
exponentials, and I mentioned
that they corkscrew one way or
corkscrew another way depending
on whether the momentum
is positive
and the particular is moving
to the positive x direction
or the particle could have
negative momentum and be moving
in the negative x direction.
But the size of this corkscrew
does not change in space.
It's completely uniform
everywhere,
and so in a momentum
eigen function,
we know absolutely
nothing about the position.
The question is if we know
something about the position
but not perfect knowledge,
we have a distribution
like this Gaussian function,
then how much, if anything,
do we know about the momentum,
and are the two things
related to each other.
And to find that out, what we
have to do is we have to figure
out how to write this smooth
Gaussian function and position
as a linear combination of
momentum eigen functions.
And then the co-efficients
of those momentum eigen
functions will tell us what is
the chance if we make a
measurement of momentum rather
than position that we
will get a certain value
for the momentum eigen value.
Of course, if we make a
measurement of momentum,
we will have changed
the wave function
in a very fundamental way,
and so it will not have
the same distribution,
and it may not be located
anywhere near x naught
after that.
It turns out that you can
kind of see what's going
to happen just by imaging
this thin wave function.
If I have a thin wave
function, and I have a bunch
of corkscrews, let's just
forget about the imaginary part
for the time being and just plot
the real part, if I have a bunch
of functions that are going
up and down and up and down,
and I want to make
something that's quite narrow
and then pretty much 0 outside,
I can't just use a
very long wave length
because that'll never
actually be very tight.
And I can't just use one wave
because that goes everywhere.
That's never going to be 0.
So what I have to do in order
to make it work is I have
to take a bunch of different
momentum eigen functions
and the, they have to at
least oscillate as fast
as this thing is dropping.
In other words, if I have a sign
wave, it has to drop at least
at that fast, and then what has
to happen outside is they have
to all kind of interfere.
They're all still there,
but when you add them up,
they cancel out to 0, which is,
of course, the beauty of waves
and which is what light
does all the time.
It may take many paths, but
many of them cancel to 0.
And so what we anticipate
in this analysis is
that the more localized the
wave function is in position,
the larger the spread
of momentum eigen functions
we're going to have to use
because we're going to
have to use something
that oscillates quickly, and
that means it has a big value
of p, and that means the total
spread can be quite wide.
And so what that means if
we measure the momentum is
that we're going to get
a wider distribution
of momentum eigen values.
Now here are a couple of
momentum eigen functions
that contribute to the sigma
equals one wave function,
and you can see from the
graph that the amounts
of these momentum eigen
functions are all small.
The p equals 0 is a flat.
That's 0 momentum.
That's just flat.
No corkscrewing at all.
P equals 1 is a very lazy
thing that's barely changing,
and p equals 10 is
going quite quickly.
But because the function
dies at around 2,
p equals 10 is not going to be
enough momentum to get a wave
that oscillates quickly enough
to actually get this peak to be
that narrow, and so we
anticipate that we're going
to need more waves than that.
And because all these
values are small,
and because there
are a lot of waves,
what that means is we get
a very wide distribution
of momentum functions,
and that's because we need
this wide distribution to build
up this localized
position, Gaussian eigen,
Gaussian position function.
What I've done here in the
next slide is show what happens
if we take our original
Gaussian wave function,
and we approximate it by
20 momentum eigen values,
and we choose the co-efficient
so that we get the best fit
between our approximation
and the true function.
Now what you can see is two
things are not so good here.
The first is that because p
equals 20 is not fast enough,
we can't make the
peak narrow enough.
So we can't bring in the skirts
of the peak quickly enough
because we don't have anything
that's dropping that fast.
The second thing that's wrong
is that we get these wiggles
that keep going outside.
Now they get smaller
and smaller,
but they go outside the
region that we want,
and this is reminiscent of the
fraction of light through a hole
and many other kinds of problems
that are encountered quite often
and is just a fundamental
property of trying
to cast this function in terms
of these sines and cosines
or equivalently in terms
of complex exponentials.
