Prof: So this is a very
exciting day for me,
because today,
we're going to start quantum
mechanics and that's all we'll
do till the end of the term.
Now I've got bad news and good
news.
The bad news is that it's a
subject that's kind of hard to
follow intuitively,
and the good news is that
nobody can follow it
intuitively.
Richard Feynman,
one of the big figures in
physics, used to say,
"No one understands
quantum mechanics."
 
So in some sense,
the pressure is off for you
guys, because I don't get it and
you don't get it and Feynman
doesn't get it.
 
The point is, here is my goal.
 
Right now, I'm the only one who
doesn't understand quantum
mechanics.
 
In about seven days,
all of you will be unable to
understand quantum mechanics.
 
Then you can go back and spread
your ignorance everywhere else.
That's the only legacy a
teacher can want.
All right, so that's the spirit
in which we are doing this.
I want you to think about this
as a real adventure.
Try to think beyond the exams
and grades and everything.
It's one of the biggest
discoveries in physics,
in science, and it's marvelous
how people even figured out this
is what's going on.
 
So I want to tell you in some
fashion, but not strictly
historical fashion.
 
Purely historical fashion is
pedagogically not the best way,
because you go through all the
wrong tracks and get confused,
and there are a lot of battles
going on.
When the dust settles down,
a certain picture emerges and
that's the picture I wanted to
give you.
In a way, I will appeal to
experiments that were perhaps
not done in the sequence in
which I describe them,
but we know that if you did
them, this is what the answer
would be,
and everyone agrees,
and they are the simplest
experiments.
All right, so today we're going
to shoot down Newtonian
mechanics and Maxwell's theory.
 
So we are like the press.
 
We build somebody up,
only to destroy them.
Built up Newton;
shot down.
Built up Maxwell;
going to get shot down.
So again, I have tried to drill
into all of you the notion that
people get shot down because
somebody else does a new
experiment that probes an
entirely new regime which had
not been seen before.
 
So it's not that people were
dumb;
it's that given the information
they had, they built the best
theory that they could.
 
And if you give me more
additional information,
more refined measurements,
something to the tenth decimal
place, I may have to change what
I do.
That's how it's going to be.
 
So there's always going to
be--for example,
in the big collider,
people are expecting to see new
stuff,
hopefully stuff that hasn't
been explained by any existing
theory.
And we all want that,
because we want some
excitement, we want to find out
new things.
The best way not to worry about
your old theories is to not do
any experiments.
 
Then you can go home.
 
But that's not how it goes.
 
You probe more and more stuff.
 
So here's what you do to find
out what's wrong with
electrodynamics,
I mean, with Maxwell's theory.
It all starts with a double
slit experiment.
You have this famous double
slit and some waves are coming
from here.
 
You have some wavelength l.
 
Then in the back,
I'm going to put a photographic
plate.
 
A photographic plate,
as you know,
is made of these tiny little
pixels which change color when
light hits them and then you see
your picture.
And that's the way to detect
light, a perfectly good way to
detect light.
 
So first thing we do is,
we block this hole or this
slit.
 
This is slit 1 and slit 2.
 
We block this and we look at
what happened to the
photographic plate.
 
What you will find is that the
region in front of it got pretty
dark,
or let's say had an image,
whereas if you go too far from
the slit,
you don't see anything.
 
So that's called intensity,
when one is open.
Then you close that guy and you
get similar pattern.
Then you open both.
 
Then I told you,
you may expect that,
but what you get instead--let's
see, I've got to pick my graph
properly-- is something that
looks like this.
Now that is the phenomenon of
interference,
which we studied last time.
 
So what's the part that's funny?
 
What's the part that makes you
wonder is if you go to some
location like this,
go to a location like this.
This used to be a bright
location when one slit was open.
It was also a bright location,
reasonably bright,
when the other slit was open.
 
But when both are open,
it becomes dark.
You can ask,
"How can it be that I open
two windows, room gets darker?
 
Why doesn't it happen there,
and why does it happen
here?"
 
The answer is that you're
sending light of definite
wavelength and the wave
function,
Y, whatever measures the
oscillation,
maybe electric field,
magnetic field,
obeys the superposition
principle.
And when two slits are open,
what you're supposed to add is
the electric field,
not the intensity.
The intensity is proportional
to the square of the electric
field.
 
You don't add E^(2);
you add E.
E is what obeys the wave
equation.
E_1 is a
solution, E_2
is a solution.
 
E_1
E_2 is a
solution.
 
No one tells you that if you
add the two sources,
I_1
I_2 is going to be
the final answer.
 
The correct answer is to find
E_1
E_2 and then
square that.
But when you do
E_1
E_2,
since E_1 and
E_2 are not
necessarily positive definite,
when you add them,
sometimes they can add with the
same sign,
sometimes they can add with
opposite signs,
and sometimes in between,
so you get this pattern.
 
So we're not surprised.
 
And I've told you many times
why we don't see it when we open
big windows,
first of all,
when you open a window,
the slit sizes are all many,
many, many million times bigger
than the wavelength of light.
Plus the light is not just one
color and so on.
So you don't pick up these
oscillations.
These oscillations are very
fine.
I draw them this way so you can
see them.
In a real life thing,
if this were really windows and
this was the back wall of your
house,
the oscillations would be so
tightly spaced,
I'll just draw some of them,
that the human eye cannot
detect these oscillations.
 
It will only pick up the
average value.
The average value will in fact
look like I_1
I_2.
 
So this is all very nice.
 
This is how Young discovered
that light is a wave.
By doing the interference,
I told you,
he could even find a
wavelength, because it's a
simple matter of geometry to see
where you've got to go for two
guys to cancel.
 
And once you know that angle,
you know where on that screen
you will get a minimum or a
maximum.
So you've got the wavelength.
 
He didn't know what was doing
it.
He didn't know it was
electromagnetic,
but you can get the wavelength.
 
So interference is a hallmark
of waves.
Any wave will do interference.
 
Water will do the same thing.
 
For example,
this is your beach house.
You've got some ocean front
property.
This is a little lagoon and you
have a wall to keep the ocean
waves out of your mansion.
 
And then suddenly one day,
there is a break in the wall.
Break in the wall,
the waves start coming in and
you're having a little boat
here, trying to get some rest.
The boat starts jumping up and
down because of the waves.
So you have two options.
 
One options is to go and try to
plug that thing,
but let's say you've got no
bricks, no mortar,
no time, no nothing.
 
You've just got a sledgehammer.
 
What you can do,
you can make another hole.
If these water waves are a
definite wavelength,
only in that case,
you can make another hole so
these two add up to 0 where you
are.
Similarly, if you don't like
the music your roommate's
playing,
if you can manufacture the same
music with a phase shift of p,
you can add them together,
you get 0.
 
But you've got to figure out
what the roommate's about to do
and be synchronous with the
person,
but get a wavelength of p--I
mean, a phase shift of p.
So you can cancel waves.
 
That's the idea behind all
kinds of noise cancelation,
but you've got to know the
exact phase of the one signal
that you're trying to cancel.
 
