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.
But what about these other
answers?
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).