How can we cure this?
Well, instead of using 20
momentum eigen functions,
suppose I use 50, and
I try the same thing,
then we get this graph here.
And now it's far, far better.
Now it pretty much tracks
the Gaussian function.
There are a few wiggles.
We're always going to have a
few wiggles unless we use an
infinite number of functions.
There are a few wiggles outside,
and the peak doesn't quite,
if you look closely, it
doesn't quite get all the way
up to the top of the Gaussian
function in the center,
and that's, that means that
we're missing a few functions,
and the functions we're
actually missing are the ones
that are also creating
these wiggles outside.
If we put in these extra
functions from 50 to infinity,
and they would be very small
amounts toward the end,
what we would find is that
we could match this Gaussian
distribution absolutely
perfectly, and we can do
that with any set
of eigen functions.
We could use position
eigen functions,
we can use momentum
eigen functions,
we can use eigen functions of
some other operator, and it,
because the eigen functions
form a basis just like any point
in a 2D plane can be written
as this much x and this much y,
and there's no escaping,
if we use enough
of these eigen functions
of any operator,
we can exactly match any kind
of wave function we're going
to encounter, and then
when we look at the amounts
of these basis functions,
that's when we find
out what a measurement of
momentum will be give us
or a measurement of
any other operator.
In fact, any reasonable
function,
and wave functions are always
reasonable because we have
to be able to differentiate them
and integrate them and so forth,
any function at all can be cast
as a sum of sines and cosines,
or equivalently as complex
exponentials, e to the i theta.
And that is actually
the principle
of Fourier series expansion,
and that is another good
mass subject to study
so that you understand exactly
how this type of thing works.
In fact, if you do study that in
a proper course in mathematics,
what you will come
to the conclusion is
that the uncertainty principle
can be seen from this aspect
as a consequence of the position
and momentum eigen
functions being related
to a Fourier transform.
And that, right away, gives
us the uncertainty principle
and makes it quantitative.
So if we have a very narrow
distribution in position,
we have a very wide
distribution and momentum.
We need those fast
momentum eigen functions
to pull the skirts in.
If we have a very narrow
distribution and momentum,
that means that the
corkscrews go way out all
over the place before they
actually interfere and go away,
and that means that we have
a very wide distribution
in position where a
particle can be anywhere,
and they are just flip
side of each other,
but we can't have
them both be narrow.
It's not possible to use one
corkscrew and make one spike
in position because they're
totally different things.
When one is narrow,
the other is wide.
Now this measurement, as I said,
if we actually measure
the momentum,
or we make a measurement on
this Gaussian wave function,
we will have changed it because
it is not an eigen function
of what we're measuring.
And with all the worry
about security and privacy,
I thought I would
do a little bit
of a topic here called
quantum cryptography.
Using quantum mechanics,
it's possible
to make an unbreakable code
that nobody can spy on you, and,
in fact, it's a very
simple and ingenious thing.
It's being used now in some
places in the United States
and in other countries, and
it's based on the observation
that if we make a measurement,
that the wave function must fall
into an eigen function
of the measured variable,
and that means if a spy makes
a measurement on something
that we're transmitting, the
spy will influence the data,
and we can pick that up.
So we can know something's
wrong,
we can know somebody's
spying as well,
and we can then just simply
stop talking, for example.
So how does this work?
Well, we need some sort of
thing to send, some particle,
and the easiest particle
to send is a photon.
We can send a photon
down a fiber optic line.
We can send a photon
through free space.
Lasers are very efficient
at making photons
that have very nice properties.
The best properties available.
Very narrow wave length
range, and so forth and so on.
And we can send lots of photons
one at a time very quickly,
and that allows us to
send a lot of information
and to establish a
method of communication.
How could you do this?
