7.6 Asymmetric Two-State Systems

Two-state systems are quantum systems for which just two states $\psi_1$ and $\psi_2$ are relevant. If the two states have different expec­tation energy, or if the Hamiltonian depends on time, the two-state system is asymmetric. Such systems must be considered to fix the problems in the description of sponta­neous emission that turned up in the previous section.

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

$$\Psi = c_1 \psi_1 + c_2 \psi_2$$ (7.33)

where $\vert c_1\vert^2$ and $\vert c_2\vert^2$ are the proba­bilities that the system is in state $\psi_1$, respectively $\psi_2$.

The coefficients $c_1$ and $c_2$ evolve in time according to

$$\fbox{$\displaystyle {\rm i}\hbar \dot c_1 = \langle{E_1... ...\hbar \dot C_2 = H_{21} c_1 + \langle{E}_2\rangle c_2 $} %$$ (7.34)

where

$$\langle{E}_1\rangle = \langle\psi_1\vert H\psi_1\rangle, \... ...uad \langle{E}_2\rangle = \langle\psi_2\vert H\psi_2\rangle$$

with $H$ the Hamiltonian. The Hamiltonian coefficients $\langle{E}_1\rangle$ and $\langle{E}_2\rangle$ are the expec­tation energies of states $\psi_1$ and $\psi_2$. The Hamiltonian coefficients $H_{12}$ and $H_{21}$ are complex conjugates. Either one is often referred to as the “matrix element.” To derive the above evolution equations, plug the two-state wave function $\Psi$ into the Schrödinger equation and take inner products with $\langle\psi_1\vert$ and $\langle\psi_2\vert$, using ortho­normality of the states.

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 $\psi_1$. That means that initially $\vert c_1\vert^2$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1 and $\vert c_2\vert^2$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0. Then in the symmetric case discussed in the previous section, the system oscillates between the two states. But that requires that the states have the same expec­tation energy.

This section addresses the asymmetric case, in which there is a non­zero difference $E_{21}$ between the two expec­tation energies:

$$\fbox{$\displaystyle E_{21} \equiv \langle{E}_2\rangle - \langle{E}_1\rangle $} %$$ (7.35)

In the asymmetric case, the system never gets into state $\psi_2$ completely. There is always some proba­bility for state $\psi_1$ left. That can be seen from energy conserv­ation: the expec­tation value of energy must stay the same during the evolution, and it would not if the system went fully into state 2. However, the system will periodi­cally return fully to the state $\psi_1$. That is all that will be said about the exact solution here.

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 $\psi_1$ and stays close to it.

That assumption results in the following proba­bility for the system to be in the state $\psi_2$, {D.38}:

$$\fbox{$\displaystyle \vert c_2\vert^2 \approx \left(\fra... ...2 \frac{\sin^2(E_{21}t/2\hbar)}{(E_{21}t/2\hbar)^2} $} %$$ (7.36)

For this expression to be a valid approxi­mation, the parenthetical ratio must be small. Note that the final factor shows the effect of the asymmetry of the two state system; $E_{21}$ is the difference in expec­tation energy between the states. For a symmetric two-state system, the final factor would be 1, (using l’Hôpital).

Key Points

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

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If the system is initially in the state $\psi_1$, it will never fully get into the state $\psi_2$.

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If the system is initially in the state $\psi_1$ and remains close to it, then the proba­bility of the state $\psi_2$ 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 $\psi_1$ is the excited atomic state. The state $\psi_2$ is the atomic state of lowered energy plus the emitted photon. These states seem states of definite energy, but if they really were, there would not be any decay. Energy states are stationary. There is a slight uncertainty in energy in the states.

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 decay processes, a bit of energy slop $E_{21}$ must be allowed between the initial and final expec­tation values of energy.

In practical terms, that means that the energy of the emitted photon can vary a bit. So its frequency can vary a bit.

