All physics
of the 19th century and earlier is called classical physics. Examples are
Newtonian mechanics, which we dealt
with this whole term, and electricity and magnetism, which you will encounter
the next term. In the early part
of this century, when we learned
about the composition of atoms, it became clear that
classical physics did not work on the very small scale
of the atoms. The size of an atom is only
ten to the minus ten meters. If you take 250 million of them
and you line them up, that's only one inch. In 1911, the English physicist
Rutherford demonstrated that almost all the mass
of an atom is concentrated in an extreme small volume
at the center of the atom. We call that the nucleus,
it's positively charged. And there are electrons
which are negatively charged, which are in orbits
around the nucleus, and the typical distances from
the nucleus to the electrons is about 100,000 times larger than the size
of the nucleus itself. As early as 1920,
Rutherford named the proton, and Chadwick discovered
the neutron in 1932, for which he received
the Nobel Prize. Now, let us imagine that
this lecture hall is an atom. And the size of an atom
is defined by the orbits, the outer orbits
of the electrons. If I scale it properly, now,
in this ratio 100,000 to 1, then the size
of the nucleus would be even smaller
than a grain of sand. And it just so happens that yesterday
I went to Plum Island, I walked for three hours
on the beach and I ended up
with some sand in my pockets. And so I will donate
to you one proton; make sure you hold onto it... Ooh, this is two protons,
that's too generous. So keep it there--
this is one proton. Just think about
what an atom is. An atom is all vacuum. You and I are all vacuum. You think of yourself as being
something, but we are nothing. You can ask yourself
the question, If you are all vacuum, why is it, then,
that I can move my hand not through the other hand, like
a ghost can walk through a wall? That's not so easy to answer, and in fact, you cannot answer
it with classical physics and I will not return
to that today. But you are all vacuum. According to Maxwell's
equations, Maxwell's law of electricity
and magnetism, an electron, because of the
attractive force of the proton, would spiral into the proton in a minute fraction
of a second, and so atoms could not exist. Now, we know that's not true. We know that atoms do exist. And so that created
a problem for physics and it was the Danish physicist
Niels Bohr who in 1913 postulated that electrons move around the
nucleus in well-defined energy levels which are distinctly separated
from each other, and that the spiraling-in of
the electrons into the nucleus does not occur, for the reason
that an electron cannot exist in between these allowed energy levels. It can jump
from one energy level to another, but it cannot exist in between. Now, Bohr's suggestion
was earth-shaking, because it would also imply that a planet
that goes around the sun cannot orbit the sun
just at any distance. You couldn't move it
just a trifle in or a trifle farther out. It would also require
discrete energy levels. It would also mean that
if you had a tennis ball and you would bounce
the tennis ball up and down, that the tennis ball
could not reach just any level above the ground, but it would only be
discrete levels, and that is very much
against our intuition. We'd like to think that
when you bounce a tennis ball, that it can reach
any level that you want to. You give it
just a little bit more energy and it will go a little higher. That, according
to quantum mechanics, would not be possible. Now, all this seems
rather bizarre, as it goes
against our daily experiences, but before we dismiss
the idea of quantization-- see, the quantization comes in when you talk
about discrete energy levels-- you have to realize
that the differences in the allowed heights
of the tennis ball and the differences between the allowed orbits
of the planets around the sun would be so
infinitesimally small that we may never be able
to measure it. In other words, quantum
mechanics really plays no role in our macroscopic world. Now, atoms are very, very small
compared to tennis balls, and the quantization effects
are much larger in the sub-microscopic world
of electrons and atoms than in our familiar world of baseballs,
pots and pans, and planets. So before we continue,
I would like to repeat to you one of the cornerstones
of quantum mechanics. And it says that the electrons
in atoms can only exist at well-defined energy levels--
and they can not exist in between. Now, when I heat a substance,
the electrons in the atoms can jump from low energy states
to high energy states, and when they do so, they can leave a hole,