Well, the first realization
of this scheme was first put
forward, as far as I know,
by Bennett and Broussard,
I've given the reference here
in slide 107, in 1984.
So quantum cryptography has
not existed for all that long.
And this scheme is called
BB84 after the two authors
and the year in which
it was invented.
There are other schemes as well
because once you realize how
to do it, there's a lot of ways
to skin a cat, as they say,
but I just want to
talk about BB84
and show you how it is possible
to establish a communication
link and an encryption strategy
where nobody can find
out what you're doing.
In this field, there, just
like in quantum mechanics,
we have psi, and we always use
psi for unknown wave function
that we're going
to find out about,
and we tend to use psi
for basis functions.
In this field of communication,
the two parties who are trying
to communicate are always
Alice and Bob by convention.
And Alice is trying to send
data to Bob, and they want
to encrypt their data, and
they don't want anybody else
to know what the
encryption key is
so they don't decrypt the data.
And the spy is traditionally
referred to by the name Eve,
which is perfect because
Eve is an eavesdropper.
And how do we do this?
Well, I've shown here two
possible states of polarization.
A polarizer is just a
filter, and you can think
of any measurement
in quantum mechanics
as some kind of filtration.
If I have a photon polarized
this way up and down,
the electric field
is going like that,
and I have a polarizer
at right angles.
The polarizer cuts out
that light, and I get 0.
And if the polarizer is
parallel, then the light goes
through completely unimpeded,
and I get 100 percent.
And if I've got one photon,
I get the one photon,
and if I don't have one,
if I have the polarizer the
other way, I get 0 photons.
And that's perfect for computing
because I can have this be 1,
and I can have this be 0.
The trick here is, though, not
to just use that, but at random
to pick instead of,
what I'll call the plus,
pick the time spaces.
The time spaces is just the plus
basis rotated by 45 degrees.
Now if the polarizer
is this way, I get a 1,
and if it's the other way,
and the photon's
this way, I get a 0.
So I get 1 and 0 again, but
I have 45 degree rotation
between this basis
and this basis.
Now what happens if I
have a photon this way,
and I have my polarizer
this way?
Well, then quantum
mechanics says at random.
So I cannot predict.
I get 50/50 1 and 0,
and after I measure it,
the photon is now instead
of polarized this way,
it's polarized either
that way or that way.
And that's the basis, then, of
a completely unbeatable strategy
to establish a key between
two anonymous people
or two anonymous computers,
connect it up, and then use
that key to transmit data.
Here's how, here's
what we do then.
Alice sends a 0 or
1, and at random,
and she also chooses
the basis at random.
How could you choose
the basis at random?
Well, one way would be, for
example, to have a very,
very weak radioactive
source nearby.
That's completely random.
If it's an odd number of
counts and a certain number
of intervals that the detector
gets, you pick the plus basis.
And if it's an even
number of counts,
you pick the time spaces.
So you pick the basis at random,
but you record it where you are,
what basis you're picking,
but you do not send
that information, and so
nobody else, not even Bob,
knows what basis
is being picked.
And then you send a
photon polarized that way,
and you can change that very
quickly with an electric field,
and you send photons
along to Bob.
Now Bob has no idea
because you don't want
to be telling what the basis
is, that defeats the purpose
of secure communication.
Bob has absolutely no idea at
all what basis Alice has picked.
Therefore, Bob doesn't
try to figure
out what basis Alice has picked.
Bob, at random, either picks
plus or times as his basis.
And then measures
the photon that comes
through with the polarizer
either oriented this way
or that way, and then
measures either a 1 or a 0,
but he has no idea whether
what he's measuring is
with absolute certainty if his
polarizer's the same as Alice's.
The theory of quantum mechanics
says if I put a photon this way,
and I measure this way, again,
and nothing's intervened,
then I get the same
value with certainty.
Or it could be this
way, and I'm getting a 1
at random, and I have no idea.
We send a whole bunch
of photons.