All right, so everything looks
good for Maxwell,
till you start doing the
following experiment.
You make the source of light,
whatever it is,
dimmer and dimmer,
okay?
So you may not be able to turn
down the brightness.
Maybe you can,
maybe you cannot,
but you can imagine moving the
source further and further back.
You move it further and further
back, you know the energy falls
like 1/r^(2),
so you can make it weak.
So here's what we try to do.
 
We put a new photographic film.
 
We take the light source way
back, then we wait for something
to happen.
 
We come the next morning,
we find there's a very faint
pattern that's taken place over
night, because the film got
exposed all through the night.
 
Now we can see a faint pattern.
 
Then you go and turn it down
even more.
You come back the next day,
you look at the film.
You find no pattern,
just two or three spots which
have been exposed.
 
If you look at the screen--so
let me show you a view of the
screen.
 
Normally you will have bright
and dark and bright and dark
patterns on the back wall if you
turn on a powerful light.
But I'm telling you,
if you have a really weak
source, you just find that got
exposed, that got exposed and
that got exposed,
that's it.
Only three points on the film
are exposed, and that is very
strange.
 
Because if light is a wave,
no matter how weak it is,
it should hit the entire
screen.
It cannot hit certain parts of
it.
Waves don't hit certain parts.
 
In fact, how can it hit just
one?
For example,
if you make it weak enough,
you can have a situation where
in the whole day,
you just get one hit.
 
So something is hitting that
screen and it's not a wave,
because a wave is spread out
over its full transverse
dimension,
but this is hitting one point
on the screen.
 
So you make further
observations and you find out
that what happens here is there
is a certain amount of momentum
and energy are delivered during
that hit.
If you could measure the recoil
of that film,
you will find it gets hit and
the momentum you get per hit
looks like ℏ x
k.
I'll tell you what it means.
 
This is 2p
h/l,
where l is your
wavelength, h is the new
constant called the Planck's
constant.
And its value is 6.6 x 10^(-34
)joule seconds.
You also find,
every time you get a hit here,
there's a certain energy
deposited here and the energy
deposited here happens to be
ℏw where
w,
as you know,
is 2pf.
 
So here's what I'm telling you.
 
If you send light of a known
frequency and known wavelength
and you make it extremely dim,
and you put a photographic
plate and you wait till
something happens,
what happens is not a thin blur
over the whole screen.
What happens is a hit at one
location.
And what comes to that location
seems to be a bundle of energy
and momentum,
i.e.
a particle, right?
 
When something hits you in the
face, it's got energy,
it's got momentum.
 
So this film is getting hit at
one point by a particle,
and what we can say about the
particle is the following - it
has a momentum.
 
It has the same momentum every
time.
You get this hit,
you get that hit,
you get that hit.
 
As long as you don't screw
around with the wavelength of
incoming light,
the momentum and energy of each
packet is identical.
 
It's more than saying light
seems to be made of particles.
Made of particles,
each one of them carries an
energy and momentum that's
absolutely correlated with a
wavelength and frequency.
 
Now let me remind you that
w = kc for light
waves.
 
We've done this many,
many times.
That means the energy,
which is
ℏw,
and the momentum is
ℏk,
are related by the relation
E = pc.
 
So these particles have a
momentum which is related to
energy by the formula E =
pc.
When you go back to your
relativity notes from the last
semester, you'll find the
following relation is true.
Any particle,
E^(2) = c^(2)p^(2)
m^(2)c^(4).
 
That's the connection between
energy and momentum.
Therefore this looks like a
particle whose m is 0.
If m is 0, E = pc.
 
So these particles are massless.
 
They have no rest mass and you
know, something with no rest
mass, if it is to have a
momentum, it must travel at the
speed of light.
 
Because normally,
the momentum of anything with
mass is mv,
in the old days,
divided by this,
after Einstein.
And if you don't want to have
an m, and yet you want to
have a p,
the only way it can happen is
that v = c.
 
Then you have 0/0,
there is some chance,
and nature seems to take
advantage of that 0/0.
These are the massless
particles.
So these photons are massless
particles.
So what is the shock?
 
The shock is that light,
which you thought was a
continuous wave,
is actually made up of discrete
particles.
 
In order to see them,
your light source has to be
extremely weak,
because if you turn on a light
source like this one,
millions of these photons come
and the pattern is formed
instantaneously.
The minute you turn on the
light, the film is exposed,
you see these dark and light
and dark and light fringes,
you think it's happening due to
waves that come instantaneously.
But if you look under the hood,
every pattern is formed by tiny
little dots which occur so fast
that you don't see them.
That's where you've got to turn
down the intensity to actually
see them.
 
When you see that,
you see the corpuscular nature
of light.
 
But here is the problem - if
somebody told you light is made
of particles and it's not
continuous,
it's not so disturbing,
because water,
which you think is continuous,
is actually made of water
molecules.
 
Everything that you think of as
continuous is made up of little
molecules and in a bigger scale,
much bigger than the atomic or
molecular size,
it looks like it's continuous.
That's not the bad news.
 
The bad news really has to do
with the fact that if you have
these particles called photons,
if it really were a particle,
namely a standard,
garden variety particle,
what should you find?
 
If you emit a particle from
here, and only one slit is open,
it will take some path going
through the slit and it will
come there.
 
So let us say on a given day,
10 photons will come here,
or let's say 4 photons will
come here, with this one closed.
Then let's close this one and
open this one.
In that, case,
maybe 3 photons will come here.
Or let's say again, 4.
 
Now I'm claiming that when both
are open, I get no photons.
How can it be that when you
open a second hole,
you get fewer particles coming
there?
Particles normally either take
path number 1 or path number 2.
Either this slit or this slit.
 
To all the guys going this way,
they don't care if this slit is
open or closed.
 
They don't even know about the
other slit.
They do their thing,
and guys going through this
slit should do their thing,
therefore you should get a
number equal to the sum,
but you don't.
In other words,
for particles,
which have definite
trajectories,
opening a second slit should
not affect the number going
through the first slit.
 
Do you understand that?
 
Particles are local.
 
They're moving along and they
feel the local forces acting on
them and they bend or twist or
turn.
They don't really care what's
happening far away,
whether a second slit may be
open or closed.
Therefore logically,
the number coming here must be
the sum of the number that would
come with 1 open and 2 open.
How can you cancel a positive
number of particles coming
somewhere with more positive
number of particles coming from
somewhere else?
 
How do you get a 0?
 
That is where the wave comes in.
 
The wave has no trouble knowing
how many slits are open,
because the wave is not
localized.
The wave comes like this.
 
It can hit both the slits and
certainly cares about how many
slits there are.
 
Because there's only one,
that wave will go.
You'll have some amplitude
here, which is kind of
featureless.
 
If that's open,
it will be featureless.
If both are open,
there'll be interference.
So we need that wave to
understand what the photon will
do,
because when you send millions
of photons and if you get the
pattern like this--
let's say you sent lots and
lots of photons and you got a
pattern like this.
 
