Video Transcript
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.
