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.