Lots and lots, and we can
do that very, very quickly.
So that, and very cheaply.
You can have something about
the size of this remote
as with the laser,
and no problem at all.
If Bob's basis is quote
wrong, in other words,
it doesn't match the basis
that Alice picked at random,
then Bob just gets
1 or 0 at random,
but the problem is
he doesn't know that.
He just gets 1 or 0, and
how is he to know what
on Earth he's getting.
He has no idea what
he's getting.
But what he does know
is he's got a list
of what basis he
picked at random,
whether he picked plus or times.
And that's enough.
Now Bob doesn't send
back what he measured.
He measured a 1 or
0 in sequence,
but what he does send back
to Alice is he sends back a 1
or a 0 depending on
whether he picked plus
or times as the basis.
So he doesn't say
what he measured.
He just says on trial 1,
I happened to pick plus.
On trial 2, I picked times.
On trial 3, it was times; 4 was
times; 5 was plus, and so on,
and I have lots of them.
And Alice, of course, has a
list of what basis she picked,
but that information
hasn't been sent,
and Bob hasn't sent
what he's measured.
And, therefore, Alice looks
at Bob's choice of basis,
and whenever his choice of
basis matches what she happened
to pick, oh, let's say with a
certain trial, they matched.
Maybe many of them don't match
because it is random, after all.
But if they have the
match, then she says, OK.
Go ahead and pick these values.
So, for example, she records
the list and where they match.
She then sends him another
message, which has nothing to do
with anything that
anybody can use,
and says why don't you
use whatever you measured
on the 7th, the 13th, the
22nd, so forth trial or photon,
and those are where the
basis have to match, and,
but nobody knows what
the measured result was
because Bob never
said what he measured.
But Alice knows what he
measured because she knows
that when the basis matched, he
got the same value that she got,
if nobody's eavesdropping.
Therefore, what they do is they
establish a sequence of 1's
and 0's that nobody
else can know about,
and that's perfect
to encrypt data.
They both know what the key is.
They take normal data.
They encrypt it with
some scheme,
with these random sequences
that only they know about,
and then they decrypt it because
each of them has the key,
but nobody else can
have the key.
Now how can they find out
if something's going on?
Well, they can establish a key,
but they can have sent many,
many, many more photons where
the basis happen to match.
Where Alice was plus,
and Bob was plus.
And what they can do
after they've established the
key is they can go back and say,
hey, you know, tell me what
you measured on trial 1,001,
1,003, 10,000, and so on.
And if not, if those
don't match,
or if they don't match
99.99 percent of the time,
then somebody might
be listening.
So you can have a threshold
that if somebody is listening
with some kind of
polarizer trying to find
out whether it's 1 or
0, you can say, look,
it seems that this line of
communication is not secure.
Maybe we've got an error in the
detection system for the photon
or something is wrong, but
we can't use this to talk
about these sensitive things
we're going to discuss.
We have to start over.
We scram the whole
thing and start over,
and then go from there.
Once you have the
encryption key,
then you just use any
old encryption scheme,
and the key is that since nobody
knew what the key was except Bob
and Alice, and by the
clever way they did it,
nobody can find out, they can
use a new key every single time
they talk.
This is not like the PIN on an
ATM card or a magnetic swipe
that everybody can steal.
For every single financial
transaction you do, you walk up,
establish a new key, and then
encrypt all the financial data
with that key that only you and
the other side of the party,
the bank know about, and
nobody else can know about it,
and every transaction
has its own key.
And so even if somebody
tries to steal the data,
it's not like they can get
in with a single password
and then look at all your stuff
because it's just
completely hopeless.
And this is, of course,
the power of computers is
to do stuff like this
to keep things tricky
for somebody trying
to steal your stuff.
Of course, the other side is
that computers can be used
to steal stuff pretty
effectively.
And so, anyway, this
has been used.
