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.