Now I'm going to send million 1
photon.
Where will it go?
 
We do not know where it will go.
 
We only know that if you repeat
the experiment a million times,
you get this pattern.
 
But on the million 1th attempt,
where it will go,
we don't know.
 
We just know that the odds are
high when the function is high,
or the intensity,
and the odds are low when the
function is small and the odds
are 0 when the function is 0.
So the role of the wave is to
determine the probability that
the photon will arrive at some
point on the screen.
And the probability is computed
by adding one wave function to
another wave function and then
squaring.
So you've got to be very clear.
 
If someone says to you,
"Is the photon a particle
or a wave?
 
Make up your mind,
what is it?"
Well, the answer is,
it's not going to be a yes or
no question.
 
People always ask you,
"Is matter made of
particles or waves,
electron particles or
waves?"
 
Well, sometimes the vocabulary
we have is not big enough to
describe what's really
happening.
It is what it is.
 
It is the following.
 
It is a particle in the sense
that the entire energy is
carried in these localized
places, unlike a wave.
When the wave hits the beach,
the energy's over the entire
wave front.
 
This wave here is not a
physical wave.
It does not carry any energy
and it's not even a property of
a beam of photons.
 
It's a property of one photon.
 
Here's what I want you to
understand - you send one photon
at a time, many,
many times, and you get this
pattern.
 
Each time you throw the die and
ask where will the photon land,
this function is waiting to
tell you the probability it will
land somewhere.
 
So we have to play this game in
two ways.
It is particles,
but its future is determined by
a wave.
 
The wave is purely mathematical.
 
You cannot put an instrument
that measures the energy due to
that wave.
 
It's a construct we use to
determine what will happen in
this experiment.
 
So we have no trouble
predicting this experiment,
but we only make statistical
predictions.
So if someone tells you,
"I got light from some
mercury vapor or something,
it's got a certain wavelength,
therefore a certain frequency.
 
I'm going to take two slits and
I'm going to send the light from
the left so weak that at a given
time,
only one photon leaves the
source and hits the screen.
What will happen?"
 
We will say we don't know what
it will do.
We don't know where it will
land.
But we tell you if you do it
enough times,
millions of times,
soon a pattern will develop.
Namely, if you plot your
histogram on where everybody
landed, you'll get a graph.
 
It's the graph that I can
predict.
And how do I predict that graph.
 
I say, "What was the
energy momentum of your
photon?"
 
If it was p,
I will introduce a wave whose
momentum is
2pℏ/p.
Oh, I'm sorry,
I forgot to tell you guys one
thing.
 
I apologize.
 
I've been writing
ℏ and h.
I should have mentioned it long
back, ℏ is
h/2p.
 
Since the combination occurs so
often, people write
ℏ.
 
So you can write l =
2pℏ/p,
or h/p.
 
It doesn't matter.
 
So I've stopped using h.
 
Most people now in the business
use ℏ,
because the energy is
ℏw,
the momentum is
ℏk.
If you want,
you can write this
2phk and k
is 2p/l.
Then you find p is
h/l.
That's how some people used to
write it in the old days,
but now we write it in terms of
ℏ and k.
Anyway, I can make these
predictions, if I knew the
momentum of the photons.
 
The photons were of a definite
momentum, therefore there's a
definite wavelength.
 
I can predict the interference
pattern.
So where is the photon when it
goes from start to finish?
We don't know.
 
I'll come back to that question
now.
But I want to mention to you a
historical fact,
which is, photons were not
really found this way,
by looking at the recoil of an
emulsion plate.
Just for completeness,
I'm going to make a five minute
digression to tell you how
photons were found.
So they were actually predicted
by Einstein.
He got the Nobel Prize for
predicting the photon,
rather than for the Theory of
Relativity, which was still
controversial at that time.
 
So he predicted the photons,
based on actually fairly
complicated thermodynamic
statistical mechanics arguments.
But one way to understand it is
in terms of what's called the
photoelectric effect.
 
If you take a metal and you say
"Where are the electrons in
the metal?"
 
As you know most electrons are
orbiting the parent nucleus.
But in a metal,
some electrons are communal.
Each atom donates one or two
electrons to the whole metal.
They can run all over the metal.
 
They don't have to be near
their parent nucleus.
They cannot leave the metal.
 
So in a way, they are like this.
 
There's a little tank whose
depth is h,
and let's say mgh I want
to call W.
So these guys are somewhere in
the bottom.
They can run around;
they cannot get out.
So if you want to yank an
electron out of the metal,
you have to give an energy
equal to W,
which is called the work
function.
So how are you going to get an
electron to acquire some energy?
We all know.
 
Electron is an electric charge.
 
I have to apply an electric
field and I know electromagnetic
waves are nothing but electric
and magnetic fields,
so I shine a light,
a source of light,
towards this.
 
The electric field comes and
grabs the electron and shakes it
loose.
 
Hopefully it will shake it
loose from the metal,
giving it enough energy to
escape.
And once it escapes,
it can take off.
So they took some light source
and they aimed it at the metal,
to see if electrons come out.
 
They didn't.
 
So what do you think you will
do to get some action?
Yes?
 
Student: 
>
Prof: So you make it
brighter.
You say, "Okay,
let me crank up--"
that's what anybody would do.
 
They cranked up the intensity
of light, make it brighter and
brighter and brighter.
 
Nothing happened.
 
Then by accident they found out
that instead of cranking up the
brightness of the light,
if you cranked up the frequency
of light,
slowly, suddenly beyond some
frequency,
you start getting electrons
escaping the metal.
 
So here's the graph you get.
 
Let me just plot it if you
like, ℏ times
the w.
 
In those days,
they didn't know too much about
ℏ.
 
You can even plot w.
 
It doesn't matter.
 
And you plot here the kinetic
energy of the emitted electron.
And what you find is that below
some minimum value,
no electrons come out.
 
There's nothing to plot.
 
And once you cross a magical
w, and anything higher
than that, you get a kinetic
energy that's linear in
w.
 
Now the kinetic energy is the
energy you gave to the electron
minus W.
 
Energy given to the electron -
W, because you paid
W to get it out of the
well, and whatever is left is
the kinetic energy.
 
So Einstein predicted photons
from independent arguments,
and according to him,
light and frequency w is
made up of particles,
each of which contains energy
ℏw.
 
So you can see what's happening.
 
If you've got low frequency
light, you're sending millions
of photons, each carries an
energy ℏw
somewhere here.
 
None of them has the energy to
lift the electron out of the
metal.
 
It's like sending a million
little kids to lift something
and they cannot do it.
 
They cannot do it,
but if you send 10 tall,
powerful people,
they will lift it out.
So what's happening with light
is that as you crank up the
w,
even if it's not very bright,
the individual packets are
carrying more and more energy
and more and more momentum,
and that's why they succeed in
knocking the electron out.
 
And in fact,
if you set the energy of each
photon,
it's ℏw,
then the kinetic energy of the
electron is the energy you gave
with 1 photon,
take away the W,
that's the price you pay to
leave the metal.
The rest of it is kinetic
energy.
So when plotted as a function
of w, K should
look like a straight line with
intercept W.
And that's what you find.
 
