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YEN-JIE LEE: So what are
we going to do today?
So today, we're going to
continue the discussion,
based on what we have learned
from the diffraction and also
other interesting phenomena.
And we are going
to make connections
to quantum mechanics
and discuss in greater
detail about this connection.
And also, if time
allows, we are going
to cover information about
gravitational waves as well.
So last time, we have
discussed diffraction pattern
coming from a laser beam.
And we discussed
about resolution.
And you can see that this is the
graph we see, last time already
in the lecture.
Basically, all the
point light source
coming through a single slit is
going to be doing interference
to each other.
So basically, you see
interesting pattern,
which you have a central peak.
And the total constructing
for interference
happens at the
center of the screen.
And at some point,
you have a deep--
which is actually where total
destructive interference
actually happened.
And we were able to understand
this with mathematics,
which we learned in 18.03.
Another interesting
result we find
is that, if we shoot a
laser beam to the moon--
by now, you should be able to
conclude that it's not going
to be a point on the moon.
Instead, it's going to be a spot
as large as the whole Missouri
state.
So that's actually
another interesting result
we found from that
discussion last time.
And finally, we are
able to put together
all the things we have learned
from the last few lectures.
Basically, you can
have, at the same time,
the effect of multi-slit
interference and also
the effect of a single
slit diffraction--
all then put together.
Then you have this complicated
but also beautiful pattern,
which you will be able
to observe on the screen.
And basically, the point is
that you are going to have,
for example, in this case,
five-slit interference pattern.
But that is actually modulated.
The intensity is
modulated by the pattern
from a single-slit diffraction.
OK, so that is
actually what we have
learned from the last lecture.
Coming back to the original
question, why do we study 8.03?
The reason is that we
would like to understand--
we would like to hear
from that universe.
So we cannot even
recognize the universe,
without using waves
and vibrations.
So that's the course in which
we have been doing here.
And we have seen
waves of matter.
So for example, here, we have
this water wave generator.
You can see water waves.
We have this Bell Lab machine.
Because you can see the
coupled oscillator--
a multiple coupled oscillator.
And they are doing
their job together
to form beautiful waves.
And you can see those
beautiful results from there.
Last time and also including
the last few lectures,
we have been using
this laser to produce
interesting interference
patterns and the diffraction
pattern.
So that's the second kind
of wave, which we encounter.
The first kind is
the waves of matters.
The second kind
is waves of what?
Waves of vector fields.
It's not matter anymore.
Because this field,
this oscillation field,
can also travel through vacuum.
So that is actually
the second kind
of wave, which we
should learn for 8.03.
They provide a pretty
adequate description
of the nature, description
of the phenomenon, which
we can actually measure, and
see from the experiment we
went over.
So what I would like to
say in the lecture today
is that there are two kinds
of completely different waves,
which we haven't talked about.
The first one is the
probability density wave,
which I will cover
that in a few moments.
The second one is
gravitational waves.
This is actually a
space-time distortion
coming from the motion
of massive objects.
And we would like to
see what we can also
learn from there, using the
existing knowledge of which we
have learned from matter waves
and the vector-field waves.
And that they are, actually,
pretty similar to each other
if you look at their behavior.
So the first thing which I
would like to discuss is light.
So far, what is light?
Light is like
electromagnetic waves.
So that is actually
what we have learned
so far from 8.02 and 8.03.
So they are like waves.
They are waves, all right.
On the other hand,
in the 20th century,
there are many, many
crises going on.
So the first thing
which is happening
is photoelectric effect.
So this experimental
result was actually
first discovered by Hertz.
So he found that, if you
want to kick one electron out
of some material or
some charged material,
it is easier if you use a
high-frequency light compared
to low-frequency light.
So that's the issue--
really strange.
Because based on waves and
also the intensity formula--
while we really
care, intensity is
proportional to the square of
the electric field amplitude.
So that's actually
what we learned
from the previous lectures.
But what Hertz was saying
is that the frequency also
matters.
OK, so that's really
a bit strange.
And Einstein actually
came in and explained
this photoelectric effect.
So what we found
is that this effect
can be explained by viewing
the light of small quantas.
And light is actually
not like waves any more,
but like quantas--
the small quantas--
discrete ones--
with energy proportional
to the frequency.
All right?
And basically, the energy
of those little quanta--
or we now call them photons--
is actually equal to h,
which is some constant--
relate the frequency and
the energy of the quanta.
And with this
explanation, the view
is that the photons
are like particles.
And therefore, he can
actually explain that, OK,
if I measure the kinetic energy,
the maxima kinetic energy
of the electrons,
which are kicked out
from this photoelectric
effect experiment.
I call it K-max.
What you are going to get
is a formula like this.
