### Video Transcript

In this video, we’re talking about the
quantization of electromagnetic radiation.

This is one of the foundational ideas of
modern physics, and its discovery in the early 1900s helped to explain a few puzzling
phenomena. As we consider how electromagnetic
radiation is quantized, we can remind ourselves that electromagnetic radiation, also known
as light, consists of oscillating electric and magnetic fields that propagate or move in a
certain direction. These oscillating fields give
electromagnetic- radiation-wave-like properties.

Among other things, this means that
electromagnetic radiation has a frequency — we can call it 𝑓 — associated with it. This frequency is a measure of how many
wavelengths — we can refer to those using the letter 𝜆 — pass a given point in space in one
second of time. Now, not only does electromagnetic
radiation have an associated frequency and wavelength, but it also transports energy from
one location to another as the wave moves along. This means that if we were to put, for
example, a light-sensitive measurement device right here, then we could measure how much
energy from electromagnetic radiation from light is incident on the plate.

And let’s say we did that. Let’s say we shined a light all with
wavelength 𝜆 on this plate and that we were able to plot the total energy the plate
received against that wavelength. For that wavelength, what we’ve called
𝜆, the graph might look something like this. After shining light of this particular
wavelength onto our photosensitive plate, over some amount of time, the plate has received
this much energy from the incoming light.

Now, in a classical physics model — and
this was generally the model that was used up until the 20th century — this amount of energy
received by the plate could be literally anything. If we wanted to make it slightly more
like this, we could do that by increasing the intensity of the incoming light on this
plate. Or if we wanted to make it slightly less
down at this level, we could do that as well by decreasing the incoming light intensity. And the point was that, in general, this
energy value recorded by our plate could be anything. This is another way of saying that at
that time it was believed that energy was continuous. And, really, this idea makes a lot of
sense as we consider experience from our everyday life.

For example, if a person is walking along
a path, then within the bounds of their upper and lower speed limits, it seems that they
could be moving along with any speed. If they were initially moving along, say,
at one meter per second, then there’s nothing that stops them from slightly increasing their
speed to, say, 1.01 meters per second or from there to, say, 1.0125 meters per second. The point is, if this person has a
maximum speed, we’ll call it 𝑣 sub 𝑚. Then, intuitively, it seems to make sense
that this person could have any speed from zero up into this maximum, that all of the speed
values between these end points are possible.

And beyond a person walking, we can think
similarly about other types of motion. And motion of objects we know is related
to energy. All this to say, the idea that the energy
that our absorbing plate could receive from incoming light is continuous, that is, could
take on any value, seems to agree with our everyday observations. The only problem was as scientists
studied data from experiments with electromagnetic radiation, they saw that their results
did not agree with this idea, that the energy of electromagnetic radiation can take on any
value. So, there was a big mismatch between what
theory said, that energy could have any value and it’s continuous, and what experimental
measurements showed.

At this point, a German physicist by the
name of Max Planck came on to the scene. Planck saw that a new theory was needed
to explain the recently gathered experimental data. And so, he came up with this idea. The energy 𝐸 of electromagnetic
radiation, Planck said, is equal to the frequency 𝑓 of that radiation multiplied by a
constant, we call it ℎ, more on that later, all multiplied by an integer value that we can
call 𝑛. In this equation, capital 𝐸 represents
the total electromagnetic energy involved. 𝑓, as we saw, is the frequency of the
electromagnetic radiation. ℎ is a constant value that came to be
called Planck’s constant. And 𝑛 is an integer value, one or two or
three and so on.

If we come back to our graph of energy
against wavelength over here, here’s what Planck was saying about the total energy
measured. He was saying that it’s not all one big
amount, but rather this total is made up of the accumulation of lots of small bits of
energy. And each one of these bits individually
has the same energy amount. Each one is Planck’s constant ℎ times the
frequency of the wave 𝑓. And at this point, we can recall that, in
general, the frequency of a wave is equal to the speed of the wave 𝑣 divided by its
wavelength 𝜆. So, for this frequency 𝑓, because we’re
working with an electromagnetic wave, light, we know that its speed is equal to the speed of
light, 𝑐. And its wavelength 𝜆 is simply equal to
the wavelength value we have here, the wavelength of our incoming wave.