It's a secure method, and it
is being put into practice,
and maybe one day, every
time we walk up to an ATM,
the card will have a little
laser, will establish this key,
and then we'll take some money
out, and it will be recorded.
OK. That's all I want to say
about quantum cryptography,
but it is an interesting
subject simply
because it really
illustrates the principles
of quantum mechanics and
how if you're clever.
Now it took from 1926 until
1984 for somebody to figure
out that you could do this,
but now it's a very potent
way to ensure privacy.
How does a wave function
evolve in time?
That's the question.
And the first point
you have to make is
that there is no
operator for time.
Which seems kind of funny,
at least it did to me
when I was a student because,
after all, we can measure time,
or we think we can, or we
can measure differences
in time, elapsed time.
It seems like something we
ought to be able to measure,
but there is no operator
for time.
It's not as if, for
example, it's,
like, position or momentum.
There is no operator for time,
and there is no expectation
value for time, and, therefore,
time in quantum mechanics is,
we're going to treat it
anyway, is just a number.
It's just a running variable
like x is in position
for classical mechanics.
It doesn't get elevated
to a higher level.
And you might say, well, why is
that, and the short answer is
that if time did
have an operator,
then it wouldn't be Hermitian.
And so then if you introduce an
operator that's not Hermitian,
you have a lot of problems.
Well, motion, if, the easiest
way to tell if time's going
by is if something's moving.
If we see a car rolling
by, we know time's elapsed.
If we see somebody drive, diving
off a diving board into a pool,
we know that some
time has elapsed.
And motion is related to
energy, and energy is,
multiplied by time has
the same units as h bar,
which has the units of action
or joules times seconds.
And that dimensional
analysis gives us a clue
because when we had momentum
and position we took
momentum times position.
It had the same units as h
bar, and now we've got time
and energy, and we
take time times energy,
and we get the same
units as h bar.
And so that gives us
a clue as to what kind
of wave function we
might want to try to put
in to start doing time
dependent phenomena.
Well, the 1d kinetic energy of a
particle is one-half mv squared
or p squared over 2m
because, remember,
for classical particle
that's not [inaudible],
p the momentum is just mv.
So p squared over 2m is the
same as one-half mv squared.
And in elementary
courses, we always use ke
for kinetic energy, but
in more advanced courses,
we just use t. That's
our notation.
And the potential energy,
which in elementary courses
we call pe, potential energy,
we use v. And the
potential energy
of the particle only
depends on its position,
and the kinetic energy only
depends on its momentum.
So the two of them are totally
different forms of energy.
Of course, they can
be interconverted,
and we do that all the time.
But potential energy might
be, like, a mass that is
out of position, and then if we
drop it, if energy is conserved,
we can figure out the
velocity of the particle
when it hits the ground when all
the energy has been converted
into kinetic energy.
The total energy, then,
is e equals t plus v,
and to cast this in
terms of operators,
all we have to do is
take our variables,
and we dress them up with hats.
And, therefore, we have e,
the energy, total energy,
will be p hat squared over 2m.
M is, again, just a variable.
It's not an operator, just
the p, and then v of x,
and v of x could be
any functional form,
including just 0, but instead
of just x, we put x hat.
And then we look at this thing,
and we say energy's conserved
over time if we have an
isolated system, and, therefore,
this thing is going
to stay the same.
It could interconvert
between one form or another,
but it can't disappear.
We can't get energy
from nowhere,
and we can't have energy
going nowhere, into nowhere.
If that could ever happen,
we'd know about it instantly
because we would have something
that just sat there and ran
and boiled water
endlessly, and didn't need
to be plugged into the wall.
And that would be very
handy, but, unfortunately,
it's very impossible as well.
If we put in the explicit
forms, then, of these operators,
the operator x hat, when it
operates on the wave function,
just returns the value x.
And, therefore, we can operate
with x hat on the wave
function, and we just get v of x
where x is now a variable.
It's been turned
into a variable.
There's no operator left.