The maxima kinetic
energy of electrons
will be equal to h
bar mu minus phi.
Phi is actually some
kind of threshold,
which you need
or, say, some kind
of energy, which you need
to overcome to kick one
electron out of the material.
So you can see from here that,
if the frequency of the light
is too low, then
this will never work.
Because the maximum kinetic
energy will be below 0.
Therefore, you cannot
kick out a electron.
But if the frequency is high,
as shown by Hertz experiment,
it is possible.
And also, you can create
energetic electron,
have them kicked
out of the material.
And this is actually
verified by experiment.
And that's essentially
why Einstein should
get the Nobel Prize in 1921.
So let me tell you
what the feeling here--
the feeling here is that, now,
things are really becoming
more and more interesting.
Because first of
all, we see really
well that the electromagnetic
wave describes
the behavior of the light.
We see diffraction.
We see interference.
All those things
can be explained
by the beautiful mathematics,
which we have been employing
to explain all those results.
But here, you can see
that, at the same time,
the photons are very good
tool or very good viewpoint
to explain the
photoelectric effect.
So that's actually
really surprising.
Because that means
you have, suddenly,
both kinds of explanation
of light, which should
be describing the same thing.
So now, one idea is to
describe them by waves.
The other idea is to
describe them by particles.
And also, at the same
time, as we actually
discussed in the
previous lecture,
we can do a two-slit
experiment with billiard balls
or with bullets.
Suppose we do this experiment.
We fire the balls or
bullets through some source.
And we have some kind
of two-slit set up here,
so that the bullets or balls
can actually pass through.
And then we were
wondering, what will
be the pattern which we
see from the detector
in the right-hand side.
And basically, what we
see is the following.
So basically, we
have a distribution
of the balls coming
from slit number one.
And we have another distribution
of balls, which is actually
coming from slit number two.
And I can call it P-1 and
P-2, which is actually
the probability
density distribution
of the experimental result.
And the final result,
or say, if you just
look at the
distribution of balls
without separating the
balls from slit number
one and slit number
two, basically you
get some distribution
like this, which
is a superpositional P-1
and the P-2 distributions.
All of these things
doesn't surprise anybody.
The experiment one,
which we perform--
now, instead of billiard balls,
we perform the same experiment
with electrons.
And again, we have
all those electrons
pass through this two-slit
two-slit experiment.
As we see before,
basically, again, we
can actually separate electrons
from the first slit, which
I called I-1, and the
intensity of electrons
coming from the second
slit, which I called I-2.
Then, basically, if you
can identify and make sure
that the electron is coming
from one of the slits,
you are going to get
distribution like this.
OK?
And of course, I can always
write these I-1 as a sine wave
function, psi-1 squared.
And I can always write this
I-2, which is the intensity
as a function of position of the
electron coming from the second
slit--
I can always write
it as psi-2 squared.
In the case of
light, it's actually
just proportional to
the electric field.
So if you accept this,
and now you actually
don't measure where
the electron actually
pass through when the
electron from the source
passed through this experiment--
we now don't measure if it
passed through one or two.
Then, the pattern
becomes like this.
You have something like
this which, essentially,
is very, very similar to
the two-slit interference
experimental result of
the laser experiment.
As you see from this
experimental result,
you see some kind of pattern.
You have the peak.
You have the valley,
as a function
of position in the detector.
And we can actually call
this I-1-2, which is actually
when you don't measure
or you don't identify
which slit the
electron goes through,
then you have the intensity,
which is actually called I-1-2.
And what we actually found
is that I-1-2 is actually
not I-1 plus I-2.
What we found is that
I-1-2 is actually psi-1
plus psi-2 squared.
So basically, we see
interference pattern.
And this will be equal to I-1
plus I-2 plus 2 squared root
of I-1 plus I-2 and
the cosine delta,
where the delta is coming from
the path length difference.
So based on this
experimental result,
this is actually
really surprising.
First of all, electrons are
arriving like a particle,
right?
Because we can see
from this slide,
if you look at the upper left
figure, you see doo-doo-doo.
Every time, you have
something hitting the screen.
And what is actually left over?
It's a single hit on the screen.
Therefore, the electrons are
arriving like a particle,
producing a hit in the detector.
On the other hand,
what we are saying here
is that, before
they hit the screen,
they are behaving like a wave.
It has interference with
itself, like a wave.
All right, so is
electron a particle?
Or is the electron a wave?
Strange.
The answer is
electron is actually
neither of them in reality.
So how about we actually
do some more experiments
to convince ourselves what
is actually really going on.
So what we could do--
as I actually mentioned.
I can still have
the electronic gun.
I have electron source
in the left-hand side.
Again, I have this
two-slit experiment here.
Then what I'm going to do is
to produce a light source here.
