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MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: And this is a
question based on where we left off on Wednesday -- we were
talking about Coulomb's force law to describe the
interaction between two particles, and good job, most
of you got this correct. So, what we're looking at here
is the force when we have two charged particles, one positive,
one negative -- here, the nucleus
and an electron. So, I know this is a simple
example and I can see everyone pretty much got it right, and
probably those that didn't actually made some sort of
clicker error is my guess. But I wanted to use this to
point out that in this class in general, any time you see
an equation to explain a certain phenomenon, such as here
looking at force, it's a good idea to check yourself by
first plugging it into the actual equation, so you can
plug in infinity and this equation here, and what you
would see is, of course, the force, if you just solve the
math problem goes to zero. But you can also look at it
qualitatively, so, if you think about the force between
the electron and the proton, you could just qualitatively
think about what's happening. If they're close together
there's a certain force -- they're attracted because they
have opposite charges, but as that gets further and further
away, that force is going to get smaller and smaller, and
eventually the force is going to approach zero. So, it's a good kind of mental
check as we go through this course to remember every time
there's an equation, usually there's a very good reason for
that equation, and you can go ahead and just use your
qualitative knowledge, you don't have to just always stick
with the math to check and justify your answers. So, we can get started with
today's lecture notes. And, as I mentioned, we left
off and as we started back here to describe the atom and
how the atom holds together the nucleus and the electron
using classical mechanics. And today we'll finish that
discussion, and, of course, point out actually the failure
of classical mechanics to appropriately describe what's
going on in an atom. So, then we'll get to turn to
a new kind of mechanics or quantum mechanics, which will
in fact be able to describe what's happening on this very,
very small size scale -- so on the atomic size scale on the
order of nanometers or angstroms, very small
particles. And the reason that quantum
mechanics is going to work where classical mechanics
fails is that classical mechanics did not take into
account the fact that matter has both wave-like and
particle-like properties, and light has both wave-like and
particle-like properties. So, we'll take a little bit of a
step back after we introduce quantum mechanics, and talk
about light as a wave, and the characteristic of waves, and
then light as a particle. And one example of this is in
the photoelectric effect. So, we just talked about the
force law to describe the interaction between a proton
and an electron. You told me that when the
distance went to infinity, the force went to zero. What happens instead when the
distance goes to zero? What happens to the force? Yeah. So, the force actually goes to
infinity, and specifically it goes to negative infinity. Infinity is the force when we're
thinking about it and our brains, negative infinity
is when we actually plug it into the equation here, and the
reason is the convention that the negative sign is just
telling us the direction that the force is coming together
instead of pushing apart. So, we can use Coulomb's force
law to think about the force between these two particles --
and it does that, it tells us the force is a function
of that distance. But what it does not tell us,
which if we're trying to describe an atom we really want
to know, is what happens to the distance as
time passes? So, r is a function of time. But luckily for us, there's a
classical equation of motion that will, in fact, describe how
the electron and nucleus change position or change
their radius as a function of time. So, that's -- does anyone know
which classical law of motion that would be? Yup, so it's going to be
Newton's second law, force equals mass times acceleration
-- those of you that are quick page-turners, have a little
one-up on answering that. And that tells us force as a
function of acceleration, we want to know it though as a
function of radius, so we can just take the first derivative
and get ourselves to velocity. So, force is equal to
mass times dv /dt. But, of course, we want to go
all the way to distance, so we take the second derivative and
we have this equation for force here. And what we can do in order
to bring the two equations together, is to plug in the
Coulomb force law right here. So, now we have our Coulomb
force law all plugged in here, and we have this differential
equation that we could solve, if we wanted to figure out
what the force was at different times t, or at
different positions of r. So, all you will have the
opportunity to solve differential equations in
your math courses here. We won't do it in this
chemistry course. In later chemistry courses,
you'll also get to solve differential equations. But instead in this chemistry
course, I will just tell you the solutions to differential
equations. And what we can do is we can
start with some initial value of r, and here I write r being
ten angstroms. That's a good approximation when we're talking