an opening, an empty energy state. But later on, they can fall back
to fill that opening. They can occupy
that energy state again. And when I keep heating
this substance, there is some kind of
a musical chair game going on. The electrons will go
to higher energy levels, they may spend there some time and then they may fall to
low energy levels. You see here a vase,
a very precious vase, and when I pick up this vase,
I have to do work. I bring it further away
from the center of the Earth. Now, is that energy lost? No. I could drop the vase, and it
would pick up kinetic energy. I will get that energy back. Gravitational
potential energy will be converted
to kinetic energy. It will crash to pieces,
and it will generate some heat. In fact, the breaking itself
of this vase would take some energy. In a similar way, the energy
that you put into electrons when you bring them
to higher energy levels is retrieved
when the electrons fall back. So there is a parallel-- dropping this vase and getting
your work back that I put in. It wouldn't be a nice thing
to do to this 500-year-old vase, but as far as I'm concerned, perfectly reasonable
to do it with Ohanian, so we can let that go, and the energy will come out
in the form of heat and also in the form
of, perhaps, some noise. When electrons fall from higher energy states to
low energy states, it's not kinetic energy
that is released, but it comes out often
in the form of light, electromagnetic radiation. Light has energy. Einstein formulated
that a light photon, the energy of a light photon,
is h times the frequency, and h is Planck's constant--
named after Max Planck-- and h is about 6.6 times 10
to the minus 34 joule-seconds. Now we've also seen in 8.01 that lambda,
the wavelength of light, equals the speed of light
divided by the frequency. And so if I eliminate the
frequency, I also can write that the energy
of a light photon equals hc divided by lambda. And so you see, the more energy
there is available, the smaller the wavelength. And the less energy
there is available, the longer the wavelength. And so if the jump from a high energy state
to a lower energy state is very high, then the wavelength
will be shorter than when the jump
is relatively small. I can make you some kind of an
energy diagram of these jumps. And these are energy levels, so
energy goes in this direction, this would be the lowest possible energy state. So these would be allowed
energy levels, allowed orbits. And if this electron had jumped
all the way here, then it could fall back
at a later moment in time and the energy could be so much that you couldn't even see
the light. It could be ultraviolet, and this jump may
still be ultraviolet, but now this jump, which is
a little less energy, that may be in the blue part
of our spectrum. So we may see this
as blue light. And this one, which is
a little less than this, this energy may generate, this jump may generate
green light. And the jump from here to here,
which is even less, may generate red light. And a jump from here to here,
which is even less, may again be invisible,
so this may be infrared. And so as the electrons fall from a high energy state
to a lower energy state, you expect very discrete
energies to come out, very discrete wavelengths, and these wavelengths that you
would see correspond, then, to these allowed transitions
between these energy levels. So if we could look at that
light and sort it out by color, we would, in a way, see
these energy levels. Now, you have
in your little envelope a piece of plastic,
which we call a grating, and the grating has the ability to decompose the light
in colors, which we call a spectrum, and we're going to shortly use
that grating to look at light from helium
and light from neon. But before we do that,
I'd like to hand out-- as a souvenir to a few people,
randomly picked-- something
that they can also use. It's not as good
as your grating, though, but it's also nice. You will see a more spectacular
result, but not as clean. It's not as clean. All right, one for you,
one for you, one for you and one for you. And you want one-- I can tell
that-- and you want one. And here, for you, for you. Oh, no, this side hasn't
had anything. I've got to walk
all the way over now. So this is really for children's
parties, which I'm handing out. Oh, George Costa, you want one,
of course. Professor Costa wants one--
I couldn't bypass him. And you want one, okay,
and you want one. So, by all means,
use your grating, but then, at the very end, you can always use
these little spectacles, which don't work nearly as well,
but, uh... this kind of thing. I'm going to light here this bulb, this light,
which has helium in it, and what you're going to see
