Video Transcript
The diagram below shows an atom containing an electron that was initially in the ground state. The atom interacted with two identical photons. The atom is shown just after these interactions were completed. Did the interaction of the atom and the photons occur in a time interval longer than the lifetime of the excited state for the electron in the atom?
To determine this, let’s look at the entire process that the atom goes through. The atom’s electron was initially in the ground state. It then interacted with two identical photons, with this diagram showing the atom just after these photon interactions were completed. Now, let’s consider the atom before any photon interactions have occurred. The electron is in the ground state, or the lowest energy level possible.
When a photon interacts with an electron, if the photon has the right energy, it can be absorbed by the electron, causing it to transition to a higher energy level or become excited. While the electron is in this higher energy level, we say that it is in an excited state. And while in this excited state, there are one of two things that can happen: a spontaneous or stimulated emission. And which one occurs depends on the time interval of the interactions in the atom.
To see why this is, let’s go through both of these step by step, starting with spontaneous emission. Spontaneous emission must begin by having an electron in the excited state, which after some time will undergo spontaneous decay. This happens because excited electrons have a limited lifetime, usually on the order of about 10 to the power of negative eight seconds long. If this period elapses without interruption, then the excited electron will transition down back to a lower energy level, releasing a photon in the process.
After all of this happens, the electron is back in the ground state, which means if a second identical photon comes in, the electron will be ready to absorb it again. Which is to say, in order for this process to occur, the time interval for the interactions between the photons would have to be longer than the lifetime of the excited state for the electron because we would have to wait for the lifetime of the excited electron to decay on its own.
So if the photon interactions cause spontaneous emission, then the answer would be yes for whether the time interval is longer than the lifetime of the excited state for the electron in the atom. So, that’s spontaneous emission. Let’s now look at stimulated emission.
Just like spontaneous, stimulated starts with an electron in the excited state. However, before this electron has time to spontaneously decay, it is stimulated by a photon, in this case the second identical photon. This electron stimulation has to occur before the electron has time to decay back down to the ground state, which means that the time interval of the interaction has to be shorter than the lifetime of the excited state for the electron in the atom, which means for a stimulated emission, no, the time interval is not longer than the lifetime of the excited state of the electron.
Now then, after the stimulation, the excited electron transitions down to a lower energy level and releases a photon in the process. So, that’s stimulated emission. And now that we know the two different emission types and thus the two different processes that could be occurring in this diagram, let’s consider what information we are given. We don’t see any photons in the original diagram, only a single electron in the excited state.
In order for this electron to be excited after all of the photon interactions, it must have absorbed a photon, which can only occur if the emission type that took place here was spontaneous, since the last photon interaction that occurs is the second identical photon coming into an electron that is in the ground state, absorbing it and becoming excited. This is in contrast to the stimulated emission, with the first photon causing the excited state and the second photon stimulating the excited electron, causing the electron to come back down to the ground state.
So, if it was a stimulated emission that occurred, we should expect the electron to be in the ground state after all photon interaction. But here we see that it is excited, which means it must have been spontaneous emission. And the only way that spontaneous emission could have occurred is if yes, the interaction of the atom and the photons does occur in a time interval longer than the lifetime of the excited state for the electron in the atom.
