Video: Spontaneous and Stimulated Emission

In this video, we will learn how to describe the processes of spontaneous emission of light and stimulated emission of light.

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Video Transcript

In this video, our topic is spontaneous and stimulated emission. These two processes describe ways that photons, packets of light, are created as electrons in atoms move from one energy level to another. We’re going to learn the steps involved in these two processes, as well as how they’re similar and different.

Let’s start off by talking about spontaneous emission. This process begins by having an electron — that’s this orange dot here — at the lowest energy level it can possibly occupy in an atom; here, we’ve called that energy level 𝐸 one. Now, as is true of physical systems in general, this one tends toward the lowest energy state it can maintain. In other words, it’s a very common thing for an electron to be occupying its lowest possible energy level. To change from this state, the electron will need a nudge, some amount of energy input. There are different ways for this to happen. But one common way — and the way that we’ll consider — is having a photon, a bit of light energy, incident on the electron.

Now, the energy of an individual photon is equal to a constant, called Planck’s constant, times the frequency of oscillation of this radiation. Now, interestingly, in order for our electron to be able to absorb this energy at all, the amount of energy the photon possesses must be equal to the energy difference between these two energy levels, 𝐸 one and 𝐸 two. So, only, a photon with a particular frequency, so that that frequency times Planck’s constant is equal to 𝐸 two minus 𝐸 one, can be absorbed by this electron. But then, assuming this does happen, the effect will be for that electron to absorb the photon’s energy and thereby move up to the higher energy level, 𝐸 two.

When an electron has moved out of its ground-energy state to some higher state, we say that it’s in an excited state. And, as we saw, this movement is enabled by the electron absorbing energy from an incoming photon. Well, we mentioned earlier that the most natural state of this system is for the electron to be in its ground energy level, 𝐸 one. And if we just give it enough time, what will happen is this electron will decay — it’s called — back down to that level. The electron doesn’t need any special stimulus for this to happen, but rather it happens, we say, spontaneously.

Now, the time an electron spends in an excited state before decaying back down to its original energy level is typically very short, on the order of 10 to the negative eighth seconds. This time is an approximate time because this process really is a spontaneous one. We can’t predict exactly when an electron will decay back down. But on average, 10 to the negative eight seconds after occupying this excited state, it will.

Now, because the energy of these two levels is different and the electron is moving from a higher to a lower energy state, that means that as it decays back down, it must give off some energy. In fact, that energy needs to be equal to 𝐸 two minus 𝐸 one, the same amount the electron absorbed earlier. The way the electron releases this energy is by giving off a photon. And actually, this photon is the same frequency and therefore the same total energy as the photon that the electron first absorbed. When the electron is returned to its lowest energy level, we say that it has relaxed or entered its relaxed state.

So for this overall process of spontaneous emission, our electron started out at a lower energy level. It then absorbed a photon of energy ℎ times 𝑓, where that energy matched the difference between 𝐸 two and 𝐸 one. And that absorption allowed our electron to move up to this excited state, 𝐸 two. But then, after a typically short amount of time, it spontaneously decayed back to its relaxed state, in the process giving off a photon also of energy ℎ times 𝑓.

Now, let’s compare for a moment these two photons, the one originally incident on our electron and then the one it gave off as it decayed. We can see that these two different photons have the same energy and therefore the same frequency. But we can also say that they’re different in a couple of ways. They move in different directions, and they also have different phases. Notice that this photon starts, we could say, at a peak in the wave, whereas this one that was incident on our electron begins at a point on the wave, where we would say its displacement from equilibrium is zero. So, these two photons are at the same energy and frequency, but different direction, and they also don’t share the same phase.

So, that’s the story for how photons relate in spontaneous emission. Now let’s consider the second process of stimulated emission. In this instance, we start off with the same two energy levels, 𝐸 one and 𝐸 two. But now, our electron is already in its excited state. It may have got there by absorbing a photon or by some other means, but in any case, it’s currently in its excited state. Now, just like this electron, the one we’re considering now is also a candidate for spontaneously decaying down to its lower energy level. We saw that that decay typically happens about 10 billionths of a second after the electron has been excited. But before that decay takes place, it’s possible for this electron to interact with a photon.

Just like before, this can happen if the photon’s energy, ℎ, times its frequency, 𝑓, is equal to the energy difference between 𝐸 two and 𝐸 one. Now, unlike before where our electron absorbed a photon and then was bumped up to a higher energy level, this interaction will actually send our electron back down to its relaxed state. We say that this transition is stimulated by the interaction with this incoming photon. And just like before, as our electron makes this transition, it must release an energy amount equal to 𝐸 two minus 𝐸 one. And it does this by giving off a photon.

