Question Video: Analyzing the Excitation of the Hydrogen Atom Physics

At an instant 𝑡₀, a hydrogen atom has just absorbed a photon, increasing the energy of its electron to 𝐸₁. A time interval Δ𝑡 ≃ 1 𝜇s then elapses, during which no other photons interact with the atom. How does 𝐸₂, the energy of the electron at a time Δ𝑡 after 𝑡₀, compare to 𝐸₁? Will any photons have been emitted at a time Δ𝑡 after 𝑡₀? Which of the following is the term used for the state of the electron at a time Δ𝑡 after 𝑡₀? [A] Relaxed [B] Stimulated [C] Spontaneous [D] Instantaneous [E] Excited

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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.