OK, I put a light source there
to shine the whole experiment.
Then, I was wondering
what is going
to happen to the distribution.
So if I first close the
lower slit, slit number two,
and only measure the intensity
coming front slit number 1,
then this is the distribution
I get, which is I-1.
If I close the upper one
and open only the lower one,
I get a distribution
which is I-2.
And this light
source, essentially
interacting with the
electron-- when electrons
pass through this slit,
what is going to happen
is that you'll see some
scattered light coming
from the slit.
Therefore, you can know which
slit the electron actually
passed through.
And if I do this
experiment result--
if I block one of the slits,
this is the distribution, I-1
and I-2 and now I'm
going to open both slits,
and you will see light
coming out of the slit
when the electron pass slit
number one or slit number two.
You can identify which slit
fora all the electrons passing
through this experiment.
And the resulting distribution
of electrons on the screen
is actually like this.
It's actually like I-1 plus I-2.
There will be no interference.
Why is that?
Now, this because you
know very precisely which
slit the electron
actually passed through,
this experiment.
And also, we can say that,
huh, the electrons are actually
disturbed.
Therefore, they now behave like
bullets or like billiard balls.
So this is actually
a bit strange.
Maybe it is because the
intensity of the light
is too large.
Therefore, it's changing the
behavior of the electron.
So what are we going to do now?
Experimental result
number three is
to lower the intensity
of the light.
So what will happen if
I lower the intensity
of the light source so that we
would like to see the behavior,
as a function of intensity?
So at some point, we will find
that some of the electrons
are not heated by a photon.
Or say, there will be no
scattered light of the electron
when it passes through
the experiment.
Because the intensity of
the light is too small.
And we already know, from
photoelectric experiments,
light is, essentially,
also like quanta.
So when the intensity
is low enough,
the effectiveness of the
light source decreases.
Then the experimental
result would be like what?
Can somebody actually
give a guess?
Is that going to be like
experimental result number one?
Or is that going to be like
experimental result number two?
Anybody want to get--
AUDIENCE: [INAUDIBLE]
YEN-JIE LEE: Very good.
So the result-- as I mentioned
before, sometimes the electrons
are detected by
the light source.
Sometimes the
electrons are lucky.
They pass through without
getting heated by a photon.
Therefore, you cannot
know which slit, actually,
this electron went through.
Therefore, the
experimental result
is that, if I just lower the
intensity of the light source,
then what I'm going to get is a
mixture of experimental result
number one and the
experimental result number two.
When I have the
intensity low enough--
going to really, really
low intensity limit--
the result will become
experimental result number one.
Because then you are
not really impacting
the position of the electrons.
Finally, you can actually
suggest something else.
So OK, now I have low
intensity of light.
One way to lower the intensity
is like what I was saying.
The rate of the
photon emission--
I can make it lower and lower.
All right?
There's another way to do this.
Experimental result
number four--
so what will happen
if I use this formula
E equal to h bar mu equal
to hc divided by lambda?
Because mu is actually
just c over lambda.
What will happen if
I, instead of lowering
the rate of photon
emission, I lower the energy
of individual photons?
How do I do that?
What I could do is to
lower the frequency
of the electromagnetic
wave of the light source
or, say, increase
the wavelengths
of the light source.
OK, this is very nice.
Because now, I can
keep this in, right?
So that I make sure
all the electrons
are bothered by the photon.
Because I can emit
very high rate.
But at the same time, I can
also lower the intensity,
so that the intensity
is very, very low.
Can anybody guess what
is going to happen?
With a result like
experimental result number one,
when I go to extremely
low intensity?
Or my result will be like
experimental result number two?
Because each electrons
are bothered,
are heated, by the emission
from the light source.
Anybody want to try?
Just guess, no?
One or two?
Or a mixture of them?
AUDIENCE: Be like two.
YEN-JIE LEE: The
guess is that it's
going to be like two, which
was actually well-motivated.
Very good try.
What essentially,
happens is that--
OK, I can say, oh!
Each electron are bothered
by many, many photons.
So those are disturbed.
Therefore, it has to look like
experimental result number two.
The answer may surprise you.
The answer is that,
if I have the limit
lambda goes to
infinity, mu goes to 0,
what is going to happen
is that, no matter how
high frequency of
photon emission I have,
I am going to get the result
of experimental number one.
Why is that?
Now, this is because
when the wavelengths
of the electromagnetic
wave or the photon is going
to infinity, that means you
cannot resolve which slit,
actually, the
electron goes through.
Because if I draw
the wavelengths here,
it's going to be like this.
If you observe some
kind of scattered light
from the electron, it
could come from both slits,
because the wavelength
is too long.
So if you go to
infinity, then it's
like you have a constant
electric field there.
It doesn't really
actually help you
to identify which slit
the electron actually
passed through.