P, on the other hand, was minus
ih bar times the derivative
with respect to x, and
I've got p squared,
and p squared is p times p.
So I put in two of them, and,
therefore, p squared over
2m becomes minus ih bar d
by dx times minus ih bar d by dx
times 1 over 2m times psi of x.
And that whole thing should then
equal to e, the energy times psi
of x, and that should
be an equation that says
that energy is conserved,
and we now have
to find the wave function that
makes the energy conserved,
and that will depend
on the potential.
Well, we can tidy this up,
remembering that i
squared is minus 1
and get minus h bar
squared over 2m.
The second derivative
with respect to x
of the wave function,
plus the potential energy
of x times the wave
function is equal to e,
some number with the units of
energy times the wave function,
and this equation is called the
one-dimensional time independent
Schrodinger equation.
This is an equation that
says energy is conserved.
If you find the wave function
that makes this thing true,
you will have found
the allowed energy,
but there's no time
in this equation yet.
Now it, depending on the
functional form of v of x,
the potential, for example,
for a molecular spring,
we might have v of x
is one-half kx squared
so that the energy
goes up quadratically.
That's a harmonic oscillator.
Depending on what this form
of this potential energy is,
electrostatic energy, various
kinds of repulsive forces
and so on, we can put all those
in, and if we have something,
then we get certain
wave functions
which if the particle
is confined
or if the potential gets big,
and the particle is stuck
like water in a cup, and it has
to stay there, or an electron
on an atom, now what we find is
that when we solve
this equation,
we can't have any old energy.
And the reason why we can't
have any old energy is
that the wave function
has to fit.
We saw that on the particle
on a ring, and it's going
to be the same no
matter how it's trapped.
The wave function has to
fit into the allowed space,
and that means it can only have
a certain kind of wave length,
not just any old thing, and that
means the energies are discrete.
The energies get labeled
with a quantum number.
So we have instead of e,
e sub n, and we label them
with a quantum number, n.
We don't have any time.
The question is how
do we introduce time.
And it's very tricky to
think how to introduce time
because we don't have
any guidance necessarily
from classical mechanics
about how to do it.
And in the case of an
isolated atom, let's say,
just sitting there in a vacuum
in its lowest energy state,
it appears to just sit
there quote unquote
and the electron distribution,
the probability distribution
of the wave function
remains constant.
It doesn't fluctuate,
and that means
that psi star psi remains
constant at all times,
and that's a big constraint then
on what the wave
function can do over time.
Because that means if we use
now the capital wave function,
the dressed up one with an
explicit function of time in it,
the psi star psi at some
time t at all values of x has
to be the same as psi star
psi at some time or we call 0
where we start looking.
We start the experiment.
And that means that there's,
whatever happens in time
to some isolated state like
that, it can't be too violent
because if it were
some strange thing
that affected the way
wave function a lot,
moved it around a lot,
made extra lumps and stuff,
what would happen
is we'd notice it.
We'd see something changing
because we, the probability
of finding the particle and
so forth would be different.
And when we do the
measurement, of course,
we've destroyed whatever
probability distribution was
there, but we can do the
measurement over and over
and over, and we can find, look,
the chance of finding the
electron is, like, a sphere,
and it doesn't change.
If I wait a minute
later, it's the same.
And, therefore, that means
that the most we can do
to this wave function
psi is multiply it
by a phase factor
e to the i theta.
Why e to the i theta?
Well, the length of e
to the i theta is 1.
It's just in the complex
plane and arrow with length 1.
If theta's 0, it is 1.
If theta is 90 degrees, it's i,
which still has the same length
of 1, then minus 1
minus i, and so on.
And when I take psi star,
instead of either
the minus i theta,
I get e to the plus i theta,
and e to the anything times e
to the minus anything is
e to the 0, and that's 1.
And that means that the
probability distribution
stays put.
So now I have a clue.