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 $\psi_2$, each with its own coefficient $c_2$. The square magnitude of that coefficient gives the proba­bility that the system can be found in that state $\psi_2$. And in the approxi­mation of time-dependent perturb­ation theory, the coefficients $c_2$ do not inter­act; the square magnitude of each is given by (7.36). The total proba­bility that the system can be found in some decayed state at a time $t_{\rm {c}}$ is then

$$P_{1\to{\rm all\ }2} = \sum_{\rm all\ states\ 2} \left(\... ...in^2(E_{21}t_{\rm {c}}/2\hbar)}{(E_{21}t_{\rm {c}}/2\hbar)^2}$$

The time $t_{\rm {c}}$ will again model the time between “collisions,” inter­actions with the surroundings that “measure” whether the atom has decayed or not. The decay rate, the number of transitions per unit time, is found from dividing by the time:

$$\lambda = \sum_{\rm all\ states\ 2} \frac{\vert H_{21}\v... ...in^2(E_{21}t_{\rm {c}}/2\hbar)}{(E_{21}t_{\rm {c}}/2\hbar)^2}$$

The final factor in the sum for the decay rate depends on the energy slop $E_{21}$. This factor is plotted graphi­cally in figure 7.7. Notice that only a limited range around the point of zero slop contributes much to the decay rate. The spikes in the figure are intended to quali­tatively indicate the discrete photon frequencies that are possible in the box that the atom is in. If the box is extremely big, then these spikes will be extremely close together.

Figure 7.7: Energy slop diagram.

Now suppose that you plot the energy slop diagram against the actual photon energy instead of the scaled energy slop $E_{21}t_{\rm {c}}$$\raisebox{.5pt}{$/$}$$2\hbar$. Then the center of the diagram will be at the nominal energy of the emitted photon and $E_{21}$ will be the deviation from that nominal energy. The spike at the center then represents the transition of atoms where the photon comes out with exactly the nominal energy. And those surrounding spikes whose height is not negligible represent slightly different photon energies that have a reasonable proba­bility of being observed. So the energy slop diagram, plotted against photon energy, graphi­cally represents the uncertainty in energy of the final state that will be observed.

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.

Figure 7.8: Schematized energy slop diagram.

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 $\pi\hbar$$\raisebox{.5pt}{$/$}$$t_{\rm {c}}$, and otherwise it is one. And it integrates to the same value as the original function. So, if the spikes are very closely spaced, they still sum to the same value as before. To be sure, if the square matrix element $\vert H_{21}\vert^2$ varied non­linearly over the typical width of the diagram, the transition rate would now sum to something else. But it should not; if the variation in photon energy is negligible, then so should the one in the matrix element be.

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 $\pi\hbar$$\raisebox{.5pt}{$/$}$$t_{\rm {c}}$ different from the initial expec­tation energy. And inside this summation range, the final factor can be dropped because it is now 1. That gives:

$$\lambda = \sum_{\scriptstyle \strut {\rm all\ states\ 2\... ...rm {c}}} \frac{\vert H_{21}\vert^2}{\hbar^2} t_{\rm {c}} %$$ (7.37)

This can be cleaned up further, assuming that $H_{21}$ is constant and can be taken out of the sum:

$$\fbox{$\displaystyle \lambda = 2 \pi \frac{\vert H_{21}\... ...}{\hbar} \frac{{\rm d}N}{{\rm d}\langle E_2\rangle} $} %$$ (7.38)

This formula is known as “Fermi’s golden rule.” The final factor is the number of photon states per unit energy range. It is to be evaluated at the nominal photon energy. The formula simply observes that the number of terms in the sum is the number of photon states per unit energy range times the energy range. The equation is considered to originate from Dirac, but Fermi is the one who named it “golden rule number two.”

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 $t_{\rm {c}}$. Atoms can have specific values for their decay rates despite the minute details of their surroundings. Shorter collision times do produce less transitions per unit time for a given state. But they also allow more slop in energy, so the number of states that achieve a significant amount of transitions per unit time goes up. The net effect is that the decay rate stays the same, though the uncertainty in energy goes up.