with your grating, if you hold
your grating properly-- you may have to rotate
it 90 degrees; you will see how that works
when you try it-- you're going to see
very, very sharp, narrow lines at various colors. I want you to realize
that the reason why you see
very sharp, narrow lines is only because my light source
are very sharp, narrow line. If you use it on something that is not
a very sharp, narrow line, then you're not going to see
through that grating very sharp, narrow lines. So don't confuse the lines
that are on the grating with the line source
that I have here. Now, when you look through
your grating very shortly, you will see, on both sides,
the wonderful lines. It's a mirror image, and we will discuss it
in a little bit more detail, but before you look
through your grating, I first want you to simply look
at it without the grating, because then it is
even more spectacular when you use the grating. Because you have no clue,
when you don't use the grating, what kind of colors
are hidden there. And the colors
that you are going to see are these electron levels. So I am going to make it dark. And I will turn this on. And this one, I believe,
is helium. I have a grating here. So we have to rotate it
so that you see vertical lines on either side. You may have to rotate it
90 degrees, no more. And if you look closely-- for instance, look on
the right side of the light-- you'll see a distinct blue line,
a few blue lines, green, very nice bright yellow one,
and you see red. And if you go further
to the right, you see a repeat. It's a little fainter,
but you see a repeat of that. That's not important right now; I just want you to see that this
light, which you have no idea that it comes out
in very discrete wavelengths, very discrete frequencies, and they correspond to these
jumps from allowed energy levels to other allowed energy levels,
but there is nothing in between. And when you look on the left
side, you'll see a mirror image of what you see
on the right side. Now, neon... excuse me, helium
has only two electrons. I'm now going to put in the neon
bulb and that makes it richer, for reasons that neon
has ten electrons, so you have many more
allowed energy levels, so many more ways, that the electrons can play
musical chair. A lot of lines in the red--
I'm not blocking you, I hope-- A lot of lines in the red-- and some beautiful lines
in the yellow. I see some in the green, I don't see much in the blue...
a little bit in the blue. But the key thing is,
I want you to see that these lines are discrete. It is not just any wavelength
that can be generated; it's only the allowed energy levels,
the musical game when the electrons jump
from one orbit to another, and that gives you
this unique discrete spectrum. Now, these light spectra
were known long before Bohr came
with his daring ideas, but before quantum mechanics, these lines were
a great mystery, but they no longer are. I suggest you use this grating and use it
when you are outside at night; look at some streetlights, particularly sodium lamps
and mercury lamps. And, of course, the neon lamps
are quite spectacular, but keep in mind, you will not
see very nice straight lines unless your light source itself is a very nice straight,
narrow light source. Now, quantum mechanics took
a big leap in the '20s, and it would be
impossible for me in the available amount of time to do justice
to all the basic concepts. However, I will discuss
some consequences that are rather nonintuitive. Prior to quantum mechanics, there was a long-standing battle
between physicists whether light consists
of particles or whether they are waves. Newton believed strongly
that they're particles, and the Dutchman Huygens
believed that they were waves. And it seemed like, in 1801, that a conclusive experiment
was done by Young, which demonstrated unambiguously
that light was waves; Huygens was right. But as time went on,
discomfort was growing, as there were also experiments that showed rather conclusively
that light really was particles. And it was one of the great
victories of quantum mechanics that it showed
that light is both. At times it behaves like waves and at other times,
it behaves like particles; it all depends on
how you do your experiment. In 1923, Louis de Broglie made
the daring suggestion that a particle can behave
like a wave, and he specified, he was very
specific, that the wavelength-- which nowadays is called
de Broglie wavelength-- is h, Max Planck's constant, divided by the momentum
of that particle and the momentum is the mass of
the particle times the velocity, as we have seen in 8.01. If the momentum is higher,
then the wavelength is shorter. A baseball will have
a very high momentum, with a ridiculously low...