So therefore, the interference
pattern reappears.
So this is really crazy thing,
if you look at all these four
experimental results.
The conclusion from these
four imaginary experiments
is that it is not
yet possible to tell
the position of the electron
and also, at the same time,
do not disturb it.
If you were able to tell the
position of the electron,
then there would be no
interference pattern.
On the other hand, if
your experimental set-up
have no ability at all to
tell if the electron's coming
from slit number one
compared to slit number two,
then you are going to
get interference pattern.
There's another thing which I
would like to make connection
to the Uncertainty Principle,
which we actually learn
from waves and vibrations.
So we have learned that
Heisenberg's Uncertainty
Principle--
this is essentially
purity coming
from the property of the
wave, if you actually
remember the deviation
which we have
done in the previous lecture.
So what is this uncertainty
principle telling us?
Is that the standard
deviation of the position
times the standard
deviation of the momentum
is going to be greater or equal
to h bar over 2 some constant.
And how do we actually
understand this
from the electron experiment.
That is actually highly related
to the single-slit experiment
with electrons.
So what we could do now is
to have a fifth experiment.
I have electron source here.
And I have a single slit.
And the width of the
slit is capital D.
And we were wondering,
what is going to happen?
What will be recorded
by the smoke detector
in the right-hand side?
And by now, it should
not surprise you
that this would give you some
kind of diffraction pattern,
which you say should be very
similar to what we actually
observe with laser experiments.
So you can see that.
The electron-- the momentum--
now, I would like to actually
define my coordinate system.
The vertical direction, pointing
upward, is my x direction.
So now, take a look at
this experimental result.
So what this is actually
telling us is the following.
We know the position
of the electron
to a accuracy of the
width of this slit, which
is D. So that is
actually telling you
about the uncertainty over the
position in the x direction.
Now, this electron goes through.
And they actually
hit the screen.
And each electron is
having a single path.
If I look at one of the paths,
the upper one, what I'm getting
is that there must be a
momentum quintessentially
in the x direction, when
this electronic goes
through the slit
and hits the screen.
One interesting thing we
learned from the deviation
from last time is that, if I
just look at the slide here--
if I look at the
left-hand side slide--
if I have a very small slit--
D is small-- what does
that correspond to?
That corresponds to delta x
goes to very small value case.
We have a small delta x value.
You are really sure
where is the electron
at some instant of time, when it
passed through the experiment.
Then what is going to happen?
If you look at the
right-hand side,
the distribution on the screen,
you have a wide distribution.
The central maximum
peak will be very wide.
So what does that mean?
That means you have
a wide distribution
of momentum in the x direction.
Therefore, that
will give you that
is actually consistent with what
we've actually written here,
Heisenberg's Principle.
Delta x times delta p will be
greater or equal to some value.
On the other hand, if I
increase the width of the slit,
the D is now large.
As you are making a D larger
and larger, what is happening
is that the central
peak, the width,
is actually going to be
narrower and narrower.
Now, this is actually
also consistent with what
we have learned from
Heisenberg's Uncertainty
Principle.
When delta x become even larger,
then the uncertainty or, say,
the distribution of the
momentum in the x direction,
becomes smaller.
So now, we actually also
understand a little bit more
about what the single
diffraction actually means.
And this issue is really closely
connected to the Uncertainty
Principle Heisenberg
actually proposed.
And if we use the
mathematics which
we learned from last
time, the C function
is going to be proportional
to integration from minus D
over 2 to D over 2.
dx exponential as ikx times x.
And if I have D
goes to infinity,
which means that you have
an infinitely wide slit--
based on the formula which
we have derived last time--
basically, we will see that C
function is a function of kx.
It's going to become
a delta function.
And this delta
function is delta kx.
So that means, if you
have absolutely no idea
about the position
of the electron,
you are going to get very, very
precise information about--
the momentum in the x
direction is actually
going to be equal to 0.
Because it's a delta function.
It's only nonzero at kx, which
is the directional propagation
equal to 0.
Any questions so far?
OK, so from those
experimental results,
we've found that the probability
of getting heat on the screen
is proportional to
psi-1 square, if I only
have the first lead there.
That means the probability, p,
is proportional to psi squared.
And this is actually
probably one
of the most crazy results in
the physics we learned so far.
In some sense, it's
kind of sad as well.
Why sad?
This means that, OK, I can
calculate those wave functions.
And the probability
of getting an outcome
at a specific position is
proportional to this wave
function squared.
But I feel, maybe,
demotivated, right?
Because originally,
we are like god.
You can predict--
OK, I have this
thing, this object.
And I have force.
And then it goes like--
oh-- like this way.
I can calculate the
trajectory of this chalk thing
all over the place,
as a function of time.