about atoms, because that's about the size
of and atom. So, let's say we start off
at the distance being ten angstroms. We can plug that
into this differential equation that we'll have and
solve it, and what we find out is that r actually goes to zero
at a time that's equal to 10 to the negative 10 seconds. So, let's think qualitatively
for a second about what that means or what the real
meaning of that is. What that is telling us is that
according to Newtonian mechanics and Coulomb's force
law, is that the electron should actually plummet into
the nucleus in 0.1 nanoseconds. So, we have a little bit
of a problem here. And the problem that we have
is that what we're figuring out mathematically is not
exactly matching up with what we're observing experimentally. And, in fact, it's often kind of
difficult to experimentally test your mathematical
predictions -- a lot of people spend many, many years testing
one single mathematical prediction. But, I think all of us right
now can probably test this prediction right here, and we're
observing that, in fact, all of us and all the atoms we
can see are not immediately collapsing in less than
a nanosecond. So, just, if you can take what
I'm saying for a moment right now that in fact this should
collapse in this very small time frame, we have to see that
there's a problem with one of these two things, either
the Coulomb force law or Newtonian mechanics. So, what do you guys think is
probably the issue here? So, it's Newtonian mechanics,
and the reason for this is because Newtonian mechanics does
not work on this very, very small size scale. As we said, Newtonian mechanics
does work in most cases, it does work when we're
discussing things that we can see, it does work even
on things that are too small to measure. But once we got to the atomic
size scale, what happens is we need to be taking into account
the fact that matter has these wave-like properties, and we'll
learn more about that later, but essentially classical
mechanics does not take that into account at all. So, we need a new kind of
mechanics, which is quantum mechanics, which will accurately
explain the behavior of molecules
on this small scale. So, as I mentioned, the real
key to quantum mechanics is that it's treating matter not
just like it's a particle, which is what we were just
doing, but also like it's a wave, and it treats light
that way, too. The second important point to
quantum mechanics is that it actually considers the fact that
light consists of these discrete packets or
particle-like pieces of energy, which are
called photons. And if you think about what's
actually happening here, this second point that light
consists of photons is actually the same thing as
saying that light shows particle-like properties, but
that's such an important point that I put it separately, and
we'll cover that separately as we go along. So, we now have this new way
of thinking about how a nucleus and an electron can
hang together, and this is quantum mechanics, and we can
use this to come up with a new way to describe our atom and the
behavior of atoms. But the problem is before we do this,
it makes sense to take a little bit of a step back and
actually make sure we're all on the same page and
understanding why quantum mechanics is so important and
how it works, and specifically understanding what we mean when
we say that light is both a particle and a wave, and that
matter is both a particle and a wave. So, we'll move on to
this discussion of light as a wave, and we really won't
pick up into going back to applying quantum mechanics to
the atom until Friday, but in the meantime, we'll really get
to understand the wave particle duality of light
and of matter. So, we'll start with thinking
about some properties of waves that are going to be applicable
to all waves that we're talking about, including
light waves. The easiest kind of waves for us
to picture are ocean waves or water waves, because we can,
in fact, see them, but they have similar properties
to all waves. And those properties include
that you have this periodic variation of some property. So, when we're talking about
water waves, the property we're discussing is just
the water level. So, for example, we have this
average level, and then it can go high where we have the peak,
or it can go very low. We can also discuss sound waves,
so again it's just the periodic variation of some
property -- in this case we're talking about density, so we
have high density areas and low density areas. So, regardless of the type of
wave that we're talking about, there's some common definitions
that we want to make sure that we're all
able to use, and the first is amplitude. And when we're talking about
the amplitude of the wave, we're talking about
the deviation from that average level. So, if we define the average
level as zero, you can have either a positive amplitude
or a negative amplitude. So, sometimes people get
confused when they're solving problems and call the amplitude
this distance all the way from the max to the
min, but it's only half of that because we're only going
back to the average level. So, what we really want to
talk about here is light waves, and light waves have the
same properties as these other kind of waves in that
they're the periodic variation of some property. So, when we're discussing
light waves, what we're talking about is actually
light or electromagnetic radiation, is what we'll
be calling it throughout the course. And that's the periodic