So, in essence, Planck said that this
chunk of energy here is equal to ℎ times 𝑐 divided by 𝜆. And so is this little chunk right here
and this little chunk right here and so on. So that if we count the total number of
these little chunks, we have one, two, three, four, five, six, seven, eight, nine of
them. Then the total energy being measured here
— we can call it 𝐸 sub 𝑡 — is equal to nine times ℎ times 𝑐 divided by 𝜆.

What Planck was saying then was that for
light of a given wavelength 𝜆, the energy of that light couldn’t be just any value, but
rather it had to be some integer multiple of ℎ times 𝑐 divided by the wavelength of the
light. This is what it means for energy to be
quantized.

Now, this idea of quantization, strange
as it may seem, is something we have experience with. Say that a person is going to climb a
flight of steps. And we’ll say further that all of the
steps have the same height. They’re all at distance 𝑑 tall. Now, we know that as this person climbs
up the steps, they can either be on the ground level, like they are now, or up on the first
step, like this, or the second step or the third and so on. And if they’re always on one step or the
other, then we can say that as they climb the stairs. They’re either a distance 𝑑 or a
distance of two 𝑑 or three 𝑑 or four 𝑑 above ground level. Four 𝑑 would be their height when
they’re on the top step. We could say then that this person’s
position above ground is quantized, that there’s always some integer multiple of the
distance 𝑑 above ground.

Planck’s theory encourages us to think
about energy this way. And part of the understanding involved
was that electromagnetic radiation doesn’t just come as a wave, but it also exists as a
packet or a particle. That is, a discreet or an incremental
amount. The name for this smallest possible chunk
of electromagnetic radiation is photon. And this term helps us to better
understand these results over here.

Understanding that light can exist in
individual packets called photons, we would say that an individual photon of light with
wavelength 𝜆 has energy ℎ times 𝑐 divided by 𝜆. And that the total energy 𝐸 sub 𝑡 that
we’ve measured with our plate is equal to the energy of nine such photons. Knowing this, let’s come back over here
and write a simplified version of this energy equation.

We’ll write an equation that’s specific
for individual photons. This just involves simplifying our
current equation a bit because now our integer 𝑛 is equal to one. That’s what it means to be talking about
the energy of one single photon. So, if 𝑛 is one, then we can drop it out
of our equation. But now, we’re no longer talking about a
total of electromagnetic energy, but rather the energy of a single photon. So, let’s call it 𝐸 sub 𝑝.

So, the energy of an individual photon
with frequency 𝑓 is equal to that frequency multiplied by this constant called Planck’s
constant. Planck’s constant, by the way, is an
incredibly small value. To a good approximation, it’s equal to
6.63 times 10 to the negative 34th joule seconds. Recall that the standard unit of
frequency, the hertz, can be expressed as an inverse second. Which means that when we multiply a
frequency in units of one over seconds times Planck’s constant in units of joules times
seconds, the units of seconds cancel out and we’re left with joules, the SI base unit of
energy.

So, while this works out nicely, just as
a side note, notice that the smallness of Planck’s constant helps us understand why it would
be hard to see that energy is not continuous. If these individual energy increments are
very, very small and they are based on Planck’s constant. Then, without very precise ways of
measuring energy amounts, it would be very hard to tell that this energy only comes in
incremental amounts.

In any case, this equation here is a
useful equation for the energy of an individual photon based on that photon’s frequency. But we’ve seen that there is also another
way to write this. That’s because the frequency 𝑓 can be
written as the speed of the wave, the speed of light, divided by the wavelength 𝜆. So, the energy of an individual photon is
equal to the frequency of that photon times Planck’s constant, which is also equal to ℎ
times 𝑐 over 𝜆. Whether we’re working with frequency or
wavelength, these are equivalent ways of calculating photon energy.

Now, one last clarifying comment before
we get to an example exercise. We’ve said that, based on Planck’s
theory, a currently accepted physical theory, electromagnetic radiation comes in discrete
amounts, and therefore the energy of such radiation is quantized. It’s helpful to see though that not all
the chunks of energy associated with light are the same size.