In fact, this is one way to
measure the work function.
How much energy do we need to
rip an electron out of a metal
depends on the metal.
 
And you shine light and you
crank up the frequency,
till something happens.
 
And just to be sure,
you go a little beyond that and
you find that the kinetic energy
grows linearly in w.
Anyway, this is how one
confirmed the existence,
indirect existence,
of photons.
There's another experiment that
also confirmed the existence of
photons.
 
Look, that's the beauty.
 
Once you've got the right
answer, everything is going to
be on your side.
 
Before I forget,
I should mention to you,
you've probably heard that
Einstein is very unhappy with
quantum mechanics.
 
And yet if you look at the
history, he made enormous
contributions to quantum
mechanics.
Even Planck didn't have the
courage to stand behind the
photons that were implied in his
own formula.
Einstein took it to be very
real and pursued it.
So when you say he doesn't like
quantum mechanics,
it's not that he couldn't do
the problem sets.
It's that he had problems with
the problems.
He did not like the
probabilistic nature of quantum
mechanics, but he had no trouble
divining what was going on.
So it's quite different.
 
It's like saying,
"I don't like that
joke."
 
There are two reasons.
 
Some guys don't get it and they
don't like it.
Some guys get it and don't
think it's funny.
So this was like Einstein
certainly understood all the
complexities of quantum
mechanics.
He said he had spent more time
on quantum,
much more on either the special
or the general theory of
relativity,
because he said that was a real
problem.
 
That's a problem I couldn't
track.
Now it turns out that even till
the end, he didn't find an
answer that was satisfactory to
him.
The answer I'm giving you
certainly works,
makes all the predictions,
never said anything wrong.
Until something better comes to
replace it, we will keep using
it.
 
Anyway, going back,
the second experiment that
confirmed the reality of
photons.
See, if you say light is made
of particles and each one has an
energy and momentum,
do you understand why the
photoelectric effect is a good
test.
It agrees with that picture.
 
Individual particles come.
 
Some have the energy to
liberate the electron and some
don't.
 
And if individually,
they cannot do it,
it doesn't matter how many you
send.
Now you may have thought of one
scenario in which all of these
tiny little kids can get
something lifted out of the
well.
 
How will they do that?
 
Maybe 10 kids together,
like ants, can lift the thing
out.
 
So if you had 10 photons which
can collectively excite the
electron,
it can happen,
but in those days,
they didn't have a light whose
intensity was enough to send
enough of these photons.
But nowadays,
it turns out that if you
really, really crank up the
intensity, you can make
electrons come out,
even below the frequency.
That's because more than one
photon is involved in ejecting
the electron.
 
So luckily, we didn't have that
intensity then,
so we go the picture of the
photons right.
Anyway, Compton said the
following thing - it turns out
that if you have an electron
here and you send a beam of
light,
it scatters off the electron
and comes off in some direction
at an angle q to the original
direction.
 
The wavelength here changes by
an amount Dl,
and Dl happens to be
2pℏ
/mc x 1 - cosine
q.
Are you with me?
 
You send light in at a known
wavelength.
It scatters off the electron
and comes at an angle q,
no longer preserving its
wavelength, having a different
wavelength.
 
And the shift in the wavelength
is connected to the angle of
scattering.
 
For example,
if q is 0, Dl is 0 in the
forward direction.
 
If it bounces right back,
that cos q is -1.
That number is 2 and you get a
huge Dl.
And you can find the l of it by
putting a diffraction grating.
Now, what one could show is
that if you took this to be made
of particles,
and each particle has an
energy, ℏw,
and each particle has a
momentum, ℏk,
and that that collides with an
electron,
then you just balance energy
conservation and momentum
conservation.
In any collision,
energy and momentum before =
energy and momentum after.
 
You set them equal and you
fiddle around,
you can find the new momentum
after scattering.
From the new momentum,
you can extract the new
wavelength and you will find
this formula actually works.
So I did that in Physics 200,
I think, so if you want,
you can go look at that,
or maybe it was done for you.
I don't know.
 
But Compton's scattering,
the scattering due to Compton,
can be completely understood if
you think of the incoming beam
of light as made up of particles
with that momentum and that
energy.
 
In other words,
you're always going to go back
and forth.
 
Light will be characterized by
a wavelength and by a momentum.
It will be characterized by a
frequency and by an energy.
When you think about the
particles, you'll think of the
energy and momentum.
 
When you think about the waves,
you'll think of frequency and
wave number.
 
So this is what really nailed
it.
After this, you could not doubt
the reality of the photons.
Okay, now I go back to my old
story.
Let's remember what it is.
 
The shock is that light,
which we were willing to
believe was waves,
because Young had done the
interference experiment,
is actually made up of
particles.
 
That's the first thing.
 
So who needs the wave?
 
If you send a single photon
into a double slit,
we don't know what it will do.
 
We can only give the odds.
 
To find the odds,
we take the photon's wavelength
and we form this wave,
and then we form the
interference pattern.
 
And we find out that whenever
it is high, it is very likely to
come.
 
Wherever it's low,
it's very unlikely,
but at 0, it won't come.
 
So to test this theory,
it's not enough to send 1
photon.
 
1 photon may come here;
that doesn't show you anything.
You've got to send millions of
photons, because if a prediction
is probabilistic,
to test it, you've got to do
many times.
 
If I give you a coin,
and I tell you it's a fair
coin, I toss it a couple of
times and I get 1 head and 1
tail, it doesn't mean anything.
 
You want to toss it 500,000
times and see if roughly half
the time it's heads and half the
time it's tails.
That's when a probabilistic
theory is verified.
It's not verified by
individuals.
Insurance companies are always
drawing pictures of when I'm
going to die.
 
They've got some plot,
and that's my average chance.
I don't know when I will be
part of that statistic,
because in fact--sorry,
it usually looks like this.
Life expectancy of people looks
like that, but doesn't mean
everybody dies at one day.
 
People are dying left and
right, so there's probability on
either side.
 
So to verify this table that
companies have got,
you have to watch a huge
population.
Then you can do the histogram
and then you get the profile.
So whenever you do statistical
theories, you've got to run it
many times.
 
I'll tell you more about
statistics and quantum
mechanics.
 
It's different from statistics
and classical mechanics and
we'll come to that later.
 
But for now,
you must understand the
peculiar behavior of photons.
 
They are not particles
entirely, they are not waves
entirely.
 
They are particles in the sense
they're localized energy and
momentum, but they don't travel
like Newtonian particles.
If they were Newtonian
particles, you'll never
understand why opening a second
slit reduced the amount of light
coming somewhere.
 
All right, so this is the story.
 
So now comes the French
physicist,
de Broglie, and he argued as
follows - you'll find his
argument quite persuasive,
and this is what he did for his
PhD.
 