variation of an electric field. So, instead of having the
periodic variation of water, or the periodic variation of
air density, here we're talking about an
electric field. We know what an electric field
is, it's just a space through which a Coulomb force
operates. And the important thing to
think about when you're talking about the fact that it's
a periodic variation, is if you put a charged particle
somewhere into an electric field, it will, of course, go in
a certain direction toward the charge it's attracted to. But you need to think about the
difference, if you have a particle here on your wave, it
will go in one direction. But remember, waves don't just
have magnitude, they also do have direction. So, if instead you put your
particle somewhere down here on the electric field, or on the
wave, the electric field will now be in the other
direction, so your particle will be pushed the other way. And from physics you know that,
of course, if we have a propagating electric field, we
also have a perpendicular magnetic field that's going
back and forth. But in terms of worrying about
using the concepts of a wave to solve chemistry problems in
this course, we can actually put aside the fact, and only
focus on the electric field part of things, because that's
what's going to be interacting with our charged particles,
such as our electrons. So, other properties of waves
that you probably are all familiar with but I just want
to review is the idea of a wavelength. If we're talking about the
wavelength of a wave, we're just talking about the distance
that there is between successive maxima, or of course,
we can also be talking about the distance between
successive minima. Basically, we can take any point
on the wave, and it's the distance to that same point
later on in the wave. So, that's what we call
one wavelength. We also commonly discuss the
frequency of a wave, and the frequency is just the number of
cycles that that wave goes through per unit time. So, by a cycle we'd basically
mean how many times we cycle through a complete wavelength. So, if something cycles through
five wavelengths in a single second, we would just say
that the frequency of that wave is five per second. We can also mathematically
describe what's going on here other than just graphing it. So, if we want to look at the
mathematical equation of a wave, we want to describe --
again as I mention, what we're describing is the electric
field, we're not worrying about the magnetic field here,
as a function of x and t that's equal to a cosine [
2 pi x over wavelength, minus 2 pi nu t ]. And note this is the
Greek letter nu. This is not a v. Where we have
E, which is equal to the electric field, what is x? STUDENT: Position. PROFESSOR: Yup, the position of
the wave. And what about t? Yeah, so we're talking about
both position and time. So what we can do if we're
talking about a wave is think of it both in terms of position
time, but if we're trying to visualize this -- for
example if we're actually to graph this out, the easiest
thing to do is keep one of these two variables constant,
either the x or the t, and then just consider the
other variable. So, for example, if we're to
hold the time constant, this makes it a lot simpler of an
equation, because what we can end up doing is actually
crossing out this whole term here. So what we're left with is just
that the electric field as a function of distance is a
times cosine of the argument there, which is now just
2 pi x over wavelength. So, what we want to be able to
do, either when we're looking at the graph or looking at the
equation up there, is to think about different properties of
the wave. For example, to think about at what point do we
have the wave where it's at its maximum amplitude? So, if we think about that, we
need to have a point where we're making this argument of
the cosine such that the cosine is going to all be equal
to one, so all we're left with is that a term. So, we can do that basically
any time that we have an integer variable that is either
zero or an integer variable of the wavelength. So, for example, negative
wavelength or positive wavelength are two times the
wavelength, because that lets us cross out the term with the
wavelength here, and we're left with some integer
multiple of just pi. So, that's sort of the
mathematically how we get to a, but we can also just look
at the graph here, because every time we go one wavelength,
we can see that we're back in a maximum. So, I mentioned we should be
able to figure out where the maximum amplitude is. You should also just looking at
an equation, immediately be able to figure out what that
maximum amplitude is in terms of the height of it just by
looking at that a-term, here we should also be able to know
the intensity of any light wave, because intensity is just
the amplitude squared. So, we should immediately be
able to know how bright or how intense a light is just looking
at the wave equation, or just by looking at a graph. We can also do a similar
thing, and I'll keep my distance from the board, but
we can instead be holding x constant, for example, putting
x to be equal to zero, and then all we're doing is
considering the electric field as a function of t. So, in this case we're crossing
out the first term there, and we're left with
amplitude times the cosine of 2 pi nu times t. And, of course, we can do the
same thing again, we can think about when the amplitude is