That photon has the same energy as the one that stimulated this emission. And it turns out that the original photon and the emitted one are similar in other ways, too. In fact, they’re nearly identical. They have the same energy, frequency, direction, and phase. This similarity is one of the greatest practical differences between photons emitted through stimulated emission and those emitted spontaneously.

Spontaneous emission, for example, is characteristic of light sources like incandescent bulbs, while stimulated emission describes the way that lasers operate. Now that we know a bit about these two processes and how they’re similar, as well as how they’re different, let’s look at an example exercise for practice.

Which of the following is the closest value to the approximate typical lifetime of an excited electron in an atom? (A) 0.1 nanoseconds, (B) 10 nanoseconds, (C) one microsecond, (D) 10 microseconds, (E) 0.1 milliseconds.

In this question, when we talk about the typical lifetime of an excited electron in an atom, we’re imagining different energy levels within this atom. We can call them 𝐸 one and 𝐸 two. And saying that there’s an electron in this higher energy state, therefore, the electron is excited. Now without doing anything to it, eventually, this electron will decay back down to 𝐸 one. And this action is spontaneous; we can’t predict exactly when it will happen. That’s said, there is a fairly typical amount of time that passes before this decay event occurs. That time is approximately 10 to the negative eighth seconds, a very small amount of time.

In this question, we want to pick which one of our answer options is closest to this typical lifetime of an excited electron. To figure this out, let’s look at this number here a bit more closely. An equivalent way to write 10 to the negative eighth seconds is one times 10 to the negative eighth seconds. As it is, this number is written in scientific notation, but like any number written in this form, we can also express this as a decimal number.

To do that, we would start with our leading value, the one, with a decimal point immediately to its right. And then, because we multiply this value by 10 to the negative eighth, we would move this decimal point eight places to the left. So far, we can see we’ve moved three spots. So here’s four, then five, then six, then seven, and then eight. This is where our decimal point ends up and we’ll fill in the blank places with zeros. And so then, we have one, two, three, four, five, six, seven of those zeros. So then, the value one times 10 to the negative eighth seconds written in decimal form is equal to 0.00000001 second.

Now, we do all this because now we can consider the conversions between milliseconds and seconds, microseconds and seconds, and nanoseconds and seconds, respectively. We can recall that 1000 milliseconds or 10 to the third milliseconds equals one second, while it’s 10 to the sixth or a million microseconds that’s equivalent to one second. And 10 to the ninth or a billion nanoseconds is equivalent to one second of time. So, considering this decimal form of our typical lifetime of an excited electron, let’s see what this time is written in units of milliseconds and then microseconds and then nanoseconds.

Considering milliseconds first, we can multiply the value by 1000 to express it in milliseconds. If we do that, we get this result here. Notice that there are three fewer zeroes to the right of our decimal point. But then, considering this result in milliseconds, we see that answer option (E) has a time also in this unit, but it has a value of 0.1 milliseconds. We see that doesn’t agree with the actual typical lifetime expressed in this unit. So, we’ll cross off option (E).

Now, let’s think about this number, this typical lifetime of an excited electron in microseconds, where one million microseconds equals one second. Multiplying our time in seconds by a million microseconds per second, we come up with this result, 0.01 microseconds. But then, looking at answer options (D) and (C), which offer choices in this unit, we can see they also don’t agree with this value we’re calculating. So, we’ll cross those out.

Finally, let’s convert our time to be expressed in units of nanoseconds. To do this, we multiply our original time value in seconds by one billion nanoseconds per second. And when we do that, the decimal point shifts nine spots to the right. And we get this result here, 10 nanoseconds. Looking at our remaining answer choices, we see this agrees with option (B). And so, this is our choice for the closest value to the approximate typical lifetime of an excited electron in an atom.

Let’s look now at a second example exercise.

At an instant 𝑡 zero, a hydrogen atom has just absorbed a photon, increasing the energy of its electron to 𝐸 one. A time interval Δ𝑡 approximately equal to one microsecond then elapses, during which no other photons interact with the atom. How does 𝐸 two, the energy of the electron at a time Δ𝑡 after 𝑡 zero, compare to 𝐸 one? Will any photons have been emitted at a time Δ𝑡 after 𝑡 zero? Which of the following is the term used for the state of the electron at a time Δ𝑡 after 𝑡 zero? (A) Relaxed, (B) stimulated, (C) spontaneous, (D) instantaneous, (E) excited.

