# Question Video: Determining the Lifetime of an Excited Electron Using Emission Types Physics

The diagram shows an atom containing an electron that was initially in the ground state and two photons emitted from the atom. The atom interacted with two identical photons before it emitted the photons shown. The atom is shown just after these interactions were completed. Did the interactions occur within a time interval longer than the lifetime of the excited state for the electron in the atom?

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### Video Transcript

The diagram below shows an atom containing an electron that was initially in the ground state and two photons emitted from the atom. The atom interacted with two identical photons before it emitted the photons shown. The atom is shown just after these interactions were completed. Did the interactions occur within a time interval longer than the lifetime of the excited state for the electron in the atom?

Before we tackle this problem, let’s make sure we understand what the question is describing.

To start, we have an atom that contains one electron. We’re told that the atom interacts with two photons and then emits two photons. The diagram shows the atom just after those photons were emitted. To answer this question, we need to work out whether the interactions described in the question took a longer time than the lifetime of the excited state for the electron. There’s a lot to break down here. So let’s separately go through each of the interactions that the question describes.

To illustrate each step of the process described in the question, we’ll clear some space and draw an example diagram of the atom that we can refer to. The diagram shows an atom with two electron energy levels. There is the ground state, which corresponds to the innermost orbital path, and there is also an excited state, which corresponds to the outer orbital path. We are told that the electron is initially in the ground state. We also know that the atom interacted with two identical photons.

Recall that if an incoming photon has the correct amount of energy, the photon will interact with an electron. This interaction can cause the electron to transition to a different energy state. But in order for this to happen, the energy of the incoming photon must be equal to the difference in energy between the electron’s initial and final state, as given by the equation 𝐸 p equals 𝐸 e minus 𝐸 g, where 𝐸 p is the photon’s energy, the energy of the ground state is 𝐸 g, and the energy of the excited state is 𝐸 e. When a photon does interact with an electron, the electron will undergo an upward or downward energy level transition, depending on the electron’s initial energy state. As we discuss this question, we will go over both possibilities.

Let’s first consider what happened when the atom interacted with the first of the incoming photons. The photon interacted with the electron, which as we know was initially in the ground state. The photon was absorbed by the electron, and so the photon’s energy was transferred to the electron. We saw earlier that in order for a photon to interact with an electron, the photon’s energy must equal the difference between two electron energy levels. Thus, when the photon was absorbed, the electron transitioned from the ground state to the excited state.

Next, let’s consider what happened when the atom interacted with the second incoming photon. When the second incoming photon reached the atom, it interacted with the electron in the excited state. Remember that the question told us the two incident photons were identical. So this second photon had the same energy as the first one.

However, this second interaction was different from the first interaction we just discussed, as in the second interaction, the electron could no longer transition to a higher energy state. Rather, the incoming photon caused the electron to move from the excited state back down to the ground state. When this happened, the electron emitted another photon. This emitted photon had to make up for the difference between the two energy levels. Thus, this emitted photon was identical to the incoming photons that the electron absorbed. This process is called stimulated emission. When the second incoming photon interacted with the electron, the photon stimulated the electron and caused it to move down to the ground state, causing a photon to be emitted in turn.

Now, the incoming photon that stimulated the emission was not changed or absorbed during the process. It just carried on traveling through the atom. This means that two identical photons must have left the atom: the second of the incoming photons and the photon that the electron emitted in its downward transition. These are the two photons shown in the diagram.

We’ve now detailed all the interactions discussed in the question. So let’s return to what the question is asking of us. This question is asking us whether this whole process took a longer time than the lifetime of the excited state for the electron in the atom. By lifetime, we mean the time it takes for an electron to spontaneously transition from the excited state to the ground state.

It’s worth noting that while many people believe spontaneous to mean happening randomly, in physics, spontaneous more accurately refers to a process that happens naturally and without external energy input. On average, an electron will spend 10 to the negative eight seconds in an excited state before it spontaneously decays. When this does happen, the electron must emit a photon to make up for the downward energy transition. This is known as spontaneous emission.

So, in this question, we’ve been given an atom that interacts with two photons such that the electron undergoes stimulated, rather than spontaneous, emission. We are asked whether this process takes longer than the lifetime of an electron that undergoes spontaneous emission. In other words, did this stimulated emission take more time than a spontaneous emission would have taken?

Well, if we think about it, the answer has to be no. If the interactions described in this question had taken longer than about 10 to the negative eight seconds, then the electron would’ve spontaneously decayed by the time the stimulated emission could’ve ever occurred. So this stimulated emission must have occurred within a time interval that was shorter than the lifetime of the excited state, or else it would never have happened at all.

So the answer to this question is no. The interactions did not occur within a time interval longer than the lifetime of the excited state for the electron in the atom.