going to be at its maximum, and it's going to be any time
cosine of this term now is equal to one. So that will be at, for example,
negative 1 over nu, or 0, or 1 over nu. And again, we can just look at
our graph to figure that out, that's exactly where
we're at a maximum. So, 1 over nu is another term
we use and we call it the period of a wave, and the period
is just the inverse of the frequency. And if we think about frequency,
that's number of cycles per unit time. So, for example, number of
cycles per second, whereas the period is how much time it takes
for one cycle to occur. And when we talk about units of
frequency, in almost every case, you'll be talking about
number of cycles per second. So, you can just write
inverse second, the cycle part is assumed. But you'll also frequently see
it called Hertz, so, Hz here. So, if you're talking about five
cycles per second, you can write five per second, or
you can write five Hertz. The one thing you want to keep
in mind though is that Hertz does not actually mean
inverse seconds, it means cycles per second. So, if you're talking about a
car going so many meters per second, you can't say it's going
meter Hertz, you have to say meters per second. So, this really just means
for frequency, it's a frequency label. Alright. So, since we have these terms
defined, we know the frequency and the wavelength, it turns
out we can also think about the speed of the wave, and
specifically of a light wave, and speed and is just equal to
the distance that's traveled divided by the time
the elapsed. And because we've defined these
terms, we have ways to describe these things. So, we can describe the distance
that's traveled, it's just a wavelength here. And we can think about how
long it takes for a wave, because waves are, we know not
just changing in position, but the whole wave is moving forward
with time, we can think about how long it takes
for wave to go one wavelength. So, one distance that's
equal to lambda. So, how much time would that
take, does anyone know? So, would it take, for example,
the same amount of time as the frequency? The period, that's right. So, it's going to take one
period to move that long. And another way we can say
period is just 1 over nu or 1 over the frequency So, now
we know both the distance traveled and the time
the elapsed. So, we can just plug it in. Speed is equal to the distance
traveled, which is lambda over the time elapsed, which is 1
over nu. so, we can re-write that as speed is equal to lambda
times nu, and it turns out typically this is reported
in meters per second or nanometers per second. So, now we have an equation
where we know the relationship between speed and wavelength
and frequency, and it turns out that we could take any wave,
and as long as we know the frequency and the
wavelength, we'll be able to figure out the speed. But, of course, there's
something very special about electromagnetic waves,
electromagnetic radiation and the speed. And it's not really surprising
for me to tell you that electromagnetic radiation has
a constant speed, and that speed is what we call the speed
of light, and typically we abbreviate that as c, and
that's from the Latin term celeritas, which means
speed in Latin. That's one of four or five Latin
words I remember from four years of high school Latin,
but it comes in handy to remember speed of light. And some of you may have
memorized what the speed of light is in high school -- it's
about 3 times 10 to the 8 meters per second. This is another example of
a constant that you will accidentally memorize in this
course as you use it over and over again. But again, that we will supply
for you on the exam just in case you forget it
at that moment. And this is a very fast speed,
of course, it's about 700 million miles per hour. So, one way to put that in
perspective is to think about how long it takes for a
light beam to get from earth to the moon. Does anyone have any guesses? Eight seconds, that
sounds good. Anyone else? These are all really good
guesses, so it actually takes 1.2 seconds for light
to travel from the earth to the moon. So, we're talking pretty
fast, so that's nice to appreciate in itself. But other than that point, we
can also think about the fact that frequency and wavelength
are related in a way that now since we know the speed of
light, if we know one we can tell the other. So, you can go ahead and switch
us to our clicker question here. So, we should be able to look
at different types of waves and be able to figure out
something about both their frequency and their wavelength,
and know the relationship between the two. So, it's up on this screen
here now, so we'll work on the other one. If you can identify which of
these statements is correct based on what you know about
the relationship between frequency and wavelength
and also just looking at the waves. Alright. So, let's give ten more
seconds on that. So, ten seconds on that. Alright. So, good job. So, most people could recognize
that light wave a has the shorter wavelength. We can see that just by looking
at the graph itself -- we can see, certainly, this is
shorter from maxima to maxima. This we can't even see
the next maxima, so it's much longer. And then, we also know that
means that it has the higher frequency, because our
relationship between wavelength and frequency
are inversely related. And also, we know the
speed of light. So, if we think about if it's a