short wavelength. Now, one of
the startling consequences is that protons and electrons, which everyone of that time
considered particles, can then also be considered
as being waves. And in 1926, the Austrian
physicist Schrodinger drove the nail in the coffin
with his famous equation-- Schrodinger's equation,
it's called now-- which is the ground pillar
of quantum mechanics and it unifies the wave and the
particle character of matter. Returning to my baseball, take a mass of the baseball
of, say, half a kilogram and give it a speed
of 100 miles per hour. Calculate the wavelength
that you would find, according to quantum mechanics. That wavelength
is so absurdly small, So it is completely meaningless. So quantum mechanics plays
no role in our macroscopic world of pots and pans and baseballs. But now take an electron. You take the mass
of the electron, 10 to the minus 30 kilograms. And you give the electron a speed of, say,
1,000 meters per second. Now you get a wavelength which
is comparable to the wavelength of visible light, red light. And now it's something
that becomes very meaningful, something that can be measured. Now, you may argue, "Gee,
what difference does it make? "Who cares
whether something is a wave or whether something
is a particle?" Well, it makes
a huge difference, because waves have crests
and they have valleys, and so if you take two sources
of waves, either water waves-- two sources, tapping up and down
on the water-- or you can take
two sound sources, then there are certain locations
on the surface of the water where the crest of one wave arrives at the same time
as the valley of the other, and so they cancel
each other out. There is nothing,
there is no motion of the water. We call that
destructive interference. Of course,
there are other places where there is
constructive interference, where they support each other. Now, if particles
can do that, too... That is very hard to imagine-- how can one particle
with another particle interfere and vanish, that the
two particles no longer exist? So if, indeed,
particles are waves, you should be able
to demonstrate that by having the interference
pattern of two particles, like the water waves, and make--
at certain locations in space-- those particles disappear,
which turn out to be possible. But that's
a very nonintuitive idea. So we think of it
too classically when we say, "Well, two particles
cannot disappear." But in quantum mechanics, you can think in waves
if you want to, and then you have no problems
with the interference pattern and the destructive interference
at certain locations. Now, there are
other remarkable consequences of quantum mechanics
in classical mechanics. If you and I are clever enough, you think that
we should be able to determine the position of an object
to any accuracy that we require, and at the same time
determine also its momentum at any accuracy that we require. It's just a matter
of how clever we are. Simultaneously,
the object is right there and that is its mass
and that is its speed. However, the German physicist
Heisenberg realized in 1927 that a consequence
of quantum mechanics is that this is not possible. Strange as it may sound to you,
Heisenberg stated that the position
and the momentum of an object cannot be measured very
accurately at the same time. And I will read to you Heisenberg's uncertainty
principle, the way we know it. It says, "The very concept
of exact position of an object "and its exact momentum,
together, have no meaning in nature." It's a profound
nonclassical idea, and it is hard for any one
of us-- you and me included-- to comprehend. But it is consistent
with all experiments that we can do to date. I want to repeat it, because it's going to be
important of what follows. "The very concept of exact
position of an object "and its exact momentum,
together, have no meaning in nature." What does it mean? First, let me write down Heisenberg's
uncertainty principle. Delta p, which is
the uncertainty in the momentum, multiplied by delta x, which is an uncertainty in
the position of that particle, is larger
or approximately equal to Planck's constant
divided by two pi-- for which, in physics,
we call that "h-bar"-- and h-bar is approximately 10
to the minus 34 joule-seconds. You see, h is 6.6 times 10
to the minus 34. If you divide that by two pi, you get about 10
to the minus 34. What does this mean, now? What it means that if the position is known
to an accuracy delta x-- we'll give you some examples-- that the momentum is ill-
determined, is not determined, to the amount delta p, larger or equal
than h-bar divided by delta x. That's what it means. And I'll give you an example which I've chosen
from a book of George Gamow. Gamow wrote a book
which he called Mr. Tompkins in Wonderland. It's about dreams. Mr. Tompkins wants
to understand the quantum world, and there is a professor-- you will see a picture
of the professor-- who takes him, in his dreams, along the various
remarkable nonintuitive effects of quantum mechanics. And in one of these dreams, the professor suggests
that we make h-bar one. And the professor takes
a triangle in the pool table and he puts the triangle
over one billiard ball, so the billiard ball is
constrained in its position and that delta x is roughly...
say, 30 centimeters, 0.3 meters. That means that the momentum
is not determined, not determined to an approximate
value of one divided by 0.3, is about 3 kilogram-meters
per second. Now, if we give the billiard
ball a mass of one kilogram, then delta p is m delta v,
and so if m is one kilogram, then the speed of that
billiard ball is undetermined, according to Heisenberg's
uncertainty principle, by at least approximately
three meters per second. Three meters per second--
that means seven miles per hour, and so that billiard ball will go around like crazy
in that triangle, and that's exactly
what happens in the dream. And I will show you here
a picture from that book. Mr. Tompkins is
always in pajamas, just to remind you
that it is a dream. And needless to say, the professor is a very old man
and has a very nice beard; it adds to the prestige. And I will read you
from this book. I will read you a very short
paragraph that deals with this. "So the professor says, "'Look, here, I'm going
to put definite limits "'on the position of this ball by putting it
inside a wooden triangle.'" "As soon as the ball
was placed in the enclosure, "of the whole inside
of the triangle "became filled up
with glittering of ivory. "'You see,' said the professor, "'I defined the position
of the ball "'to the extent of the
dimensions of the triangle. "'This results in considerable
uncertainty in the velocity "'and the ball is moving rapidly
inside the boundary. "'Can't you stop it?'
