Video Transcript
In this video, we’re talking about black body radiation. This is a topic that helped to move our understanding in physics from a classical approach to a quantum one. We’ll see how that happened in a bit, but right now let’s consider just what a black body is.
First off, an ideal black body absorbs all light that’s incident on it. No matter the wavelength of light or its angle of incidence, a perfect black body will absorb it all. And second, an ideal black body also emits light. This emission has to do with only one factor. And that’s the black body’s temperature. So, an ideal black body absorbs all light, that is, all electromagnetic radiation that’s incident on it. And it also emits radiation, where that emission depends only on the temperature of the black body.
Now, there’s no real object that follows these two conditions perfectly. This is, after all, an idealized or perfect black body. But to a good approximation, many objects are like this. For example, it’s possible to build a simple black body. To do this, we can take a box. Say that this is a cutaway view of the box. We paint the inside of the box black. And then, we open up a small hole in the side.
With our box sitting there, let’s imagine that a ray of light enters through the hole we’ve created. This ray will enter the box and then land on one of the sides. Some of the ray will be absorbed and some reflected. And then, the reflected portion will land on another part of the box, again with some being absorbed and some reflected. And this will continue on as the light bounces around on the inside. Each time the ray touches one of the walls, part of the ray’s energy will be absorbed. Until eventually, after enough reflections, effectively, all of the light’s energy will be absorbed by the box.
Since the probability is low that an incident ray of light will enter the box and also leave it before all of its energy is absorbed, we can say that our box is approximately absorbing all light incident on it. Or in other words, effectively, all of the light that goes in the box is absorbed. The energy absorbed by the box will cause it to heat up and begin to radiate on its own. This is the emission aspect of a black body.
When our box begins to emit radiation, only certain kinds of light waves are allowed inside. This has to do with the dimensions of the physical space available to these waves. We can understand this using an analogy. Say that we have a stretchy rope and that both ends of the rope are fixed to walls. It’s possible to move this rope so that standing waves are produced on it. We know, though, that any standing waves on this rope are constrained by the rope’s boundary conditions, the fact that both its ends are fastened in place.
This means that one of the possible standing waves for the rope looks like this, where the rope oscillates back and forth between these two positions. And then, another possible standing wave looks like this, where, again, both ends of the rope are fixed in place. As we add more possible standing waves, we see over and over that the endpoints, the boundary conditions, affect what waves are possible. Well, it works the same way with electromagnetic waves that are created inside our box.
If we imagine light waves traveling in this dimension, then we could have a wave that looks like this or one that looks like this or like this and so on. Just like with our rope fixed between two walls, the boundaries of our cavity determine what electromagnetic waves are possible inside the box. These allowed light waves, we could call them, inside our box are called cavity modes. And each individual standing wave is referred to as a cavity mode. All this means that if we were to put our eye just outside of our black body and look at the light that comes out of it, we would only see certain frequencies of light, the frequencies that agree with the dimensions of our cavity.
Now, we said earlier that a black body emits light according to its temperature. If our box here was at a relatively lower temperature, then we would still only see a cavity mode making its way out to our eye. But that mode would have a lower frequency than it would if the box’s temperature was higher. That is, for a lower-temperature black body, we’d be more likely to observe a wave with a lower frequency, like this wave, than we would to observe emitted radiation like this with a relatively higher frequency.
Note that both of these waves are modes of our cavity. It’s just that higher-frequency emitted waves become more likely as our black body heats up. To get a sense for the way that black body temperature, the emitted radiation intensity, and the emitted radiation wavelength all fit together, we can consider a graph. On this graph, let’s label the horizontal axis 𝜆 for the wavelength of the radiation given off by the black body. And on the vertical axis, let’s show the radiant intensity of this radiation.
Now, if we take a given black body and we keep its temperature fixed at some constant value, then if we consider the intensity of the light that black body gives off versus the wavelength of that light, then the curve will look something like this. Note that there’s radiation given off at essentially every wavelength within this range. This reflects the fact that different cavity modes in a black body can be nearly identical in wavelength. So, the radiation emitted is effectively continuous over the range we’re looking at.