He said, "If light,
which I thought was a
particle--
I'm sorry, which I thought was
a wave,
is actually made up of
particles, perhaps things which
I always thought of as
particles,
like electrons,
have a wave associated with
them."
And he said,
"Let me postulate that
electrons also have a wave
associated with them and that
the wavelength associated with
an electron of momentum p
will be 2pℏ
/p;
and that this wave will produce
the same interference pattern
when you do it with electrons,
as you did with light."
So what does that mean?
 
It means if you did a double
slit experiment,
and you sent electrons of
momentum p,
one at a time,
and you sit here with an
electron detector,
or you have an array of
electron detectors,
he claims that the pattern will
look like this,
where this pattern is obtained
by using a certain wavelength
that corresponds to the momentum
of the incoming beam of
electrons.
Now there the shock is not that
the electron hits one point on
the screen.
 
It supposed to;
it's a particle.
What is shocking is that when
two slits are open,
you don't get any electrons in
the location where you used to
get electrons.
 
That is the surprising thing,
because if an electron is a
Newtonian particle and you used
to go like that through hole 1,
and you used to go like that
through hole 2,
if you open the two holes and
two slits,
you've got to get the sum of
the two numbers.
You cannot escape that,
because in Newtonian mechanics,
an electron either goes through
slit 1 or through slit 2.
And therefore,
the number coming here is
simply the sum of the ones that
went here, the ones that went
here.
 
Now sometimes people think,
"Well,
if you have a lot of electrons
coming here,
maybe these guys bumped into
these guys and collided and
therefore didn't hit the screen
at that point."
That's a fake.
 
You know you don't have much of
a chance with that explanation,
because if there are random
collisions,
what are the odds they'll form
this beautiful,
repeatable pattern?
 
Not very big.
 
Furthermore,
you can silence that criticism
by making the electron gun that
emits electrons so feeble that
at a given time,
there's only one electron.
There's only one electron in
the lab.
It left here,
then it arrived there.
And it cannot collide with
itself.
And yet it knows two slits are
open.
A Newtonian particle cannot
know that two slits are open.
So it has an associated wave,
and if you do this calculation
and you find the interference
pattern, that's what electrons
do.
 
Originally, it was not done
with a double slit.
It was done with a crystal.
 
I have given you one homework
problem where you can see how a
crystal of atoms regularly
arranged can also help you find
the wavelength of anything.
 
And you shine a beam of
electrons on a crystal,
you find out that they come out
in only one particular angle,
and using the angle,
you can find the wavelength,
and the wavelength agrees with
the momentum.
The momentum of the electron is
known,
because if you accelerate them
between two plates with a
certain voltage,
V, and the electron
drops down the voltage,
it gains an energy eV,
which is ½
mv^(2),
which you can also write as
p^(2)/2m.
So you can find the momentum of
an electron before you send it
in.
 
Okay, so this is the
peculiarity of particles now.
Electron also behaves like a
particle or a wave.
So now you can ask yourself the
following question.
Why is it that microscopic
bodies--first of all,
I hope you understand how
surprising this is.
Suppose it was not electrons.
 
Suppose this was not an
electron gun,
but a machine gun,
okay?
And these are some concrete
barriers.
The barrier has a hole in it
and that's you.
They've tied you to the back
wall and they're firing bullets
at you,
and you're of course very
anxious when a friend of yours
comes along and says,
"I want to help you."
 
So let me do that.
 
So you know that that's not a
friend, and if you do it with
bullets, it won't help.
 
You cannot reduce the number of
bullets.
And why is it with
electrons--if instead of the big
scenario,
we scale the whole thing down
to atomic dimensions,
and you're talking about
electrons and slits which are a
few micrometers away,
why is it that with electrons,
you can do that?
Why is it with bullets you
don't do that?
The answer has to do with this
wavelength p.
If you put for p,
m x v and you put
for m the mass of a
cannonball or a bullet,
say 1 kilogram,
you will find this wavelength
is 10^(-27 )something.
 
That means these oscillations
will have maybe 10^(20
)oscillations per centimeter and
you cannot detect that.
So oscillation,
the human eye cannot detect
that, and everything else looks
like you're just adding the
intensities, not adding the wave
function.
It looks like the probabilities
are additive,
and you don't see the
interference pattern.
Now there's another very
interesting twist on this
experiment, which is as follows.
 
You go back to that experiment,
and you say,
"Look, I do not buy this
notion that an electron does not
go through one slit.
 
I mean, come on.
 
How can it not go through one
particular slit?"
So here's what I'm going to do.
 
I'm going to put a light bulb
here.
I'm going to have the light
bulb look at the slit,
and when this guy goes past,
I will see whether the guy went
through this slit or through
that slit.
Then there's no talk about
going through both slits or not
going through a definite slit or
not having the trajectory.
All that's wrong,
because I'm going to actually
catch the electron in the act of
going through one or the other
by putting a light source.
 
So you put a light source,
and whenever it hits an
electron, you will see a flash
and you will know whether it was
near this hole or that hole.
 
You make a tally.
 
So you find that a certain
number went through hole 1,
a certain number went through
hole 2.
You add them up,
you get the number,
you cannot avoid getting the
number.
Let's imagine that of our 1,000
electrons, about 20 got by
without your seeing them.
 
It can happen.
 
When you turn the light,
you don't see it;
it misses.
 
Then you will find a pattern
that looks like this.
There'll be a 2 percent wiggle
on top of this featureless
curve.
 
In other words,
the electrons that you caught
and identified as going through
slit 1 or slit 2,
their numbers add up the way
they do in Newtonian mechanics,
but the electrons you did not
catch,
who slipped by,
pretend as if they went through
both the slits,
or at least they showed the
interference pattern.
 
That's a very novel thing,
that whether you see the
electron or not,
makes such a difference.
That's all I did.
 
In one case,
I caught the electron.
In the other case, I slipped by.
 
And whenever it's not observed,
it seems to be able to somehow
be aware of two slits.
 
And this was a big surprise,
because normally when we study
anything in Newtonian mechanics,
you say here's a collision,
ball 1 collides and goes there,
you do all the calculations.
Meanwhile, we are watching it.
 
Maybe we are not watching it.
 
Who cares?
 
The answer doesn't depend on
whether we are watching or not?
For example,
if you have a football game,
and somebody throws the pass,
and you close your eyes,
which sometimes my kids do,
because they don't know what's
happening,
that doesn't change the outcome
of the experiment.
 
It follows its own trajectory.
 
So what does seeing do to
anything?
And you can say maybe he didn't
see it, but maybe people in the
stadium were looking at the
football.
So turn off all the lights.
 
Then does the football have a
definite trajectory from start
to finish?
 
It does, because it's colliding
with all these air molecules.
To remove all the air
molecules, of course,
first you remove all the
spectators, then you remove all
the air molecules.
 
Then does it have a definite
trajectory?
You might say,
"Of course it does.
What difference does it
make?"
But then you would be wrong.
 
You would be wrong to think it
had a trajectory,
because the minute you said it
had a trajectory,
you will never understand
interference,
which even a football can show.
 
But the condition is,
for a football to show this
kind of quantum effects,
it should not be disturbed by
anything.
 
It should not be seen.
 