asked Mr. Tompkins. "'No, it is
physically impossible. "'Anybody in an enclosed space
possesses a certain motion. "'We physicists call it
zero point motion, "'such as, for example, the motion of electrons
in any atom.'" So here you see
quantum mechanics at work when h-bar is one. This is
a very nonclassical idea, because you and I would think-- and we've always dealt
with that in 8.01-- that you can take an object
and place it at location "a," and we say at time t zero
it is at "a" and it has no speed
and we know the mass, so we know both the momentum
and the position to an infinite accuracy. But according to quantum
mechanics, that's not possible. So let's now return to the real
world, where h-bar is not one, but where h-bar is
10 to the minus 34, and let's now put a billiard
ball inside this triangle. Now, delta x is the same, but since h-bar is
10 to the minus 34, delta p is, of course,
10 to the 34 times smaller, and so the velocity is 10
to the 34 times smaller. This undeterminedness... degree to which the velocity
is now undetermined, is so ridiculously small-- it is 3 times 10 to the minus
34 meters per second-- that if you allowed that ball
to move with that speed, in 1 billion years, it would move only 1/100
of a diameter of a proton, so it's meaningless again. And so again, you see that
quantum mechanics plays no role in our daily macroscopic world of baseballs and basketballs
and billiards and pots and pans. And therefore, it is
completely okay for us to say, "I have a billiard ball
which is at point 'a,' and its mass is one kilogram
and it has no speed." That is completely kosher,
completely acceptable, and quantum mechanics
has no problems with that. Let's now turn to an atom. Take a hydrogen atom. The diameter of a hydrogen atom is about 10
to the minus 10 meters. So the electron is confined to a delta x of about 10
to the minus 10 meters. That means the momentum of that
electron becomes undetermined-- according to Heisenberg's
uncertainty principle-- to about 10 to the minus 34,
divided by 10 to the minus 10, is about 10 to the minus 24
kilogram-meters per second. What is the mass of an electron? That's about 10
to the minus 30 kilograms. So this, delta p,
is also m delta v. So it means that delta v-- that means the velocity
of the electron-- is undetermined, according
to Heisenberg's principle, by an amount which is at least
10 to the minus 24, which is this delta p divided
by the mass of the electron, which is 10 to the minus 30. And that is about 10
to the six meters per second-- that is one-third of a percent
of the speed of light. So the electron is moving only because of the fact
that it is confined. That's what quantum mechanics
is all about. The electron's motion
is dictated exclusively by quantum mechanics. I'm going to show you
an experiment in which I want to convey to you how nonintuitive Heisenberg's
uncertainty principle is. I have here a laser beam, and this laser beam is going to
be aimed through a narrow slit-- I'll make a drawing,
I'll turn this light off-- and that slit,
which is a vertical slit, can be made narrow
and can be made wider. Here is this light beam and here
is this opening, this slit. It's only going to be confined
in this direction, not in this direction. And so
the light will come out here, and then, on a screen, which
is going to be that screen, at large distance capital L, we're going to see
that light spot, due to the light beam
going through the slit and this separation, capital L. I start off
with the slit all the way open and so you're going to see
this light spot like this. And then I'm going to make
the slit narrower and narrower, and as I'm going to cut
into the light beam, what you're going to see
is exactly what you expect. You expect
that this light disappears, and when I cut in further,
you see exactly what you expect, that this light disappears. And so the light spot
there on that screen will become narrower
and narrower and narrower. But then there comes a point
that Heisenberg says, "Uh-uh, careful now, because
your delta x, your knowledge, "the accuracy in this direction
where the light goes through "is now so high
that now I'm going to introduce "an uncertainty
in the momentum of that light. "The momentum of that light is now no longer determined
to infinite accuracy." And what that means,
if you start fooling around with the momentum of that light
in the x direction, it no longer
goes through straight but it goes off at an angle,
and I will make you a more quantitative calculation
for that. So let's look at this slit
from above. Here's the slit, and the slit
has an opening, delta x. And this delta x we're going