And I know what is
going to happen.
I have the full control
of all the objects
which I have in my
hand in my experiment.
But now, quantum mechanics
or this experimental result
tells me that we
can only predict
the probability, the odd,
instead of the outcome.
You see my point?
I can only pretend the wave
function, the distribution
of the wave function.
And the probability of
getting a result here
is proportional to the
wave function squared.
But I cannot predict the outcome
before I do the experiment.
That's really a big
change in your view
or, they say, in our current
understanding of the physics.
You can say that, well,
maybe Yen-Jie's not
working hard enough.
Maybe all those electrons which
are emitted from the electron
source already
made up their mind
where this electron is going to.
For example, electron number
one is doing this-- rrrrr--
and going to here.
And electron number two
already made up his mind.
He's just going to do this.
And the electron
number three is-- uh--
maybe do this--
vwooo-do-do-do-do.
And then all those trajectories
are already determined.
And they are heating variables
which Yen-Jie doesn't know.
Therefore, he screwed this
up and said, oh, come on.
We can only predict
the probability.
But the thing is that, from the
experimental result number two,
experimental result number
three and number four,
you can see that the electrons
cannot make up their mind when
they are emitted.
Because when they got
heated by that light--
electrons cannot know in advance
that it is going to be heated
by a light.
And the light can be a very,
very mild, very, very small
energy.
So that it should not
affect the predetermined
path of the electron.
Do you get this?
So that doesn't makes sense.
So it is not because
Yen-Jie is not trying hard.
It is really because
nobody can really
tell before the experimental
result is actually shown
or the measurement
is already done.
If you can find
any case, maybe you
will win another
100 Nobel Prize.
Because you are showing that the
whole understanding of quantum
mechanics is not
correct, really.
Please tell me when you actually
have done this experiment.
I will be very proud
of you, for sure.
So now, we are
entering a position
to discuss this result.
So now, actually, we
can also make use of
this understanding
and predict what would be
the particle probability
distribution in
a potential well.
Suppose I have an experiment,
which I have a well,
where I have potential
goes to infinity
in the left-hand side
or right-hand side
edge of this well.
And I will define
my coordinate system
so that the well is
actually equal to 0,
or x equal to L. So by
now, when you see this,
this looks really
familiar to you.
In the center part,
you have some kind
of translation symmetry.
And the boundary-- those
are forbidden regions.
You cannot actually
have particles there,
because the potential
is infinity.
Therefore, this is
actually giving you
boundary conditions
of the wave function,
describing the state of the
particle inside this box.
So the boundary
condition would be psi 0.
It will be equal to 0,
because it's actually
at the left-hand side
edge of the well,
where you have
infinite potential.
And also, you can have psi
L. This will be equal to 0,
because the
right-hand side edge,
you also have infinitely
high potential.
Therefore, when you see
this, your immediate reaction
will be, how do I
know what is this?
This is actually psi m.
The solution to
this problem must
be something like
psi m of x should be
the normal mode of this system.
And it's going to
be A-m sin km-x,
where km will be equal
to m-pi divided by L,
where m is a number.
It can be 1, 2, 3-- it
goes to infinity, right?
By far, you have
actually learned
all these practical calculations
from the previous examples
we had.
Therefore, what would be the
psi m, x as a function of time?
Then what I am going
to get is A-m sin km
x exponential minus i omega-m t.
Of course, we can also really
plot all those results.
So for example, n is equal to 1.
Basically, what you are
getting is like this.
Doesn't surprise you.
This is actually psi
as a function of x.
And n equal to 2--
this will correspond
to the situation
where you have one node in the
middle, et cetera, et cetera.
You can't have many, many
higher m value solutions.
And what would be
the probability
to find the particle
in a specific location?
That's why we mentioned
before, the probability is
proportional to a
wave function squared.
Therefore, the probability will
be proportional to sin squared
k-m x.
And what we are going
to get is like this.
If I plot m equal to 1,
using P as a function of x,
the probability of getting a
particle at a specific place,
if we are looking at the
situation in normal mode number
one, that is like this.
You are much more likely
to find a particle
in the middle of the box.
On the other hand, we can
also plot the probability
as function of x, where
m equals to 2 k's.
If we are actually operating
in a second normal mode,
then basically you have some
distribution-- looks like this.
In this situation, it
is forbidden-- or say,
there's zero
probability you will
find the particle in
the middle of the box,
et cetera, et cetera.
You can actually calculate all
those corresponding probability
distributions as a
function of m value
and as a function of position.
OK?
Sounds like a very good story.
But there's something
missing, right?
What is actually missing?
You have the normal modes.
You have the k-m.
What is missing is
the wave equation.
The wave equation
is missing, right?
You don't have the
dispersion relation.
