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.