Nothing can collide with it.
 
The minute you interact with a
quantum system,
it stops doing this wishy-washy
business of "Where am
I?"
 
Till you see it,
it's not anywhere.
Once you see it,
it's in a different location.
Till you see it,
it's not taking any particular
path.
 
To assume it took this or that
path is simply wrong.
But the act of observation
nails it.
So why is observation so
important?
You have to ask how we observe
things.
We shine light.
 
You've already seen,
the light is made of quanta,
and each quantum carries a
certain momentum and certain
energy.
 
If I want to locate the
electron with some waves,
with some light,
I want the momentum of the
light to be weak,
because I don't want to slam
the electron too hard in the act
of finding it.
So I want p to be very
small.
If p is very small,
l,
which is 2pℏ
/p becomes large,
and once l's bigger than
the spacing between the slits,
the picture you get will be so
fuzzy,
you cannot tell which slit it
went through.
In other words,
to make a fine observation in
optics, you need a wavelength
smaller than the distances
you're trying to resolve.
 
So you've got to use a
wavelength smaller than these
two slits.
 
So this p should be such
that this l is comparable
to this slit,
or even smaller.
But then you will find the act
of observing the electron
imparts to it an unknown amount
of momentum.
Once you change the momentum,
you change the interference
pattern.
 
So the act of observation,
which is pretty innocuous for
you and me--
right now, I'm getting slammed
by millions of photons,
but I'm taking it like a man.
But for the electron,
it is not that simple.
One collision with a photon is
like getting hit by a truck.
The momentum of the photon is
enormous in the scale of the
electron.
 
So it matters a lot to the
electron.
For example,
when I observe you,
I see you because photons
bounce back and forth.
Suppose it's a dark room and I
was swinging one of those things
you see in Gladiator.
 
What's that thing called?
 
Trying to locate you.
 
So the act of location,
you realize it will be
memorable for you,
because it's a destructive
process.
 
But in Newtonian mechanics,
we can imagine finding gentler
ways to observe somebody and
there's no limit to how gentle
it is.
 
You just say make the light
dimmer and dimmer and dimmer
till the person doesn't care.
 
But in quantum theory,
it's not how dim the light is.
If the light is too dim,
there are too few photons and
nobody catches the electron.
 
In order to see the electron,
you've got to send enough
photons.
 
But the point is,
each one carries a punch which
is minimum.
 
It cannot be smaller than this
number, because if the
wavelength is bigger than this,
you cannot tell which hole it
went through.
 
That's why in quantum theory,
the act of observation is very
important, and it can change the
outcome.
Okay, so what can we figure out
from this.
Well, it looks like the act of
observing somehow affects the
momentum of the electron.
 
So people often say that's why,
when you try to measure the
position of the electron,
you do something bad to the
momentum of the electron.
 
We change it,
because you need a large
momentum to see it very
accurately.
But that statement is partly
correct but partly incomplete
and I'll tell you what it is.
 
The trouble is not that you use
a high momentum photon to see an
electron precisely.
 
That's not a problem.
 
The problem is that when it
bounces off the electron and
comes back to you,
it would have changed,
the momentum by an amount that
you cannot predict,
and I'll tell you why that is
the case.
So I told you long back that if
you have a hole and light comes
in through it,
it doesn't go straight,
it fans out,
that the profile of light looks
like this.
 
It spreads out and the angle by
which it spreads out obeys the
condition dsinq =
l.
Remember that part from wave
theory of light.
Now here is the person trying
to catch an electron,
which is somewhere around this
line.
And he or she brings a
microscope that looks like this.
Here's the opening of the
microscope, and you send some
light.
 
This opening of the microscope
has some extent d.
Let's say it's got a sharp
opening here of width d.
The light comes,
hits an electron,
if it is there,
and goes right back to the
microscope.
 
If I see a flicker of reflected
light, I know the electron had
to be somewhere here,
because if it's here,
it's not going to collide with
the light.
So you agree,
this is a way to locate the
electron's position with an
uncertainty, which is roughly
d, right?
 
The electron had to be in front
of the opening of the microscope
for me to actually see that
flash.
So I make an electron
microscope with a very tiny
hole,
and I'm scanning back and
forth, hoping one day I will hit
an electron and one day I hit
the electron,
it sends the light right back.
This has momentum p.
 
It also sends back with
momentum p,
but there's one problem.
 
You know that light entering an
aperture will spread out.
It won't go straight through.
 
This is this process.
 
So if you think of this light
entering your microscope,
it spreads out.
 
If it spreads out,
it means the photon that
bounced back can have a momentum
anywhere in this cone.
And we don't know where it is.
 
All we know is it re-entered
the microscope,
entered this cone,
but anywhere in this cone is
possible,
because there's a sizeable
chance the light will come
anywhere into this diffracted
region.
 
That means the final photon's
momentum magnitude may be
p_0,
but its direction is indefinite
by an amount q.
 
Therefore the photon's momentum
has a horizontal part,
p_0sine q,
which is an uncertainty in the
momentum of the photon in the
x direction.
This is my x direction.
 
So now you can see that
Dpx =
p_0sine q.
 
Sine q is l over the
width of the slit.
And l was
2pℏ
/p_0
over d.
You can see that these
p_0's cancel,
then you get d x
Dpx =
2pℏ.
 
By the way, another good news
is I'm going to give you very
detailed notes on quantum
mechanics.
I'm not following the textbook,
and I know you have to choose
between listening to me and
writing down everything.
So everything I'm saying here,
you will find in those notes,
so don't worry if you didn't
get everything.
You will have a second chance
to look at it.
But what you find here is that
d x Dpx is
2pℏ,
but d is the uncertainty
in the location of the electron,
so you get Dx,
Dpx, I'm not going to
say =,
roughly of order,
ℏ.
Forget the 2p's and everything.
 
This is a very tiny number,
10^(-34), so we don't care if
there are 2p's.
 
But what this tells you is that
in the act of locating the
electron--so let's understand
why.
It's a constant going back and
forth between waves and
particles, okay?
 
That's why this happens.
 
I want to see an electron and I
want to know exactly where I saw
it.
 
So I take a microscope with a
very small opening,
so that if I see that guy,
I know it has to be somewhere
in front of that hole.
 
But the photon that came down
and bounced off it,
if you now use wave theory,
the wave will spread out when
it re-enters the cone by minimal
angle q,
given by dsine q =
l.
That means the photon will also
come at a range of angles,
spread out, but if it comes at
a tilted angle,
it certainly has horizontal
momentum.
That extra horizontal momentum
should be imparted to the
particle, because initially,
the momentum of this thing was
strictly vertical.
 
So the photon has given a
certain horizontal momentum to
the electron and you don't know
how much it has given.
And smaller your opening,
so the better you try to locate
the electron,
bigger will be the spreading
out, and bigger will be the
uncertainty in the reflected
photon and therefore uncertainty
in the electron after collision.
So before the collision,
you could have had an electron
with perfectly well known
momentum in the x
direction.
 