This solution is incomplete.
You don't know what is
actually the omega value.
Because you don't have
dispersion relation.
So what is actually
the wave equation
for the quantum mechanics?
So it is actually
Schrodinger's equation.
So Feynman once
commented on the origin
of the Schrodinger's Equation.
It's from where?
It's not possible to derive
it from anything you know.
It's just coming out of the
might Schrodinger, actually.
So there's no reason.
And it works.
That's the beautiful part.
So what is, actually,
this equation?
So this is the equation
Schrodinger actually
writes down.
It's like i h-bar--
Plank's constant,
partial/partial t, psi xt.
And this will be equal
to minus h bar squared
over 2m partial square, partial
x square, plus v xt, psi xt.
So this is really nice.
And it works and matches
with experimental results.
And now, I have already
the normal mode.
I can plug that
into this equation
to see what I can
actually learn from there.
So what I am going to do
is to plug in psi-m xt
into this equation, to get
the dispersion relation.
So what this issue, the
dispersion relation.
So here, I have
partial/partial t.
So I extract one omega
minus i omega out of this.
Then, basically, I get
h bar omega-m psi-m xt
in the lambda psi.
OK, plugging in psi-m into this
equation and see what happens.
Then this will be equal to--
I also know that, in the middle
of the box, v, essentially,
the potential--
the potential is
0 inside the box.
The potential is infinity
at the edge of the box.
Therefore, I can safely ignore
this term, to be equal to 0.
So you have a free path
to go inside the box.
And what, essentially,
this term--
this term will give you
minus h-bar square over 2m.
OK, I have a double
differential of x.
And therefore, I get, basically,
minus km squared, right?
Because it's psi zeta, right?
So basically, I get
minus k squared.
So therefore, I cancel
this minus sign.
I have km squared out
of this calculation.
And I still have psi-m xt.
I can cancel this too.
Then, what I'm getting is that
omega-m is equal to km squared,
h-bar--
I cancel one of the h-bar here--
divided by 2m.
This-- essentially, dispersion
relation of the wave function.
De Broglie proposed that
wavelengths of the matter wave,
essentially, highly related
to the momentum of the matter.
So basically, he propose that
p, the momentum of the particle
is actually equal to h-bar
k, where k is the wave
number of the matter wave.
If you accept de Broglie's
interpretation, basically what
we are getting is something
really, really interesting.
If we put together this
dispersion relation
and the de Broglie's
interpretation of matter wave--
what I am going to do is to
calculate the group velocity
of this dispersion relation.
So I can now calculate--
group velocity, V-g, will
be equal to d omega, dk.
And I know that omega is equal
to h-bar k squared divided
by 2m.
This is essentially equal
to h-bar k divided by m.
Everybody is following?
OK.
And this is equal to what?
This is equal to p divided by
m, if I use de Broglie's matter
wave.
Therefore, you have p
equal to m times ng.
Wow!
Look at this.
What are we getting here?
What we are getting here is that
the group velocity of the wave
equation of the
waves is actually
the classical velocity, p
equal to m times v. Now,
everything actually is
becoming more and more clear.
We know and we learned
already, from 8.03, what is
the meaning of group velocity.
The meaning of
the group velocity
is the speed of the propagation
of a wave package, right?
Remember our discussion
before about a AM radio?
So what is actually the speed
of propagation of a wave packet
is the group velocity.
So now we have
solved the problem--
why electron can be a
particle, at the same time,
also like waves.
It's essentially described
by wave functions.
The classical behavior
we see on the electron
is because it is,
as you described,
by these wave packages.
It's pretty localized.
And the motion of this wave
package in a free space
is actually the speed
of the propagation--
is the group velocity.
Therefore, there
is no contradiction
between the
classical calculation
and the wave interpretation
of the electron.
So that really
surprised me very much.
And you can see that, given
the dispersion relation, also,
this is a rather
dynamical result.
The real part of the wave
function is actually blue.
And the imaginary
part is actually red.
It's actually
oscillating up and down.
And the oscillation
frequency, by now,
you know is governed by
that dispersion relation.
OK, now actually, everything
seems to make sense now--
really, really, very cool.
So on the other
hand, we also have
to live with
probability density.
So you cannot tell the exact
position of a particle any
more.
You cannot tell the exact
outcome of an experiment
anymore.
And that is actually to do
with this interpretation.
And all of those phenomena,
at a very, very small scale,
is actually described by
quantum mechanics, which we
will learn some more in 8.04.
And also, in the
future, you will
be governed by the quantum
field theory, which
is actually a father future.
And what is actually the life
living with quantum mechanics
and quantum field theory?
So this is a very simple--
[LAUGHTER]
--Lagrangian of
the standard model.
And it describes everything
except the gravity.
OK?