shorter wavelength, we'll be able to get a lot more
wavelengths in, in a given time, than we would for
a longer wavelength. So, we can switch back to the
notes and think about what this means, and what this means
when we're talking about all the different kinds of light
waves we have, and I've shown a bunch here, is that if
we have the wavelength, we also know the frequency
of these wavelengths. So, for example, radio waves,
which have very long wavelengths have very
low frequencies. Whereas where we go to waves
that have very short wavelengths, such a x-rays or
cosmic rays, they, in turn, have very high frequencies. So, it's important to get a
little bit of a sense of what all these different kinds
of lights do. You're absolutely not
responsible to memorize what the wavelengths of the different
types of lights are, but you do want to be
able to know the general order of them. So, if someone tells you they're
using UV light versus x-ray light, you know that the
x-ray light is, in fact, at a higher frequency. So that's the important
take-away message from this slide. If we think about these
different types of lights, microwave light, if it's
absorbed by a molecule, is a sufficient amount of frequency
and energy to get those molecules to rotate. That, of course, generates
heat, so that's how your microwaves work. If we talk about infrared light,
which is at a higher frequency here and a shorter
wavelength, infrared light when it's absorbed by molecules
actually is enough to cause molecules
now to vibrate. If we move up to the more
high-frequency and divisible light and all the way into UV
light, if you shine UV light at certain molecules, it's going
to have enough energy to actually pop those electrons
in that molecule up to a higher energy level, which will
make more sense once we talk about energy levels in
atoms, but that's what UV light can do. And actually, that's responsible
for fluorescence and phosphorescence that
you see where typically UV light comes in. So, if you use a black lamp or
something and you excite something up to a higher energy
level and then it relaxes back down to its lower
energy state, it's going to emit a new wavelength of light,
which is going to be visible to you. X-rays are at even a higher
frequency, and those are sufficient to actually be
absorbed by a molecule and pop an electron all the way
out of that molecule. You can see how that would be
damaging to the integrity of that molecule, that's why x-rays
are so damaging -- you don't want to have electrons
disappearing for no good reason from your molecules that
can cause the kind of mutations we don't want to
be seeing in ourselves. And then also as we go
higher, we have gamma rays and cosmic rays. Within the visible range of what
we can see, you also want to know this relative order
that's pretty easy -- most of us have memorized that
in kindergarten, so that should be fine. Just remembering that violet is
the end that actually has the shortest wavelength, which
means that it also has, of course, the highest frequency. So, just an interesting fact
about this set of light, which we're most familiar with, if we
think about our vision, it turns out that our vision's
actually logarithmic and it's centered around this
green frequency. So, if instead of a red laser
pointer here, I had a green one, you'd actually, to our
eyes, it would seem like the green one was brighter, even if
the intensity was the same, and that's just because our
eyes are centered and logarithmic around this
green frequency set. So, using the relationship
between frequency and wavelength, we can actually
understand a lot about what's going on, and pretty soon
we'll also draw the relationship very soon to
energy, so it will be even more informative then. But I just want to point out one
of the many, many groups at MIT that works with different
fluorescing types of molecules, and this is Professor
Bawendi's laboratory at MIT, and he works
with quantum dots. And quantum dots are these just
very tiny, tiny crystals of semiconductor material. They're on the order of one to
ten nanometers, and these can be shined on with UV light --
they have a lot of different interesting properties, but one
I'll mention is that if you excite them with UV light,
they will have some of the electrons move to a higher
energy state, and when they drop back down, they actually
emit light with a wavelength that corresponds with the size
of the actual quantum dot. So, from what we know so far,
we should be able to look at any of these quantum dots,
which are depicted as a cartoon here, but here we have
an actual picture of the quantum dots suspended in some
sort of solution and shone on with UV light, and you can see
that you can achieve this whole beautiful range of colors
just by modulating the size of the different dots. And we should be able to know if
we're looking at a red dot -- is a red dot, it's going to
have a longer wavelength, so is this a higher or
lower frequency? Yeah, and similarly, if someone
tells us that their dot is blue-shifted, that should
automatically in our heads tell us, oh it shifted
to a higher frequency. And these dots are really
interesting in that you can, I'm sure by looking at this
picture, already imagine just a whole slew of different
biological or sensing applications that you
could think of. For example, if you were trying
to study different protein interactions, you could
think about labeling them with different colored