The other problem remains; the evaluation of the matrix element $H_{21}$ requires relativistic quantum mechanics. But it is not hard to guess the general ideas. When the size of the periodic box that holds the system increases, the electro­magnetic field of the photons decreases; they have the same energy in a larger volume. That results in smaller values for the matrix element $H_{21}$. On the other hand, the number of photons per unit energy range ${\rm d}{N}$$\raisebox{.5pt}{$/$}$${\rm d}\langle{E}_2\rangle$ increases, chapter 6.3. The net result will be that the decay rate remains finite when the box becomes infinite.

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 $j$ that is not zero. In that case, there are $2j$ + 1 atomic states that differ only in magnetic quantum number. The magnetic quantum number describes the component of the angular momentum in the chosen $z$-direction. Now if the atom is in empty space, the direction of the $z$-axis should not make a difference. Then these $2j$ + 1 states will have the same energy. So you cannot include one and not the other. If this happens to the initial atomic state, you will need to average the decay rates over the magnetic states. The physical reason is that if you have a large number $I$ of excited atoms in the given energy state, their magnetic quantum numbers will be randomly distributed. So the average decay rate of the total sample is the average over the initial magnetic quantum numbers. But if it happens to the final state, you have to sum over the final magnetic quantum numbers. Each final magnetic quantum number gives an initial excited atom one more state that it can decay to. The general rule is:

Sum over the final atomic states, then average over the initial atomic states.

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

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 $H_{21}t_{\rm {c}}$$\raisebox{.5pt}{$/$}$$\hbar$ is small is automati­cally satisfied, because the matrix element $H_{21}$ goes to zero for infinite box size. But then the number of states 2 goes to infinity. And if you look a bit closer at the analysis, {D.38}, the requirement is really that there is little proba­bility of any transition in time inter­val $t_{\rm {c}}$. So the time between collisions must be small compared to the lifetime of the state. With typical lifetimes in the range of nano­seconds, atomic collisions are typically a few orders of magnitude more rapid. However, that depends on the relative vacuum.

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 $t_{\rm {c}}$ is large compared to the inverse of the photon frequency. For emission of visible light, that means that the collision time must be large when expressed in femtoseconds. Collisions between atoms will easily meet that requirement.

The width of the energy slop diagram figure 7.7 should give the observed variation $E_{21}$ in the energy of the final state. The diagram shows that roughly

$$E_{21} t_{\rm {c}} \sim \pi \hbar$$

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,” $t_{\rm {c}}$. But you could assume that this time is comparable to the mean lifetime $\tau$ of the state. Essentially that supposes that inter­actions with the surroundings are infrequent enough that the atomic evolution can evolve undisturbed for about the typical decay time. But that nature will definitely commit itself whether or not a decay has occurred as soon as there is a fairly reasonable proba­bility that a photon has been emitted.

That argument then leads to the definition of the typical uncertainty in energy, or “width,”of a state as $\Gamma$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\hbar$$\raisebox{.5pt}{$/$}$$\tau$, as mentioned in section 7.4.1. In addition, if there are frequent inter­actions between the atom and its surroundings, the shorter collision time $t_{\rm {c}}$ should be expected to increase the uncertainty in energy to more than the width.

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

$$\frac{1}{1+(E_{21}/\Gamma)^2} %$$ (7.39)

That is commonly called a [Cauchy] “Lorentz[ian] profile” or distribution or function, or a “Breit-Wigner distribution.” Hey, don’t blame the messenger. In any case, it still has the same inverse quadratic decay for large energy slop as the diagram figure 7.7. That means that if you start computing the standard deviation in energy, you end up with infinity. That would be a real problem for versions of the energy-time relationship like the one of Mandelshtam and Tamm. Such versions take the uncertainty in energy to be the standard deviation in energy. But it is no problem for the all-powerful energy-time uncertainty equality (7.9), because the standard deviation in energy is not needed.

Key Points

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Some energy slop occurs in decays.

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