And it's really simple.
It's called Standard Model.
And look at this part.
This is governing the
Higgs decay to Z boson.
And the experimental result--
we don't really know what
is the mass of the Higgs.
It's a missing
observable before.
And on the other hand,
as a particle physicist
or as a high energy
nuclear physicists,
I have no idea about what will
happen in the next collision.
Why is that?
You know the reason now, right?
Because we cannot
predict the exact outcome
of our experiment.
It's all governed
by wave functions.
Therefore, what we are
doing is the following.
We are doing the brute force.
So we collide like crazy--
collide, collide, collide,
collide like crazy,
until something
interesting pops out.
That's actually
what we're really
doing as a particle physicist.
And this is your beautiful event
from proton-proton collisions
at the Large Hadron Collider--
is a Higgs to the boson event.
And one of the Z bosons
becomes the two red lines.
It's actually the two muons.
And the other
decays to electrons,
which are detected by
the kilometer as the two
blue things there.
As a high-energy
nuclear physicist,
I am interested
in the production
of quark-gluon plasma
from lead ion collisions.
So I am now putting together
two ions, have them collide.
And I hope that,
by chance, I can
deposit a huge amount of
energy in a very small volume.
And then I would like to
see this crazy matter,
actually, gradually expand
and become a lot of particles.
And I study those
particles to understand,
what would be the nature
of this material, which
starts to exist in
the very early part
of the whole universe history?
Just one microsecond
after the Big Bang,
we have the whole universe
filled by this crazy material.
And we are creating
this in the experiment.
And we will only be able to
hope that, OK, by chance, I
have the collision happen.
By chance, I have a very
high-density environment.
Somehow, multiple quanta decide
to scatter on each other.
And they deposit the energy
in a very small volume.
And then we collect all
those spectacular events
to study the properties of
all those little Big Bangs.
So that is actually the
consequence of this wave
function interpretation.
So now, coming back to the
Standard Model, this really
simple one, you can't see,
that is a theory of almost
everything, except the gravity.
Really sad.
So if you can actually
put them all together,
then you will also
win the Nobel Prize.
And giving you all
those ideas, so that I
can have a very good student
winning the Nobel Prize.
Of course you will.
And now, I would like to discuss
and use the remaining, maybe,
10 minutes to discuss
with you the gravity.
So here is actually
something related to gravity.
So Einstein actually
predicted that the distortion
of the space-time
generated by objects
can travel through the space.
I don't have the
derivation here,
because it would take another,
maybe, two hours to do this.
But I would like to
ask you to trust me.
This is actually a result
coming from general relativity.
And you can see
that we can actually
generate gravitational waves.
And I can actually do
the generation here,
like this and rotating.
I'm generating
gravitational waves.
And that student in the
back is also generating.
Yes, you are.
Yeah, you are generating.
Everybody's generating.
Ah, you are also generating.
Yeah, very good.
But the problem is that
the space-time distortion
is really small for people who
are not very massive, like me.
So that's a problem.
So I can generate.
I'm doing the demo here.
But it doesn't help.
You cannot really detect them.
And even Einstein himself thinks
it's impossible to detect them,
maybe, in our lifetime.
And what would be the
outcome of the calculation?
The outcome of the calculation--
if you have gravitational
wave passing toward you--
so what it does
is the following.
So basically, the space
is distorted in a way such
that it first expands
in this direction
and then expanding the other
direction, perpendicular
to the original distortion.
And if you put a
ray of particles,
the circular array of particles,
and look at what is going on,
when the gravitational
wave pass through it,
it pass through the array
in this direction, what
you are going to get
is effect like that.
Of course, this
is actually highly
exaggerated in this set-up.
You don't really see this
kind of sizable distortion
when Yen-Jie is dancing around.
OK, so how about we actually
visualize this thing.
The problem is that we
cannot really see the space
distortion.
But what we can, as
you see, is the light
which actually pass through
those little distortions.
So this is a stimulation
from LIGO Collaboration.
They are simulating the merging
of the two massive black holes.
And you can see that
they are rotating
with respect to each other.
They are radiating energy
out of this two-body system.
Let's take a look at this again.
So this is actually a simulation
of the event observed by LIGO.
So both black holes have a mass
roughly 30 times of our sun.
It's very massive.
And they are rotating with
respect to each other.
And that generates
space-time distortion.
And you can see that the
space-time distortion stops
after they merge each other.
And we were hoping that
we can detect those.
How crazy is that?
So how do we detect them?
Actually, you already
have the knowledge
to design the experiment to
detect this kind of effect.
So remember, the effect
of the gravitational wave
is like this.
So you have
distortion like this.
What we actually--
MIT and Cal Tech and the
many other collaborators
designed the LIGO experiment.