But after you saw it,
you don't know its x
momentum very well,
because the photon's x
momentum is not known.
 
I want you to appreciate,
it's not the fact that the
photon came in its large
momentum that's the problem;
it is that it went back into
the microscope with a slight
uncertainty in its angle,
that comes from diffraction of
light.
 
It's the uncertainty of the
angle that turns into
uncertainty in the x
component of the momentum.
So basically,
collision of light with
electrons leaves the electron
with an extra momentum whose
value we don't know precisely,
because the act of seeing the
photon with the microscope
necessarily means it accepts
photons with a range of angles.
 
Okay, so now I want to tell you
a little more about the
uncertainty principle in another
language.
The language is this - here is
a slit.
Okay, here's one way to state
the uncertainty principle.
I challenge you to produce for
me an electron whose location is
known to arbitrary accuracy and
whose momentum,
in the same dimension,
same direction,
is also known to arbitrary
accuracy.
I dare you to make it.
 
In Newtonian mechanics,
that's not a big deal.
So let's say this is the
y direction,
and you say,
"I'll give you an electron
with precisely known y
coordinate,
and no uncertainty in y
momentum by the following trick.
I'll send a beam of electrons
going in this direction,
in the x direction,
with some momentum
p_0 and I put a
hole in the middle.
The only guys escaping have to
come out like this.
So right outside,
what do I have?
I have an electron whose
vertical momentum is exactly 0,
because the beam had no
vertical momentum,
whose vertical position = the
width of the slit.
It's uncertain by the width of
the slit, and I can make the
width as narrow as I like.
 
I can make my filter finer and
finer and finer,
till I'm able to give the
electrons a perfectly well
defined position and perfectly
well defined momentum,
namely 0.
 
That's true in Newtonian
mechanics,
but it's not true in the
quantum theory,
because as you know,
this incoming beam of electrons
is associated with a wave,
the wave is going to fan out
when it comes out.
 
And we sort of know how much
it's going to fan out.
That's why I did that
diffraction for you.
It fans out by an angle
q, so that dsine
q = l.
 
That means light can come
anywhere in this cone to your
screen.
 
That means the electrons
leaving could have had a
momentum in any of these
directions.
So the initial photon at a
momentum p_0,
the final one has a momentum of
magnitude p_0,
but whose direction is
uncertain.
The uncertainty in the y
momentum, simply
p_0sine
q.
You understand?
 
Take a vector
p_0.
If that angle is q,
this is
p_0sine
q.
And we don't know.
 
Look, it's not that we know
exactly where it's going to
land.
 
It can land anywhere inside
this bell shaped curve,
so it can have any momentum in
this region.
So the electrons you produce,
even though the position was
well known to the width of the
slit,
right after leaving the slit,
are capable of coming all over
here.
 
That means they have momenta
which can point in any of these
allowed directions.
 
So let's find the uncertainty
in y momentum as this.
The uncertainty in y
position is just the width of
the slit.
 
So take the product now of D py.
 
Let me call it Dy.
 
That happens then to be
p_0sine q times
d but dsine
q is l and l is
2pℏ/p
_0.
Cancel the
p_0,
you get some number.
 
Forget the 2p's that look like
ℏ.
So this is the uncertainty
principle.
So the origin of the
uncertainty principle is that
the fate of the electrons is
determined by a wave.
And when you try to localize
the wave in one direction,
it fans out.
 
And when it fans out,
the probability of finding the
electron is not 0 in the
non-forward direction.
It's got a good chance of being
in the range of non-forward
directions.
 
That means momentum has a good
chance of lying all the way from
there to here.
 
That means the y
momentum has an uncertainty.
And more you make the purchase
smaller to nail its position,
broader this will be,
keeping the product constant.
So it's not hard mathematically
to understand.
What is hard to understand is
the notion that somehow you need
this wave, but it was forced
upon us.
The wave is forced upon us,
because there's no way to
understand interference,
except through waves.
So when people saw the
interference pattern of the
electrons, they said there's got
to be a wave.
They said, "What is the
role of that wave?"
That's what I want you to
understand.
With every electron now--so
let's summarize what we have
learned.
 
When I say electron,
I mean any other particle you
like, photon,
neutron, doesn't matter.
They all do this.
 
Quantum mechanics applies to
everything.
Therefore, with every electron,
I'm going to associate a
function,
Y(x)--or
Y(x,y,z),
so that if you find its
absolute value,
that gives--or absolute value
squared,
that gives the odds of finding
it at the point x,
y, z.
Let me say it's proportional.
 
This function is stuck.
 
We are stuck with this function.
 
And what else do we know about
the function?
We know that if the electron
has momentum p,
then the function Y has
wavelength l,
which is 2pℏ
/p.
This is all we know from
experiment.
So experiment has forced us to
write this function Y.
And the theory will make
predictions.
Later on we'll find out how to
calculate the Y in every
situation.
 
But the question is,
what is the kinematics of
quantum mechanics compared to
kinematics of classical
mechanics?
 
In classical mechanics,
a particle has a definite
position, it has a definite
momentum.
That describes the state of the
particle now.
Then you want to predict the
future, so you want to know the
coordinate and momentum of a
future time.
How are you going to find that?
 
Anybody know?
 
How do you find the future of
x and p?
In Newtonian mechanics;
I'm not talking about quantum
mechanics.
 
Student: 
>
Prof: Which one?
 
Just use Newton's laws.
 
That's what Newton's law does
for you.
It tells you what the
acceleration is in a given
context.
 
Then you find the acceleration
to find the new velocity.
Find the old velocity to find
the new position a little later
and keep on doing it,
or you solve an equation.
So the cycle of Newtonian
mechanics is give me the
x and p,
and I know what they mean,
and I'll tell you x and
p later if you tell me
the forces acting on it.
 
Or if you want to write the
force as a gradient of a
potential, you will have to be
given the potential.
In quantum mechanics,
you are given a function
Y.
 
Suppose the particle lives in
only one dimension,
then for one particle,
not for a swarm of particles,
for one particle,
for every particle there can be
a function Y associated
with it at any instant.
That tells you the full story.
 
Remember, we've gone from two
numbers, x and p,
to a whole function.
 
What does the function do?
 
If you squared the function at
this point--
square will look roughly the
same thing--
that height is proportional to
the odds of finding it here,
and that means it's a very high
chance of being found here,
maybe no chance of being found
here and so on.
That's called the wave function.
 
The name for this guy is the
wave function.
So far we know only one wave
function.
In a double slit experiment,
if you send electrons of
momentum p,
that wave function seems to
have a wavelength l
connected to p by this
formula, this.
 
This is all we know.
 
So let's ask the following
question - take a particle of
momentum p.
 
What do you think the
corresponding wave function is
in the double slit experiment?
 
Can you cook up the function in
the double slit experiment at
any given time?
 
So I want to write a function
that can describe the electron
in that double slit experiment,
and I'll tell you the momentum
is p.
 
So what can you tell from wave
theory?
Let's say the wave is
traveling, this is the x
direction.
 
What can you say about
Y(x) at some given
time?
 
