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 0.1 nanoseconds 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? (A) 𝐸 two is greater than 𝐸 one. (B) 𝐸 two is less than 𝐸 one. (C) 𝐸 two is equal to 𝐸 one.
We’ll look at the other parts of this question later, but for now let’s focus on the one about energy. Whenever we talk about the energy of an electron that is still within an atom, we should immediately think of the different electron energy levels, because this is where electrons exist when within an atom. This means that 𝐸 one and 𝐸 two must be referring to specific electron energy levels. But let’s be careful with our notation.
We may be tempted to say that 𝐸 one is the first energy level and 𝐸 two is the second energy level. But this isn’t the case here. We are told that the electron has successfully absorbed a photon and increased its energy, meaning it cannot be in the ground state. We are not told exactly what electron energy level it’s in. We just know it can’t be the first one. 𝐸 one is referring to the electron’s energy right after successfully absorbing a photon, not to the first energy level. In the diagram we’ve drawn, we actually have 𝐸 one represented as the second energy level. Depending on the energy of the photon that the electron absorbed, it could actually be the third or fourth energy levels. But to save on space, we’ll just continue to show it as the second energy level.
Now, 𝐸 two refers to the energy of the electron at a time Δ𝑡 after 𝑡 zero. And we know that during this time Δ𝑡, there are no other photons that interact with the atom. This means no photons that can cause a stimulated emission or be absorbed by the electron. If there is an energy difference between 𝐸 one and 𝐸 two, the only thing that can cause it is a spontaneous decay down to a lower energy level, meaning that the electron that was at a higher energy level in 𝐸 one should be at a lower energy level for 𝐸 two, right?
Well, this is what will happen eventually. But the time interval we’re given of Δ𝑡 is about 0.1 nanoseconds. And the typical electron lifetime of an electron in a higher energy level when there is a space available in a lower energy level is on the scale of about 10 to the power of negative eight seconds. When expressed in scientific notation, 0.1 nanoseconds is equal to one times 10 to the power of negative 10 seconds, which means that the time interval Δ𝑡 is shorter than the typical electron lifetime of an electron in a higher energy level by a factor of about 100.
Practically, this means that the time Δ𝑡 is so short that there wouldn’t be enough time for the electron to decay at all. So this means that the electron at time 𝑡 zero and a time Δ𝑡 after 𝑡 zero should be in the same energy level, which means the electron has the same energy. And thus the energy 𝐸 one is equal to the energy 𝐸 two. So the correct answer for the first part of this question is (C). 𝐸 two is equal to 𝐸 one.
Now, let’s look at the second part.
Will any photons have been emitted at a time Δ𝑡 after 𝑡 zero? (A) Yes, (B) no.
For this part, let’s remember that there are no photons that interact with the atom besides the initial one that was absorbed at time 𝑡 zero. And we know that the time Δ𝑡 is shorter than the typical electron lifetime of an electron in a higher energy level when there is a lower energy level spot available. These two facts mean that there can’t be photon emission since stimulated emission requires a photon for the higher energy level electron to interact with. And spontaneous emission requires the higher energy level electron to spontaneously decay to a lower energy level. But as we already saw, it doesn’t have enough time to do so in the time interval Δ𝑡. So neither stimulated nor spontaneous emission can occur in the time interval Δ𝑡 after 𝑡 zero, which means no photons will have been emitted. The correct answer for the second part of this question is (B) no.
Now the last part of the question.
Which of the following is the term used for the state of the electron at a time Δ𝑡 after 𝑡 zero? (A) Relaxed, (B) excited, (C) stimulated, (D) spontaneous, (E) instantaneous.
Let’s work backwards, starting with (E). An electron cannot be described as instantaneous. While we say that the transition between the different electron energy levels is instantaneous, the electron itself isn’t. And also even though its lifetime in a higher energy level when there is a lower energy level spot available is very small, only being on the order of 10 to the power of negative eight seconds, it is still not instantaneous. (E) is not correct.
Looking now at (C) and (D), neither of these can describe the state of an electron either. While an electron can spontaneously decay to a lower energy level and produce a photon through spontaneous emission, we cannot say that the electron itself is spontaneous. And similarly, while an electron can become stimulated by interacting with a photon, the stimulation is an act that happens to it. It is not an inherent state or change in state. Even though stimulated emission causes a change in state in the electron by making it decay to a lower energy level, it is not an actual state to say that it is stimulated.
Now, let’s look at (B) excited. Excited is a valid state for an electron. And we say that electrons are excited when they’re in a higher energy level while there are still lower energy level spaces available. These excited electrons will eventually decay back down to a lower energy level on a scale of 10 to the power of negative eight seconds. So they can’t stay in that state for long. As for the specific electron in this question, we know that at time 𝑡 zero, the electron has just successfully absorbed a photon. So it must be in a higher energy level, which is to say it is excited. And as we’ve already seen, at a time interval Δ𝑡 after 𝑡 zero, the electron does not actually have enough time to decay back down to a lower energy level. It has the same energy, which means it must be at the same energy level, which means that it still must be excited. So we know that (B) is it. But just to wrap everything up, let’s look at (A).
We can say that an electron is in a relaxed state when it is in the lowest energy level available to it, which is to say just not excited, which is not what’s happening at a time Δ𝑡 after 𝑡 zero. The best way to describe the state of the electron then is (B) excited.