To see this, let’s imagine that we have a
photon of red light. Light like this has a wavelength of
approximately 700 nanometers, and we’ll call this wavelength 𝜆 sub one. Considering our equation describing the
energy of such a photon, if we call that energy amount 𝐸 one, we can see that it’s equal to
ℎ times 𝑐 divided by 𝜆 sub one. But now, let’s say we have a photon not
of red light, but of blue light. And this has a wavelength we’ll call 𝜆
sub two. The associated energy of this photon is
equal to ℎ times 𝑐 divided by 𝜆 sub two.

So, we can see that the energy of a
photon of one wavelength is not the same as the energy of a photon of another
wavelength. And recall that the energy of an
individual photon is what we could call the step size of energy for light of that
wavelength. So, if we had, say, five of these red
photons with wavelength 𝜆 sub one, then the total energy of those five would be equal to
five times the energy quanta we could call it of a single red photon. Here, then, we have five of these quanta
stacked on top of one another to give our total amount of energy for these red photons.

But then, let’s say that we have five
blue photons. The energy quanta for that we could call
it is ℎ times 𝑐 divided by 𝜆 sub two. And this is a greater amount than ℎ times
𝑐 divided by 𝜆 sub one. This means that if we plotted out these
individual packets of energy on our graph, they would all have the same size because they
all correspond to photons with the same wavelength. But we can see that that amount is
greater than the corresponding amount for red photons. All this to say that not all quantized
amounts of energy are the same. The amount depends on the wavelength or,
correspondingly, the frequency of the electromagnetic radiation we’re considering.

Okay, having said all this, let’s now
consider an example exercise.

A laser emits four times 10 to the 20th
photons, each with a frequency of six times 10 to the 14th hertz. What is the total energy radiated by the
laser? Use a value of 6.63 times 10 to the
negative 34th joules seconds for the Planck constant. Give your answer in joules to three
significant figures.

Okay, so, in this exercise, we have a
laser. Let’s say that this is our laser. And we’re told that, over some amount of
time, this laser emits four times 10 to the 20th photons. This, of course, is a huge number over a
1,000,000,000 billions of photons. And we’re told that each of these photons
has a frequency — we’ll call that 𝑓 — of six times 10 to the 14th hertz. Based on this, we want to calculate the
total energy radiated by the laser.

The way we can think of doing this is by
calculating first the energy of one of these four times 10 to the 20th photons and then
multiplying that amount by the total number of photons we have. So, let’s start by calculating the energy
of one of these photons. We can recall that the energy of an
individual photon — we can call it 𝐸 sub 𝑝 — is equal to a constant value, known as
Planck’s constant, times the frequency of that photon. And we’re told that we can use a value of
6.63 times 10 to the negative 34th joule seconds for that constant.

So then, the energy of one of the photons
emitted by our laser is equal to Planck’s constant multiplied by the frequency of that
photon. And so, the total amount of energy
radiated by the laser — we can call it 𝐸 — is equal to the total number of photons, four
times 10 to the 20th, times the energy of a single photon. Before we multiply these three numbers
together, let’s take a quick look at the units involved.

In Planck’s constant, we have joules
times seconds. And in frequency, we have a unit of
hertz. We know that our hertz though indicates a
number of cycles completed per second, which is equivalent to the unit of one over
seconds. Written this way, we can see that the
unit of seconds in our Planck’s constant will cancel out with the units of one over seconds
in our photon frequency.

This means we’ll be left simply with
units of joules when we calculate this figure. And that’s perfect because we’re
calculating an energy. When we compute this product to three
significant figures, the result is 159 joules. That’s the total amount of energy
radiated by the laser.

Let’s summarize now what we’ve learned
about the quantization of electromagnetic radiation. Starting out, we saw that electromagnetic
radiation, also called light, has a frequency and wavelength. And this radiation also transmits energy
as it travels. Further, we learned that light energy
comes in increments. And the amount of light possessing a
quantum of this energy is called a photon. We learned that this amount of energy,
the energy possessed by a single photon, is equal to a constant ℎ called Planck’s constant,
named after its developer, Max Planck, multiplied by the frequency of the photon 𝑓.

Or equivalently, because wave frequency
is equal to wave speed divided by wavelength, the energy of a photon moving at the speed of
light is equal to Planck’s constant times 𝑐 divided by the wavelength 𝜆. Lastly, we learned that while light is
quantized, not all the energy increments have the same size. The size of the energy increment depends
on the light’s frequency or equivalently its wavelength.