Are what is, actually, LIGO?
It's a Laser Interferometer
Gravitational wave Observatory.
It is actually always good
to have a very good name
of your experiment.
So this is actually LIGO.
So what, actually, it
does is the following.
So basically, it emits a laser.
And you split the
laser into two pieces.
And there were mirrors in
the very far end, reflect
those lasers.
And they come together.
And there is a
photo-detector, which
detects the interference pattern
of these two optical path
lengths.
Wow!
Sounds familiar to you, right?
Hey, you already know how to
explain this to your friends
already.
Really cool.
And in order to have
redundant management--
for example, if you only
have a single experiment,
maybe one graduate
student is like, oh,
doing dancing next
to a detector.
Then you see some fake signal.
And that's not
going to be helpful.
So what it does is
that-- basically, we
have two experiments.
One is actually in Hanford.
The other one is
actually in Livingston.
And they are actually
3,000 kilometers away
from each other.
So that there should not be any
correlation between the signal
coming from a
earthquake or dancing
of the graduate
students or whatever.
So they can you use that to
suppress any coincidence which
is actually not related
to gravitational waves.
So what does this do?
So now, we have,
oh, the knowledge
to actually explain
this phenomena.
So what this does
is the following.
So you meet the laser.
And when the gravitational
waves come in,
then it does this
space-time distortion.
And then the
interference pattern
of the waves going through
different optical path lengths
is going to change.
So you can see that,
originally, the experiment
is designed so that you
have complete cancellation.
But when the gravitational
waves is hitting the side,
you will be able to see that.
Really, you have constructive
interference at some point,
because of the movement
of the mirrors.
And those mirrors are
really, really far away
from the sources.
Each of them is, like,
four kilometers away
from the mirror.
And due to the
incredible precision
which were achieved
by this experiment,
we will be able to detect this
signal of gravitational waves.
So you can see,
again, from here--
so basically, when the
gravitational wave comes in,
first you split the light
source into two pieces,
have them hit the mirror, which
is actually four kilometers far
away from each other.
And they come back.
And initially, the experiment
is designed so that, very
precisely, there
will be no amplitude
detected by the photo-detector.
And when the gravitational
waves come in,
you actually really
change the length
between a splitter
and the mirror.
Therefore, you see light,
constructive interference even,
from the photo-detector.
So that is really cool.
And this is actually
the experimental result.
Look at this.
You actually can see
that light actually
achieve the sensitivity.
The gravitational wave was first
observed on September 14, 2015.
And the LIGO is actually
announcing that February 11--
earlier this year.
And what we actually see from
here is that this is actually--
as I mentioned to you--
there are two
measurements, two sides.
The left hand side is actually
the measurement from Hanford.
And the other one is
actually the measurement
from Livingston.
So they are two
different curves.
And they all have almost
exactly the same pattern.
Of course, there's time
has actually shifted,
because they are in different
places on the earth.
They are 3,000 kilometers apart.
So therefore, there
will be a shift in time.
And this is actually
a time-shifted result.
And you can see, also,
the calculation below,
which is actually
what you should
expect if you have the merger
of the two massive black holes.
So you can see that they are
actually rotating with respect
to each other.
One of them is actually
29 times larger
than the mass of the sun.
And the other one
is 36 times larger
than the mass of our sun.
It's not precision.
And they generate,
theoretically,
this kind of pattern.
And this is actually
really detected
by both LIGO experiments, which
lay at 3,000 kilometers apart
from each other.
So I think this is really
a historical moment--
that we actually were very
lucky to live in this moment.
What does that mean?
That means we have a
new way to really hear
about what the universe is
actually trying to tell us.
We have a new way
to detect phenomena,
which is actually really
happening very, very far away
from the Earth.
How far is,
actually, this event?
This event, according
to calculation,
is actually something like
1.3 billion light years
away from the Earth.
And we can detect that.
And we even know the mass
of the two black holes.
Wow!
What does that mean?
This is really crazy to
me and really exciting,
because we are opening up--
OK, I have two ears.
And it's opening up
another ear in my brain.
And that is actually the way
to hear the gravitational wave
with the experiment
we've performed on Earth.
So I hope, until now,
I have convinced you
that this is really not the end
of vibration of waves, which
is actually the end of 8.03.
Instead, this is actually
just the beginning.
You have a lot more
to explore when
you take general or
special relativity course.
You have a lot, really,
more to explore when
you take quantum mechanics.
And I hope you really enjoy
the content of this course.
Personally, I really
enjoy that very much.
I love this course.
And I hope you also love
it and understand something
from my lecture.
And thank you very much.
And the next time,
we are going to have
a review of all concepts we have
learned from 8.03 next Tuesday.
Thank you.
[APPLAUSE]
Thank you very much.