When I have a state
with constant energy
like that that's just
sitting there, then it must be
that all I'm doing in
time is multiplying
by this thing that
has same length.
And so what I can imagine
is I have a distribution,
and the distribution could
be moving in time somehow,
but rather than thinking
of it moving like Jello
and moving around, what I
should do is just color it.
And it starts out white
and then it turns grey
and then it turns black,
or maybe it starts out red
and then goes through
the rainbow over time.
But it's shape doesn't change.
And the only thing we can
measure is not the color
but just the shape when
we make the measurement.
Now how could we get a phase
factor e to the i theta?
Well, we know that anything
in an exponential
can't have any units,
and we think it should
depend on time.
So we could put in
the exponential e
to the minus i times something,
let's call it epsilon,
times time.
And then because we got p times
x, let's try epsilon times time,
and epsilon must have the units
of inverse time,
which is not energy.
But even though the probability
distribution is stationery,
we'd expect the phase
to depend on energy.
In other words, although
this thing's staying put,
if the energy's high, the colors
are really flashing like crazy,
and if the energy is low, the
colors are hardly moving at all.
Very slow throbbing.
And taking a cue from
the presence of h bar
in the momentum eigen functions,
we could guess the following.
That capital psi at time t is
capital psi at time 0 times e
to the minus iet over h bar.
And that would be a very good
guess as it turns out because
if we put this guess into
the Schrodinger equation,
we can make a connection
then with a time derivative.
So now we take e psi, which
was the time independent part,
which was just the
energy, and we say, well,
if the wave function
is time dependent,
we'll write that as
e psi at 0 times e
to the minus iet over h bar.
But if that's true, then
the other way of writing
that to get rid of the e is to
write ih bar, just like we did
with momentum only
it was minus ih bar.
Ih bar time derivative of psi.
Because when I take the time
derivative, out comes minus iet
over h bar without the t, of
course, just the constant.
And so the ih bar and minus i
over h bar cancel, and I get e,
the energy, and that's
exactly what I want to have.
So using this on the
right-hand side of the equation,
now we have a proper
time dependent equation,
which is very much
like f equals ma.
The acceleration is the second
derivative with respect to time
of position, and what we have
now is we have the kinetic
energy, the wave function
minus h bar squared
over 2m times the
second derivative
of psi plus the potential
energy,
which is just the
potential energy times psi,
and that should equal e psi,
but if it's time dependent,
then it becomes ih bar times
the time derivative of psi.
And this is, in fact,
the one-dimensional time
dependent Schrodinger equation,
which is the basis for all kinds
of time-dependent calculations
that people carry out.
The wave function in this case
depends on both x and time.
And so to be mathematically
rigorous,
we have to use the funny d's.
Writing a capital D is not
proper because t could depend
on x, and then I'd have to
figure out dtdx is and dxdt
and so forth, and that's
not what I mean in this.
We're taking the derivative
with respect to space only
on one side, and
on the other side,
we're taking the derivative
with respect to time only, and,
therefore, we have to use
the funny d's, and that means
that this becomes a partial
differential equation
which can be very, very,
very difficult to solve.
These guys are bears, and
you take a course in PD's,
and you learn how to solve them.
When there are three spatial
dimensions rather than just one,
then the kinetic energy
adds up separately.
Px squared, py squared,
pz squared.
No big deal, and the potential
then becomes a function of x, y,
and z, which we usually
write as just r. Some vector
which tells you where you are,
and then we get a
fearsome looking equation
because we get minus
h bar squared over 2m,
then this upside down triangle,
the del squared operator
on the wave function
plus the potential,
and then we have the
time derivative of psi
on the other side
as the same thing
because there's still
only one time dimension.
The del operator or this
triangle thing is just a
shorthand because we get so
sick of writing d by dx squared,
d by dy squared, and then
you get carpal tunnel,
and you give up.
So just write this
triangle with three sides.
It's the three derivatives,
and the second derivative,
del squared, is just
del dot del.