And then, if we were to decrease the temperature of our black body and then measure its curve at a new lower temperature, that curve would look a bit like this one. And if our black body cooled even more, the curve would look a bit like this. We can see that the general trend is that the cooler our black body gets, the more this curve flattens out and the more its peak moves to the right, to higher wavelength.
Now, all this would be fine, except that in the late 19th century, theory predicted that a black body’s radiant-intensity-versus-wavelength curve should look like this. If we compare this predicted green curve with the actual experimental data of the other curves, we see that there’s fairly good agreement as wavelength gets larger and larger. But as the wavelength of the radiation emitted by the black body decreases, physical theory at the time predicted an ever-increasing radiant intensity.
In fact, as 𝜆 approached zero, this theory predicted that the radiant intensity would go to infinity. To see what that would mean physically, let’s say that we have a black body right over here. We color it black and its name is black body largely because a black object absorbs all visible radiation. Note though that an ideal black body will absorb all radiation not just visible light.
Anyway, so we have this black body. And let’s say that it’s in thermal equilibrium with its surroundings. In other words, it’s at a fixed temperature 𝑇. Classical physical theory at the time predicted that this constant-temperature black body would radiate away an infinite amount of energy. But that doesn’t make sense since the temperature of this black body is fixed. It should be constant. Which means that if it radiates away infinite energy, it would need to be absorbing that much too to keep a constant temperature.
And besides all that, the idea that a finite-size object could be absorbing and then emitting away an infinite amount of energy didn’t make sense either. This green curve that was predicted to show how black bodies would behave was based on an assumption that energy is a continuous property. That is, if we imagine an energy maximum and an energy minimum value for some system, the belief at the time was that it was possible for the system to have any amount of energy between these two points. There were no off-limit values, we could say.
And we can see how this approach would make intuitive sense. And yet, this understanding of energy led to the problem that we’re seeing. It predicted that a black body of finite temperature should emit infinite energy. In wrestling with this issue, a scientist named Max Planck came up with an idea. Planck said, what if energy cannot exist continuously, but rather it can only take on certain values within a given system?
It would be a bit like walking up or down a staircase. A person standing on the staircase can be on one stair or on another stair, but they can’t be in between. That was Plank’s idea when it came to the energy of a system. This view that energy only comes in certain discrete sizes — that is, is quantized — became the basis for quantum mechanics. But all that came later. What Planck was really trying to do was to reconcile the theory at the time with the experimental data collected from black body radiation. And when Planck worked out his energy quantization hypothesis and saw what it would mean for these black body curves, he saw that his theoretical predictions assuming energy quantization agreed with the experimental results.
In essence, Planck said this. He said, imagine that we have a system that is oscillating at a frequency 𝑓. Now, this system, as we’ve called it, could be something very simple. It could even be as simple as a single electromagnetic wave. This system, Planck said, can have certain energy values. These values are equal to an integer 𝑛 multiplied by a constant, called ℎ, times the frequency of oscillation of the system. So, given a system oscillating at a certain frequency 𝑓, there are certain of what we could call allowed energy levels for that system. That’s the energy 𝐸 in this equation.
As we said, that’s equal to an integer value. That integer could be zero or one or two and so on multiplied by this constant ℎ that came to be called Planck’s constant times the frequency of the system. If we look at Planck’s constant multiplied by 𝑓, the frequency of our system, this product is equal to what we can call a quantum of energy of this system. That is, it’s the smallest chunk of energy, we could call it, that the system can possess.
Going back to our staircase analogy, we can think of ℎ times 𝑓 as the energy difference between each consecutive step. As we step onto the first step, we gain ℎ times 𝑓 amount of energy. And then, as we step to the second, we gain that same amount more and so forth all the way up to the top of the stair. The point is, regardless of which step we’re standing on, our total energy will always be some integer multiple of this basic amount, ℎ times 𝑓.