dots, or there's also a bunch of different fluorescent
techniques that you could apply using these dots, or you
could think of in-vivo sensing, how useful these could
be if you could think of a way to get them into your body
without being too toxic, for example. These are all things that the
Bawendi group is working on. What they are real experts in is
synthesizing many different kinds of these dots, and they
have a synthetic scheme that's used by research groups
around the world. The Bawendi group also
collaborates with people, both at different schools
and at MIT. One example, on some of their
biochemistry applications is with another Professor at MIT,
Alice Ting and her lab. So really what I want to point
out here is as we get more into describing quantum
mechanics, these quantum dots are one really good example
where a lot of the properties of quantum mechanics
apply directly. So, if you're interested, I put
the Bawendi lab research website onto your notes. And also, Professor Bawendi
recently did an interview with "The Tech." Did anyone see that
interview in the paper? So, three or four -- a few of
you read the paper last week. So, you can either pick up an
old issue or I put the link on the website, too. And that's not just about his
research, it's also about some of his memories as a student
and advice to all of you. So, it's interesting to read and
get to know some of these Professors at MIT a
little bit better. So, one property that was
important we talked about with waves is the relationship
between frequency and wavelength. Another very important property
of waves that's true of all waves, is that you can
have superposition or interference between
two waves. So, if we're looking at waves
and they're in-phase, and when I talk about in-phase, what I
mean is that they're lined up, so that the maxima are in the
same position and the minima are in the same position, what
we can have a something called constructive interference. And all we mean by constructive
interference is that literally those two waves
add together, such as the maxima are now twice as high,
and the minima are now twice as low. So, you can also imagine a
situation where instead of being perfectly lined up, now we
have the minima being lined up with the maxima here. So, if we switch over to a
clicker question maybe on this screen -- okay, can it be
done up there to switch? So, we're still settling
in with the renovations here in this room. So, why don't you all go ahead
and tell me what happens if you combine these two waves,
which are now out of phase? So, let's -- okay, so, why
don't you all think about would happen -- we'll start
with the thought exercise. You can switch back to my
lecture notes then if this isn't going. Alright. So, hopefully what everyone came
up with is the straight line, is that what
you answered? STUDENT: Yeah. PROFESSOR: OK, very good. And I didn't make you try
to draw the added, the superimposed positive
construction in your notes, but I think everyone can handle drawing a straight line. So, you can go ahead and draw
what happens when we have destructive interference. And destructive interference, of
course, is the extreme, but you can picture also a case
where you have waves that are not quite lined up, but they're
also not completely out of phase. So in that case, you're either
going to have the wave get a little bigger, but not
twice as big or a little bit smaller. So, I think the easiest way to
think about interference is not actually with light, but
sometimes it's easiest to think about with sound,
especially when you're dealing with times where you have
destructive interference. Has anyone here ever been in a
concert hall where they feel like they're kind of in a dead
spot, or you don't quite hear as well, and if you move down
just two seats all of a sudden it's just blasting at you --
hopefully not in this room. But have people experienced
that before? Yeah, I definitely experienced
it, too. And really, all you're
experiencing there is destructive interference
in a very bad way. Halls, they try to design halls
such that that doesn't happen, and I show an example of
a concert hall here -- this is Symphony Hall in Boston,
and I can pretty much guarantee you if you do go to
this Symphony Hall, you will not experience a bad seat
or a dead seat. This is described as actually
one of the top two or three acoustic concert halls
in the whole world. So, it's very well designed such
that they've minimized any of these destructive
interference dead sounds. So, it's nice, on a student
budget you can go and get the worst seat in the house and you
can hear just as well as they can hear up front, even
if you can't actually see what's going on. So, another example of
destructive interference is just with the Bose headphones. I've never actually tried these
on, but you see people with them, and what happens here
is it's supposed to be those noise cancellation
headphones. All they do is they take in
the ambient noise that's around it, and there's actually
battery in the headphones, that then produces
waves that are going to destructively interfere with
that ambient noise. And that's how it actually gets
to be so quiet when you have on, supposedly, these quite
expensive headphones. So, that's light as a wave, and
the reason -- well, that was sound as a wave, but light
as a wave is the same idea. And it was really established
by the early 1900s that, in fact, light behaved as a wave.