So del with an arrow over it
or bold-faced del is a vector.
It takes the derivative with
respect to a certain position.
So, for example, if
you're on a mountain,
it might be that if
you walk this way,
but it's, the slope is 0.
It's the path, or
it's very slight.
And if you go this way,
the slope is very fast,
and you fall off a cliff.
And, likewise, the derivative
when you have a
multi-dimensional function can
depend on which way you're
looking, like, x or y,
and so you have to do that.
When you square an operator,
all you do is multiply
with it again, and that means
you operate with it again.
Just as if you multiply
by 2 twice,
you've operated by
2 squared or 4.
Now suppose we have
a free particle.
That means free particle,
there's no potential energy.
It's all kinetic.
And, for example, a neutron
on a nuclear reactor,
which is uncharged and
has no electric forces,
it's a neutral particle,
may be considered
until it hits something,
like a moderator,
to be a free particle.
It's going, it's very tiny.
It's going through.
It's mostly vacuum.
Remember atoms are
mostly empty space.
If you don't have
any electric charges,
you don't notice anything,
and you go right through,
which is why, one reason
why it's kind of tough
to shield neutrons
in some cases.
In this case, I've put
in explicitly here the
kinetic energy's the same,
h bar squared over 2m.
The potential energy is 0.
So I put a big 0, and
that's equal to e psi.
And given e, the kinetic energy
of the neutron, we want to find
out the wave function
psi of the neutron.
This is a second order
differential equation,
and may not be easy to solve if
you haven't seen them before,
but we know what the kinetic
energy of the neutron should be.
It should be p squared
over 2m as long
as it's not relativistic.
And we know that the derivative
of an exponential function,
any number of times gives
back an exponential function,
and we know when we operate
twice with the derivative,
we get the wave function
back times the number.
So we try a guess, a
solution like psi of x is e
to the ipx upon h bar.
And if we substitute our
solution for the solution
of the differential equation,
n, then what we find is
that it appears to work.
So I've just worked
it out here for you.
The first derivative of that
trial wave function gives a
factor of ip over h bar times
the same way function back.
The second derivative
gives it twice.
And so we get minus p squared
over h bar squared, and,
of course, the kinetic
energy operator, excuse me,
has h bar squared over 2m.
So if we have p squared
over h bar squared,
then we get p squared over 2m.
So that happens to work.
And so we find the
solution that works.
That's p squared over
2m is the energy.
The wave function is this
e to the ipx over h bar.
And, of course, if we
only expect the particle
to have kinetic energy,
and that's what we found.
The total energy is
p squared over 2m.
What we didn't find is
that there could have been
a minus i in the exponent.
So I picked e to the plus
ipx over h bar, but it turns
out that I could pick e to
the minus ipx over h bar.
Well, that's just the thing
moving the other direction.
The weird thing, though, is
that the general solution
is some part, A let's say,
times e to the ipx over h bar
plus another part B times e
to the minus ipx over h bar.
And if I put those in, it works.
And one part's a particle
moving to the right
with momentum p. The
other's moving to the left
with linear momentum p. And
it's perfectly acceptable
for the wave function to be
moving both ways at once,
and you might say, well, surely,
the particle can't possibly
be moving both ways at once.
What does that mean?
And the answer is it means
we're doing quantum mechanics
because the particle
actually can.
Until you measure it, it's
up to itself what it
wants to decide to do.
And so the same way that
the electron can be slipping
through both slits, the particle
can be moving both ways, and,
in fact, that's one of
the interesting things
about these kinds of systems.
In the next lecture,
then, what I want to talk
about is some one-dimensional
model problems
with certain well-defined
potentials,
and use these model problems to
show you with these equations
that we've built up exactly why
it is that atoms and molecules
and various other systems,
quantum dots have
quantized energy levels
with discrete energies.
So we'll leave it there, and
pick it up in Lecture Five.
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