Now, it’s important to point out that when Planck generated this equation, he was thinking specifically of electromagnetic radiation. And moreover, the amount of electromagnetic radiation that possesses a quantum of energy came to be called a photon. This was handy because a photon can be characterized by a single specific frequency. Let’s take a look now at how this equation applies to photons for photons that have two very different frequencies.
Let’s say that we have one photon here with a frequency we’ll call simply 𝑓. And then, say we have a second photon with a frequency three times as great. Now, if we were to plot the energy possessed by each one of these two different photons. Then, using Planck’s equation for photon energy, since we have one photon of each type, in both cases, 𝑛 would be equal to one. And so, the total photon energy is just Planck’s constant, ℎ, multiplied by the photon frequency.
So then, our photon with frequency 𝑓 would have an energy level at this point. And the one with frequency three 𝑓 would have three times as much energy. On this plot, we’re seeing the quantum of energy for each one of these photons. For the lower frequency photon, that amount is ℎ times 𝑓. And for the higher frequency one, it’s three times that.
Now, what if we were to add photons to our system? Let’s say that instead of having one of the photons with frequency 𝑓, we now have three. In that case, on our graph, we would have one and then two and then three of these energy quanta added up. And so then, the energy of all three lower-frequency photons added together would equal the energy of the single higher-frequency one. And if we added one more lower-frequency photon, then we would add one more energy unit to its amount.
We can see then that though, as Planck claimed, energy is quantized — it can only take on certain values — not all energy quanta are created equal. Photon energy quanta are larger or smaller depending on the photon’s frequency.
Now, getting back to our black body radiation curve, the idea that energy was quantized rather than continuous was only part of the solution. Because imagine again that we have a black body and that it absorbs a very high-energy photon. Say that it’s a gamma ray. In other words, the wavelength of this radiation is incredibly small. That would mean the frequency of this photon is very large. And therefore, its quantum of energy is relatively large too.
Now, if our black body absorbed a very high-energy photon and then right away emitted a photon with the same amount of high energy. We wouldn’t have solved the problem that according to our prediction as wavelength gets smaller and smaller, the emitted energy gets larger and larger without bound.
But here’s how this discrepancy was finally explained. For a black body at any temperature, it is more probable for energy to be emitted by a large number of lower-energy photons than by a single high-energy photon. That is, even if a black body absorbs a lot of energy at once through a high-energy photon landing on it, probabilistically, it will not then emit a similar high-energy photon. Instead, it’s much more likely for the black body to emit many low-energy photons whose energy adds up to the total energy it has absorbed from the one high-energy photon.
Going back to our graph of photon energy versus frequency, a black body that absorbed a photon with this much energy would be more likely to then emit one, two, three photons with lesser energy that adds up to the total it received. This explains why it is that our black body radiation curves go to zero as the wavelength goes to zero. Even if a black body absorbed a very high-energy single photon, it would be unlikely to release that energy through the emission of a high-energy photon. Rather, it would probably do it by emitting many lower-energy photons. And just as this end of the curve goes to zero, the other end, where wavelength gets larger and larger, goes to zero too.
We can understand this by considering that wavelength and wave energy are inversely related. As our black body emits cavity modes of larger and larger wavelength, the energy of those waves gets smaller and smaller. So, even though a black body does emit very long-wavelength radiation, the energy of that radiation becomes negligibly small at very large wavelengths. With that understanding of this plot, let’s know summarize what we’ve learned about black body radiation.
Getting started, we saw that an ideal black body absorbs all incident radiation, and it also emits light based on the black body’s temperature. We then saw that a black body made from a cavity, like our box, emits radiation in what are called cavity modes. These are wave forms that are permitted based on the cavity’s geometry.
And lastly, we learned that Planck’s idea that energy is quantized. Telling us that the energy of a single photon is equal to the frequency of that photon times Planck’s constant. And that the energy of a collection of identical photons is equal to ℎ times their frequency times the number of photons there. This notion helped us understand black body radiation and reconcile the difference between the predictions of theory and experimental data.
