7.6 Asymmetric Two-State Systems

Two-state systems are quantum systems for which just two states

The wave function of a two state system is of the form

(7.33) |

The coefficients

with

matrix element.To derive the above evolution equations, plug the two-state wave function

It will be assumed that the Hamiltonian is independent of time. In that case the evolution equations can be solved analytically. To do so, the analysis of chapter 5.3 can be used to find the energy eigenstates and then the solution is given by the Schrödinger equation, section 7.1.2. However, the final solution is messy. The discussion here will restrict itself to some general observations about it.

It will be assumed that the solution starts out in the state

This section addresses the asymmetric case, in which there is a
nonzero difference

The remainder of this section will use an approximation called
“time-dependent perturbation theory.” It assumes that the
system stays close to a given state. In particular, it will be
assumed that the system starts out in state

That assumption results in the following probability for the system to
be in the state

Key Points

- If the states in a two-state system have different expectation energies, the system is asymmetric.

- If the system is initially in the state
, it will never fully get into the state.

- If the system is initially in the state
and remains close to it, then the probability of the state is given by (7.36)

7.6.1 Spontaneous emission revisited

Decay of excited atomic or nuclear states was addressed in the previous section using symmetric two-state systems. But there were some issues. They can now be addressed.

The example is again an excited atomic state that transitions to a
lower energy state by emitting a photon. The state

Since there is, clearly it does not make much sense to say that the initial and final expectation energies must be the same exactly.

In practical terms, that means that the energy of the emitted photon can vary a bit. So its frequency can vary a bit.In decay processes, a bit of energy slopmust be allowed between the initial and final expectation values of energy.

Now in infinite space, the possible photon frequencies are infinitely close together. So you are now suddenly dealing with not just one possible decay process, but infinitely many. That would require messy, poorly justified mathematics full of so-called delta functions.

Instead, in this subsection it will be assumed that the atom is not in infinite space, but in a very large periodic box, chapter 6.17. The decay rate in infinite space can then be found by taking the limit that the box size becomes infinite. The advantage of a finite box is that the photon frequencies, and so the corresponding energies, are discrete. So you can sum over them rather than integrate.

Each possible photon state corresponds to a different final state

The time

collisions,interactions with the surroundings that

measurewhether the atom has decayed or not. The decay rate, the number of transitions per unit time, is found from dividing by the time:

The final factor in the sum for the decay rate depends on the energy
slop

Now suppose that you plot the energy slop diagram against the actual
photon energy instead of the scaled energy slop

Normally, the observed uncertainty in energy is very small in physical terms. The energy of the emitted photon is almost exactly the nominal one; that allows spectral analysis to identify atoms so well. So the entire diagram figure 7.7 is extremely narrow horizontally when plotted against the photon energy.

That suggests that you can simplify things by replacing the energy
slop diagram by the schematized one of figure 7.8. This
diagram is zero if the energy slop is greater than

Using the schematized energy slop diagram, you only need to sum over
the states whose spikes are equal to 1. That are the states 2 whose
expectation energy is no more than

This can be cleaned up further, assuming that

Actually, the original sum (7.37) may be easier to handle
in practice since the number of photon states per unit energy range is
not needed. But Fermi’s rule is important because it shows that
the big problem of the previous section with decays has been resolved.
The decay rate does no longer depend on the time between collisions

The other problem remains; the evaluation of the matrix element

That is verified by the relativistic analysis in addendum {A.24}. That addendum completes the analysis in this section by computing the matrix element using relativistic quantum mechanics. Using a description in terms of photon states of definite linear momentum, the matrix element is inversely proportional to the volume of the box, but the density of states is directly proportional to it. (It is somewhat different using a description in terms of photon states of definite angular momentum, {A.25}. But the idea remains the same.)

One problem of section 7.5.3 that has now disappeared is
the photon being reabsorbed again. For each individual transition
process, the interaction is too weak to produce a finite reversal
time. But quantum measurement

remains required to
explain the experiments. The time-dependent perturbation theory used
does not apply if the quantum system is allowed to evolve undisturbed
over a time long enough for a significant transition probability (to
any state) to evolve, {D.38}. That would affect the
specific decay rate. If you are merely interested in the average
emission and absorption of a large number of atoms, it is not a big
problem. Then you can substitute a classical description in terms of
random collisions for the quantum measurement process. That will be
done in derivation {D.41}. But to describe what
happens to individual atoms one at a time, while still explaining the
observed statistics of many of such individual atoms, is another
matter.

So far it has been assumed that there is only one atomic initial state
of interest and only one final state. However, either state might
have a net angular momentum quantum number

The averaging over the initial states is typically trivial. Without a preferred direction, the decay rate will not depend on the initial orientation.Sum over the final atomic states, then average over the initial atomic states.

It is interesting to examine the limitations of the analysis in this
subsection. First, time-dependent perturbation theory has to be
valid. It might seem that the requirement of (7.36) that

Second, the energy slop diagram figure 7.7 has to be narrow
on the scale of the photon energy. It can be seen that this is true
if the time between collisions

The width of the energy slop diagram figure 7.7 should give
the observed variation

Note that this takes the form of the all-powerful energy-time uncertainty equality (7.9). To be sure, the equality above involves the artificial time between collisions, or

measurements,

That argument then leads to the definition of the typical uncertainty
in energy, or width,

of a state as

Note that the wavy nature of the energy slop diagram figure
7.7 is due to the assumption that the time between
collisions

is always the same. If you start averaging
over a more physical random set of collision times, the waves will
smooth out. The actual energy slop diagram as usually given is of the
form

Key Points

- Some energy slop occurs in decays.

- Taking that into account, meaningful decay rates may be computed following Fermi’s golden rule.