# Question Video: Finding Which Diagram More Correctly Shows Changes in an Atom Physics

The diagram shows an atom that interacts with two identical photons in a time interval longer than the lifetime of the excited state of an electron in the atom. Which of the following more correctly shows the changes that have occurred just after the interaction between the atom and the second photon? [A] Diagram A [B] Diagram B [C] Diagram C [D] Diagram D

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### Video Transcript

The diagram below shows an atom that interacts with two identical photons in a time interval longer than the lifetime of the excited state of an electron in the atom. Which of the following more correctly shows the changes that have occurred just after the interaction between the atom and the second photon?

If we look at the two times in this diagram, 𝑡 zero and 𝑡 one, they appear to be exactly the same, except for what number photon is interacting with the atom. At 𝑡 zero, it is photon one that is being absorbed by the ground state electron. And at 𝑡 one, it is photon two that is being absorbed by the ground state electron. At both 𝑡 zero and 𝑡 one, this electron is in the lowest energy level possible or the ground state. This may seem odd, since we know that when a photon of sufficient energy is absorbed by an electron, the electron moves up to a higher energy level. And we say that it is in an excited state.

So why don’t we see an excited state at 𝑡 one after we know that the electron has absorbed this photon? Well, it has to do with the first sentence of this question, which tells us that the atom interacts with two identical photons in a time interval longer than the lifetime of the excited state of an electron in the atom. This means that at 𝑡 zero, the photon is absorbed by the electron and transitions up to a higher energy level, becoming excited. But we never get to see this electron at 𝑡 one because the time interval between 𝑡 zero and 𝑡 one is longer than the lifetime of the excited state of an electron in the atom, which means that at some point between 𝑡 zero and 𝑡 one, the excited electron must be spontaneously decaying back down to the ground state.

Electrons spontaneously decay from excited states because electrons tend towards the lowest energy most stable positions. When electrons decay transitioning down to a lower energy level, they release a photon in a process called spontaneous emission, which is that an excited electron releases a photon when it decays spontaneously. But because of this time interval between 𝑡 zero and 𝑡 one, we actually missed the entire spontaneous emission process. By the time we look at 𝑡 one, the electron has already decayed back down to the ground state. So it looks like nothing has happened. It is not unrealistic to have trouble observing spontaneous emission since excited electrons decayed very fast, typically on time scales of 10 to the power of negative eight seconds.

With all of this in mind, let’s look at what the question is asking us. Which of the diagrams more correctly shows the changes that have occurred just after the interaction between the atom and the second photon? We have to be careful with the wording that we’re given here. All of these diagrams show that they occur at a time interval of 𝑡 two. But the time difference between 𝑡 one and 𝑡 two is not the same as the time difference between 𝑡 zero and 𝑡 one. 𝑡 zero to 𝑡 one is longer than the lifetime of the excited state of an electron in the atom, whereas 𝑡 two occurs just after the interaction between the atom and the second photon.

What this is telling us is that there will not be a spontaneous emission between 𝑡 one and 𝑡 two, because we’re looking at the step that occurs immediately after the interaction in 𝑡 one, which is for a photon being absorbed by an electron in excitation of the electron to a higher energy level. So what we should expect to see just after the interaction between the atom and second photon is that electron in an excited state, which happens to be exactly what we see in answer (D), an excited electron that has yet to decay.

Well, for answer (A), we see that there are two photons traveling away from the atom. And there are two reasons why this is wrong. The first is, of course, that it has the wrong timing. In order for there to be two photons, one of them has to be emitted from the atom after 𝑡 one. But 𝑡 two occurs just after the interaction. So there wouldn’t be enough time for an emission. And the second reason why this is wrong is that we see the photons traveling in the same direction. In spontaneous emission, the emitted photons have the same energy as the photons that caused the initial energy level transition. But they should have a different direction from the initial photons, which is not what we’re seeing here with these two perfectly lined-up photons. So for these reasons, answer (A) is not it.

But now, let’s look at (B), which has two photons going off in different directions, which means that we’re seeing a correct display of spontaneous emission, though just like with (A) because we’re able to see this emission occurring, it has the wrong timing. However, if the time intervals between 𝑡 zero, 𝑡 one, and 𝑡 two were all the same, which is to say longer than the lifetime of the excited state of an electron, then (B) would’ve been the correct answer.

But because we see the emission, it is not, which brings us to answer (C), which we still know is incorrect due to seeing this emitted photon over here, telling us once again that we have the wrong timing. But we can also say that it has the wrong number of photons since we only see one. If this time interval is occurring after the lifetime of an excited state of an electron, then we should expect to see two photons: one of them from photon one and the other one from photon two. Or perhaps, we’re seeing only the photon that was emitted because of photon one.

But if this were the case, then photon two should cause an excitation of this electron in 𝑡 two. But because it’s in the ground state, the time interval must have been after emission. Or possibly, we’re just not seeing the emission that photon one caused. Whatever the case is, we know that answer (C) cannot be correct because of the wrong timing.

Now looking back at answer (D), shouldn’t we expect to see an emitted photon, since we know that one must have been emitted between 𝑡 zero and 𝑡 one? But this question is asking us for the diagram that more correctly shows the changes that have occurred. So we should be expecting to see an electron in the excited state. But instead, we see that for answers (A), (B), and (C), the electron is in the ground state.

So even though no emitted photon is shown, the diagram that more correctly shows the changes that have occurred just after the interaction between the atom and the second photon is diagram (D).