Okay, several parts to this question here, and let’s look at them one by one. Now, in this scenario, we start out with a hydrogen atom; that’s an atom that has one single electron. And we’re told that at this particular instant in time, 𝑡 zero, the electron absorbs a photon and increases its energy level. So, let’s say this pink squiggly line is a photon that the electron absorbs. And by so doing, its energy level is bumped up to an energy we can call 𝐸 one. So, what we have at time 𝑡 zero then is an excited electron. And we’re then told that a time interval, we’re calling it Δ𝑡, where this time interval is about equal to one microsecond passes by. And during this interval, there are no other photon interactions with this electron.

The first part of our question asks, how does 𝐸 two, the energy of the electron at a time Δ𝑡 after 𝑡 zero, compare to 𝐸 one? To answer this question, it will be helpful to recall that when an electron is in an elevated energy state, an excited state, even if it doesn’t interact with any other photons, it doesn’t tend to stay at that energy level. Rather, it’s likely to decay — it’s called — down to a lower energy state spontaneously. And the process really is spontaneous; we can’t predict exactly when it will occur.

But nonetheless, a reasonable average lifetime — we could call it — for an electron to be in an excited state before it decays back down is 10 to the negative eighth seconds. This amount of time, by the way, is equal to 10 nanoseconds. So, this electron at energy level 𝐸 one at the instant in time 𝑡 zero has approximately 10 nanoseconds before it will spontaneously decay back down to a lower energy state.

Now, in this first part of our question, we want to know how the energy of the electron after a time of Δ𝑡, where Δ𝑡 recall is equal to about one microsecond, passes. So, here’s the question. If we wait one microsecond after this electron has been excited to energy level 𝐸 one, is it more likely to have remained in that energy level or to have decayed down to a lower energy state? Taking a look at our typical lifetime for an electron to remain in an excited state, we see that that’s 10 nanoseconds, whereas one microsecond is equal to 1000 nanoseconds. So, in other words, if we wait a time amount of Δ𝑡, then that means we’re waiting about 100 times longer than the typical lifetime of an excited electron.

It’s highly likely, then, that one microsecond after the time 𝑡 zero, that our electron will have spontaneously decayed down to a lower energy state. In our question, the energy of the electron at this time is called 𝐸 two. And what we’re saying is it’s highly probable that 𝐸 two will be less than 𝐸 one. And the reason we’re saying that is because we’ve given our excited electron much more time than it typically takes for it to decay to a lower energy state. So, our claim then is that 𝐸 two, the energy of the electron at a time Δ𝑡 after 𝑡 zero, is less than 𝐸 one.

Now, let’s look at the next part of our question, which asks, will any photons have been emitted at a time Δ𝑡 after 𝑡 zero? Now, assuming that over this time interval of Δ𝑡 after 𝑡 zero, our electron really has decayed back down to a lower energy state. We need to ask ourselves, what is the mechanism by which this transition occurs? That is, if the electron started out with a higher energy level, 𝐸 one, and then ended up with a lower energy level, 𝐸 two, where did that energy difference go?

The answer is that in this process of spontaneous emission, the electron emits a photon, a particle of light. That is that means by which it transitions from 𝐸 one to 𝐸 two. So, our answer to the second part of our question is yes, a photon will have been emitted at a time Δ𝑡 after 𝑡 zero.

Now, let’s look at the last part of our question. This asks, which of the following is the term used for the state of the electron at a time Δ𝑡 after 𝑡 zero? Now, looking over these answer options, we can say that option (E) excited describes the energy state of the electron after it has absorbed the photon. That’s the name for its initial energy state. But that’s not the name of the state it ends up in. Recall we’ve said that the electron drops back down to a lower energy level. This process happens spontaneously; that’s option (C). But that term describes the process, but not the state of the electron. Once the electron has dropped back down to energy level 𝐸 two, we say that it has relaxed. This makes sense as an opposite of excited, the name of the electron state after it had absorbed a photon.

Let’s now take a moment to summarize what we’ve learned about spontaneous and stimulated emission. In this lesson, we saw that two processes describing how atoms emit photons are spontaneous and stimulated emission. In spontaneous emission, an electron that’s been excited to a higher energy state, regardless of how that happens, spends an average of 10 to the negative eighth seconds in this state before spontaneously decaying back down to its original energy level. In the process, a photon with energy equal to 𝐸 two minus 𝐸 one is emitted. The two photons we see in this sketch have the same energy but different phase and direction.

In stimulated emission, on the other hand, once again, we start with an electron in an excited state. But then, through interaction with an incident photon, this electron is stimulated to drop down to its relaxed state, in the process emitting a photon with an identical frequency and therefore energy as the one that caused this emission. These photons, we noted, do have the same phase and direction. This is a summary of spontaneous and stimulated emission.

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