And the reason that it was so certain that light was a wave
was because we could observe these things -- we could see,
for example, that light defracted, and we could see that
light constructively or destructively could interfere
with other light waves, and this was all confirmed
and visualized. But also, around the time that
Thomson was discovering the electron, there were some other
observations that were going on, and the most
disturbing to kind of the understanding of the universe
was the fact that there were some observations about light
that didn't make sense with this idea that light
is a particle. And the photoelectric effect
is maybe the most clear example of this. So, the photoelectric effect is
the effect that if you have some metal, and you can pick
essentially any metal you want, and you shine light of a
certain frequency onto that metal, you can actually pop off
an electron, and you can go ahead and measure what the
kinetic energy of that electron that comes off is,
because we can measure the velocity and we know that
kinetic energy equals 1/2 m b squared, and thanks to
Thomson we also know the mass of an electron. So, this is an interesting
observation, and in itself not too disturbing, yet but the
important thing to point out is that there's this threshold
frequency that is of the metal, and each metal has a
different threshold frequency, such as if you shine light on
the metal where the frequency of the light is less than the
threshold frequency, nothing will happen -- no electron will
pop off of that metal. However, if you shine a light
with a frequency that's greater than the threshold
frequency, you will be able to pop off an electron. So, people were making this
observation, but this wasn't making any sense at all because
there was nothing in classical physics that
described any sort of relationship between the
frequency of light and the energy, much less the energy of
an electron that would get popped off of a metal that would
basically come off only when we're hitting this
threshold frequency. So, what they could do was
actually graph what was happening here, so we can also
graph what was happening, and what they found was that if we
were at any point below the threshold frequency and we were
counting the numbers of electrons that were popping off
of our metal, we weren't seeing anything at all. But if you go up the threshold
frequency, suddenly you see that there's some number of
electrons that comes off, and amazingly, the number of
electrons actually had no relationship at all to the
frequency of the light. And this didn't make a lot of
sense to people at the time because they thought that the
frequency should be related to the number of electrons that
are coming off, because you have more frequency coming in,
you'd expect more electrons that are coming off
-- this wasn't what people were seeing. So, what they decided to do
is just study absolutely everything they could about the
photoelectric effect and hope, at some point, someone
would piece something together that could explain what's going
on or shed some light on this effect. So, one thing they did, because
it was so easy to measure kinetic energy of
electrons, is plot the frequency of the light against
the kinetic energy of the electron that's coming
off here. And in your notes and on these
slides here, just for your reference, I'm just pointing
out what's going to be predicted from classical
physics. You're not responsible for
that and we won't really discuss it, but it just gives
you the contrast of the surprise that comes up when
people make these observations. And the first observation was
that the frequency of the light had a linear relationship
to the kinetic energy of the electrons
that are ejected here. This made no sense at all to
people, and again they saw this effect where if you were
below that threshold frequency, you saw
nothing at all. So, that was frequency
with kinetic energy. The next thing that they wanted
to look at was the actual intensity of the light
and see what the relationship of intensity to kinetic
energy is. So, what we would expect is that
there is a relationship between intensity in kinetic
energy, because it was understood that however intense
the light was, if you had a more intense light,
it was a higher energy light beam. So that should mean that the
energy that's transferred to the electron should be greater,
but that's not what you saw at all, and what you
saw is that if you kept the frequency constant, there was
absolutely no change in the kinetic energy of the electrons,
no matter how high up you had the intensity
of the light go. You could keep increasing the
intensity and nothing was going to happen. So, we could also plot the
number of electrons that are ejected as a relationship to the
intensity, so that was yet another experiment
they could do. And this is what they had
expected that there would be no relationship, but instead
here they saw that there was a linear relationship not to the
intensity and the kinetic energy of the electrons, but
to the intensity and the number of electrons. So, none of these observations
made sense to any scientists at the time, and really all of
these observations were made and somewhat put aside for
several years before someone that could kind of process
everything that was going on at once came along, and that
person was Einstein, conveniently enough -- if anyone
could put it together, we would hope that he
could, and he did. And what he did in a way that
made sense when all of us look at it, is he plotted all of
these different metals on the same graph and made
some observations. So, for example, here we're
showing rubidium and potassium and sodium plotted where we're
plotting the frequency -- that's the frequency of that
light that's coming into the metal versus the kinetic energy
of the electron that's ejected from the surface
of the metal. And what he found here, which is
what you can see and we can all see pretty clearly, is the
slope of all of these lines is the same regardless of what
the type of metal is. So, he fit all these to the
equation of the line, and what he noticed was the slope was
specifically this number, 6.626 times 10 to the negative
34, joules times seconds. And he also found that the y