It's got some amplitude and
it's oscillating,
so it's cosine
2px/l.
Forget the time dependence.
 
At one instant of time,
it's going to look like this.
This is the wavelength l
for anything.
But now I know that l is
connected to momentum as follows
- l is 2pℏ
/p,
so let's put that in.
 
So 2p/l
= Acosine p
over ℏx.
 
This has the right wavelength
for the given momentum.
In other words,
if you send electrons of
momentum p,
and you put that p into this
function exactly where it's
supposed to go,
it determines a wavelength in
just the right way,
that if you did interference,
you'll get a pattern you
observe.
 
But this is not the right
answer.
This is not the right answer,
because if you took the square
of the Y, it's real.
 
I don't care whether it's
absolute square or square,
you get cosine squared
px over
ℏ,
and if you plot that function,
it's going to look like this,
the incoming wave.
I'm talking not about
interference but the incoming
wave, if I write it this way.
 
But incoming wave,
if it looks like this,
I have a problem,
because the uncertainty
principle says Dx
Dpx is of
order ℏ.
 
It cannot be smaller,
so the correct statement is,
it's bigger than
ℏ over some
number.
 
Take this function here.
 
Its momentum is exactly know,
do you agree?
The uncertainty principle says
if you know the position well,
you don't know the momentum too
well.
If you know the momentum
exactly, so Dpx is 0,
Dx is infinity,
that means you don't know where
it is.
 
A particle of perfectly known
momentum has perfectly unknown
position.
 
That means the probability of
finding it everywhere should be
flat.
 
This is not flat.
 
It says I'm likely to be here,
not likely to be here,
likely to be there,
so this function is ruled out.
Because I want for
Y^(2),
for a situation where it has a
well defined momentum,
I want the answer to look like
this.
The odds of finding it should
be independent of where you are,
because we don't know where it
is.
Every place is equally likely.
 
And yet this function has no
wavelength.
So how do I sneak in a
wavelength, but not affect this
flatness of Y^(2)?
 
Is there a way to write a
function that will have a
magnitude which is constant but
has a wavelength hidden in it
somewhere,
so that it can take part in
interference?
 
Pardon me?
 
Any guess?
 
Yes?
 
Student: 
>
Prof: A complex function.
 
So I'm going to tell you what
the answer is.
We are driven to that answer.
 
Here's a function I can write
down,
which has all the good
properties I want -
Y(x) looks like
some number Ae^(ipx/
ℏ).
 
This is just cosine(px
i) sine(px/ℏ).
It's got a wavelength,
but the absolute value of
Y is just A^(2),
because the absolute value of
this guy is 1.
 
Ae to the thing looks
like this.
This is the number A,
this is px/ℏ is the
angle.
 
That complex number Y at
a given point x has got a
magnitude which is just A^(2).
 
So we are driven to the
conclusion that the correct way
to describe an electron with
wave function,
with a momentum p,
is some number in front times
e^(ipx/ℏ),
because it's got a wavelength
associated with it,
and it also has an absolute
value that is flat.
 
Do you understand why it had to
be flat?
The uncertainty principle says
if you know its momentum
precisely,
and you seem to know it,
because you put a definite
p here,
you cannot know where it is.
 
That means the probability for
finding it cannot be dependent
on position.
 
Any trigonometric function you
take with some wavelength will
necessarily oscillate,
preferring some points over
other points.
 
The exponential function,
it will oscillate and yet its
magnitude is independent.
 
That's a remarkable function.
 
It's fair to say that if you
did not know complex
exponentials,
you wouldn't have got beyond
this point in the development of
quantum mechanics.
The wave function of an
electron of definite momentum is
a complex exponential.
 
This is the sense in which
complex functions enter quantum
mechanics in an inevitable way.
 
It's not that the function is
really cosine px/ℏ and
I'm trying to write it as a real
part of something.
You need this complex beast.
 
So the wave functions of
quantum mechanics.
There are electrons which could
be doing many things,
each one has a function
Y.
Electron of definite momentum
we know is a reality.
It happens all the time.
 
In CERN they're producing
protons of a definite momentum,
4 point whatever,
3 point p tev.
So you know the momentum.
 
You can ask what function
describes it in quantum theory;
this is the answer.
 
This is not derived.
 
In a way, this is a postulate.
 
I'm only trying to motivate it.
 
You cannot derive any of
quantum mechanics,
except looking at experiments
and trying to see if there is
some theoretical structure that
will fit the data.
So I'm going to conclude with
what we have found today,
and it's probably a little
weird.
I try to pay attention to that
and I will repeat it every time,
maybe adding a little extra
stuff.
So what have you found so far?
 
It looks like electrons and
photons are all particles and
waves,
except it's more natural to
think of light in terms of waves
with the wavelength and
frequency.
 
What's surprising is that it's
made up of particles whose
energy is
ℏw,
and whose momentum is
ℏk.
Conversely, particles like
electrons, which have a definite
momentum, have a wavelength
associated with them.
And when does the wavelength
come into play?
Whenever you do an experiment
in which that wavelength is
comparable to the geometric
dimensions,
like a double slit experiment
at a single slit diffraction,
it's the wave that decides
where the electron will go.
The height squared of the wave
function is proportional to the
probability the electron will
end up somewhere.
And also, in a double slit
experiment, it is no longer
possible to think that the
electron went through one slit
or another.
 
You make that assumption,
you cannot avoid the fact that
when both slits are open,
the numbers should be additive.
The fact they are not means an
electron knows how many slits
are open, and only a wave knows
how many slits are open because
it's going everywhere.
 
A particle can only look at one
slit at a time.
In fact, it doesn't know
anything, how many slits there
are.
 
It usually bangs itself into
the wall most of the time,
but sometimes when it goes
through the hole,
it comes up.
 
And so what do you think one
should do to complete the
picture?
 
What do we need to know?
 
We need to know many things.
 
Y(x)^(2) is the
probability that if you look for
it, you will find it somewhere.
 
Instead of saying the particle
is at this x in Newtonian
mechanics,
we're saying it can be at any
x where Y doesn't
vanish and the odds are
proportional to the square of
Y at that point.
Then you can say,
what does the wave function
look like for a particle of
definite momentum?
Either you postulate it or try
to follow the arguments I gave,
but this simply is the answer.
 
This is the state of definite
momentum.
And the uncertainty principle
tells you this is an agreement
to the uncertainty principle
that any attempt to localize an
electron in space by an amount
Dx leads to a spread in
momentum in an amount
Dpx.
That's because it's given by a
wave.
If you're trying to squeeze the
wave this way,
it blows up in the other
direction.
And the odds for finding in
other directions are non 0,
that means the momentum can
point in many directions coming
out of the slit.
 
That's the origin of the
uncertainty principle.
So I'm going to post whatever I
told you today online.
You should definitely read it
and it's something you should
talk about,
not only with your analyst,
because this can really disturb
you,
talk about it with your
friends, your neighbors,
talk about it with senior
students.
The best thing in quantum is
discussing it with people and
getting over the weirdness.
 
 
 