to make smaller and smaller, and let us start with a delta x
of about 1/10 of a millimeter, which is 10
to the minus 4 meters. I have light, I know
the wavelength of the light, and I know that lambda
equals h divided by p, according to De Broglie. I know the wavelength, I know h, and so I can calculate
the momentum of that light. I have done that,
take my word for it. It is about 10 to the minus 27
kilogram-meters per second. That's the momentum of
the individual light photons. Think of them as particles, which you can do,
according to de Broglie. So now I have a delta p, the degree to which the momentum
is undetermined, according to Heisenberg, is going to be 10 to the minus
34 divided by delta x-- which is 10 to the minus 4, so that is 10
to the minus 30, very small. But the momentum itself
is 10 to the minus 27, so it's only one part
in a thousand. So what will happen? If the light comes
through here... And I now make
a classical argument. I say, "This is
the momentum of the light as it comes straight in." When it has to be squeezed
through this narrow opening, Heisenberg's uncertainty
principle demands that it is going
to be undetermined, the momentum in this direction by roughly 10 to the minus 30,
or more. Remember, it is always
larger or equal. In other words, if I introduce,
for instance, in this direction
or in this direction, delta p, then I would expect that some of that light
goes off in this direction. It is this change in momentum, this undeterminedness
in momentum, that makes it go off at an
angle, only in the x direction. If I have the slit like this, don't expect this to happen
in this direction, because the uncertainty
in the y direction, that's not the problem. Delta y is not very small,
it's delta x that is very small, so it's this direction that's
going to give you trouble. It's only in this direction that you know precisely
where that light goes through. This direction is not the issue. So this angle theta can now
be calculated very roughly. Theta is obviously delta p
divided by p, so theta is very roughly
10 to the minus 3 radians, which is a fifteenth
of a degree, and if you have
at a distance L-- if this distance here is L-- if you have here a screen, then the spot on this screen...
if I call that x at location L, then x at location L
is obviously theta times L. And if theta is
10 to the minus 3-- and let's assume this is
about 10 meters away from us, so L is about 10 meters-- then you get 10
to the minus 2 meters. That is one centimeter. One centimeter in this direction and one centimeter in that
direction-- two centimeters. But when I make the slit width
10 times smaller, if I make the slit width
only 1/100 of a millimeter, then this becomes
10 centimeters, because now I know delta x
10 times better, and so delta p is 10 times
more uncertain. So now I expect to see here at least a smear
of 20 centimeters and at least a smear
of 20 centimeters there. So the absurdity is that a
teeny-weeny little light source which in the beginning you
will see as a very small spot... When I make this slit
narrower and narrower, indeed, you will see
that you will lose photons, and you will see this getting
narrower and narrower, and then all of a sudden,
it begins to spread out, and it begins to spread out, and by the time I'm close
to a tenth of a millimeter, the light spot will be yay big. Very nonintuitive. You make the slit smaller,
and the photons spread out. And I want to show
that to you now. I have to make it very dark. And I need my flashlight,
turn on the laser beam. There you see it. The slit is now
all the way open. Yeah, it's all the way open, and I'm going to close
the slit now slowly. And if you look closely,
you will see that the... Let me also get my red laser,
then I can point something out. You will see that the light will get squeezed
in the horizontal direction. You can see already
at the left side, has a very sharp
vertical cut-off, and the right side also. It's getting narrower,
it's getting narrower. Getting narrower, but I'm nowhere nearly
a tenth of a millimeter yet. It's getting clearly narrower. You see, it's getting narrower,
it's getting narrower. If I look here... oh, I'm not yet at the
tenth of a millimeter, but I'm getting there. I'm going slowly, squeezing it. I'm squeezing those photons. Those photons now are forced to go through
an extremely narrow opening and Heisenberg is
very shortly going to jump in and says, "You are going
to pay a price for that. "You know too well where those
photons are in the x direction. "The price you pay-- "that nature will now make
the momentum undetermined in the x direction." And you begin to...