intercept for each one of these metals was equal to
basically this number here, which was the slope times the
minimum frequency required of each specific metal, so that's
of the threshold frequency. And he actually knew that this
number had popped up before, and a lot of you are familiar
with this number also, and this is Planck's constant. Planck had observed this number
as a fitting constant years earlier when he looked at
some phenomena, and you can read about in your book, such
as black body radiation. And what he found was he needed
this constant to fit his data to what was observed. And this is the same thing that
Einstein was observing, that he needed this fitting
constant, that this constant was just falling right out of,
for example, this slope and also the y intercept. So he decided to go ahead and
define exactly what it is, this line, in terms of these new
constants, this constant he's calling h, which is
Planck's constant. So, on the y axis we have
kinetic energy, so we can plug that in. If we talk about what the x
axis is, that's just the frequency of the light
that's coming in. We know what m is,
m is equal to h. And then we can plug in what b
is, the y intercept, because that's just the negative
of h times that threshold frequency. So we have this new equation
here when we're considering this photoelectric effect,
which is that the kinetic energy is equal to h nu minus
h nu threshold of the metal. And what Einstein concluded
and observed is that well, kinetic energy, of course,
that's an energy term, and h times nu, well that has to be
energy also, because energy has to be equal to energy --
there's no other way about it. And this worked out with units
as well because we're talking about joules for kinetic energy,
and when we're talking about h times nu, we're talking
about joules times second times inverse seconds. So, the very important
conclusion that Einstein made here is that energy is equal to
h times nu, or that h times nu is an actual energy term. And this kind of went along
with two observations. The first is that energy
of a photon is proportional to its frequency. So this was never recognized
before that if we know the frequency of a photon or a wave
of light, we can know the energy of that light. So, since we know that there's
relationship also between frequency and wavelength, we can
do the same thing -- if we know the wavelength, we can know
the energy of the light. And I use the term photon here,
and that's because he also concluded that light must
be made up of these energy packets, and each packet has
that h, that Planck's constant's worth of energy in
it, so that's why you have to multiply Planck's constant
times the frequency. Any frequency can't have an
energy, you have to -- you don't have a continuum of
frequencies that are of a certain energy, it's actually
punctuated into these packets that are called photons. And, as you know, Einstein made
many, many, many very important contributions to
science and relativity, but he called this his one single most
important contribution to science, the relationship
between energy and frequency and the idea of photons. So this means we now have a new
way of thinking about the photoelectric effect, and that
is the idea that h times nu is actually an energy. So, it's the energy of an
incident photon if we're talking about nu where we're
talking about the energy of the photon going in, so we can
abbreviate that as e sub i, energy of the incident photon. We can talk about also h times
nu nought, which is that threshold frequency. So this is a term we're going
to see a lot, especially in your problem sets, it's called
the work function, and the work function is the same
thing as the threshold frequency of a metal, except,
of course, that it's multiplied by Planck's
constant. So, it's the minimum energy that
a certain metal requires in order to pop a photon out of
it -- in order to eject an electron from the surface
of that metal. So this is our new kind of
schematic way that we can think about looking at the
photoelectric effect, so if this is the total amount of
energy that we put into the system, where here we have the
energy of a free electron. We have this much energy going
in, the metal itself requires this much energy, the
work function, in order to eject an electron. So that much energy is
going to be used up just ejecting it. And what we have left over is
this amount of energy here, which is going to be
the kinetic energy of the ejected electron. So, therefore, we can rewrite
our equation in two ways. One is just talking about it in
terms only of energy where our kinetic energy here is going
to be equal to the total energy going in -- the energy
initial minus this energy of the work function here. We can also talk about it in
terms of if we want to solve, if we, for example, we want to
find out what that initial energy was, we can just
rearrange our equation, or we can look at this here where the
initial energy is equal to kinetic energy plus
the work function. So before we go we'll try to
see if we can do a clicker question for you on this,
and we can, very good. So, everyone take those clickers
back out and tell me, if a beam of light with a
certain energy, and we're going to say four electron volts
strikes a gold surface, and here we're saying that the
gold surface has a work function of 5.1 electron volts,
what is the maximum kinetic energy of the electron
that is ejected? So why don't you go ahead and
take ten seconds on that. And if you don't know, that's
okay, just type in an answer and give it your best shot. And let's see what we
come up with here. Alright. So, it looks like some of you
were tricked, but many of you were not, so no electrons
will be ejected. The reason for that is because
this is the minimum amount of energy -- hold off a sec on
the packing up, so in case someone doesn't understand --
this is the minimum amount of energy that's required from the
energy going in in order to eject an electron. So if the incident energy is
less than the energy that's required, absolutely nothing
will happen. That's the same thing we
were talking about with threshold frequency. All right, now you
can pack up and we'll see you on Wednesday.