you see it now. You really begin to see that the center portion
is widening. Even photons appear. Here, you see some dark lines, which I will not
further discuss today, but notice that the light
is spreading. Of course, when I squeeze this
slit, when I make it narrower, it's obvious that I lose light, because the light
that hits the side of the slit is not going through, so the light intensity
will go down. That's just inevitable. I used fewer photons. But look at this. There are photons here,
there are photons there. It's at least 10 centimeters,
this portion. From here to here is
at least one foot. I squeeze more-- this is more
than half a meter now. I squeeze more-- this is
about one meter already. Now, not only have you seen
quantum mechanics at work, in terms of electrons jumping
between allowed energy levels, but you now have also seen one other
very interesting consequence of quantum mechanics, which is Heisenberg's
uncertainty principle. Now, the spreading of this light
can very easily be explained without Heisenberg's
uncertainty principle. In fact, it was known,
even in the previous century, to a high degree of accuracy,
why this happens, and the dark lines were
very accurately explained. All I wanted to show is that the spreading of the
light is entirely consistent with Heisenberg's uncertainty
principle, and it better be, because it would
not be possible, it would be inconceivable
that you could do any experiment that would violate Heisenberg's
uncertainty principle. And if this light that you would
see on the screen there, if that light spot would get
narrower and narrower and narrower and narrower
all the time, as we would think classically,
that would have been a violation of Heisenberg's
uncertainty principle, and that is not possible. Now, there is no way in advance to predict
which photons end up where. All you can do
with quantum mechanics is to do the experiment
with lots of photons and then you will get
a certain distribution and the distribution will be
exactly as you saw there. Quantum mechanics
can never predict, on an individual photon,
where it will end up. We saw that bright spot
in the center. So if you did this experiment
with one photon per day-- one photon per day
going through this slit-- and you had
a photographic plate there, and you would keep it there
for months, and you would develop it, you would see the same pattern
that you see there. This photon arrives today. Here arrives one tomorrow. Here arrives one the day
after tomorrow. Here one the day after that,
the day after that, the day after that,
the day after that, the day after that, the day
after that, and slowly are you
beginning to see that pattern that you saw. So don't think that this interference pattern
that you saw is the result of two photons going through the slit
simultaneously-- not at all. You can do it
with one photon at a time and you would see
exactly the same thing. Now, this idea-- that you
cannot in advance predict what a particular photon
will do-- is a very nonclassic idea, and
it rubs us all the wrong way because our classical way
of thinking is-- and you are no different from
my own feeling in this respect-- that if you do an experiment a hundred times
in a controlled way, you should get a hundred times
exactly the same result. Not so, says quantum mechanics. All that quantum mechanics
will tell you is what the probability is
that something will happen. No guarantees, but it is very good
in predicting probabilities. Now, Einstein had great problems with this idea of not knowing
precisely what would happen, and he had endless discussions
with Bohr and others in which
he tried to convince them that because you couldn't
predict what happened, that something had to be wrong
with quantum mechanics, and Einstein's famous words
were, "God does not throw dice." This was the way,
was his way of saying, "It is ridiculous
that the outcome of a well-controlled
experiment is uncertain." Now, almost nine decades
have gone by since the beginning
of quantum mechanics, and we now know that God-- if
there is one-- does throw dice. However, God is bound to
the rules of quantum mechanics and cannot violate Heisenberg's
uncertainty principle. The light could not go straight
through without spreading when I made the slit as narrow
as I did. So quantum mechanics is
a bizarre world that we rarely experience
in our daily lives, because we are used to basketballs,
baseballs, tennis balls. But yet it is the way
the world ticks, and atoms and molecules
can only exist because of quantum mechanics. That means you and I
can only exist because of quantum mechanics. I hope that this will give you
something to think about, but I warn you in advance, because if you
start thinking about this, it will give you headaches and it will give you
sleepless nights. And it has given me countless
sleepless nights in the past, and even today, when I think
about the consequences-- the bizarre consequences
of quantum mechanics-- I still cannot comprehend it,
I still cannot digest it and I still have headaches
and sleepless nights. But it may be necessary to go
through these sleepless nights if you want to eventually evolve as an independent
thinking scientist, and I hope
that someday all of you will. Thank you. (class applauds)
