$$\require{cancel}$$

# 8.6: Sudden Perturbations

Consider, for example, a constant perturbation that is suddenly switched on at time :

where is time-independent, but is generally a function of the position, momentum, and spin operators. Suppose that the system is definitely in state at time . According to Equations (795)-(797) (with ),

 (802) (803)

giving

 (804)

for . The transition probability between states and can be written

 (805)

where

 (806)

The sinc function is highly oscillatory, and decays like at large . It is a good approximation to say that is small except when . It follows that the transition probability, , is small except when

 (807)

Note that in the limit only those transitions that conserve energy (i.e., ) have an appreciable probability of occurrence. At finite , is is possible to have transitions which do not exactly conserve energy, provided that

 (808)

where is the change in energy of the system associated with the transition, and is the time elapsed since the perturbation was switched on. This result is just a manifestation of the well-known uncertainty relation for energy and time. Incidentally, the energy-time uncertainty relation is fundamentally different to the position-momentum uncertainty relation, because (in non-relativistic quantum mechanics) position and momentum are operators, whereas time is merely a parameter.

The probability of a transition that conserves energy (i.e., ) is

 (809)

where use has been made of . Note that this probability grows quadratically with time. This result is somewhat surprising, because it implies that the probability of a transition occurring in a fixed time interval, to , grows linearly with , despite the fact that is constant for . In practice, there is usually a group of final states, all possessing nearly the same energy as the energy of the initial state . It is helpful to define the density of states, , where the number of final states lying in the energy range to is given by . Thus, the probability of a transition from the initial state to any of the continuum of possible final states is

 (810)

giving

 (811)

where

 (812)

and use has been made of Equation (805). We know that in the limit the function is only non-zero in an infinitesimally narrow range of final energies centered on . It follows that, in this limit, we can take and out of the integral in the above formula to obtain

 (813)

where denotes the transition probability between the initial state and all final states that have approximately the same energy as the initial state. Here, is the average of over all final states with approximately the same energy as the initial state. In deriving the above formula, we have made use of the result

 (814)

Note that the transition probability, , is now proportional to , instead of .

It is convenient to define the transition rate, which is simply the transition probability per unit time. Thus,

 (815)

giving

 (816)

This appealingly simple result is known as Fermi's golden rule. Note that the transition rate is constant in time (for ): i.e., the probability of a transition occurring in the time interval to is independent of for fixed . Fermi's golden rule is sometimes written

 (817)

where it is understood that this formula must be integrated with to obtain the actual transition rate.

Let us now calculate the second-order term in the Dyson series, using the constant perturbation (801). From Equation (797) we find that

 (818)

Thus,

where use has been made of Equation (803). It follows, by analogy with the previous analysis, that

 (820)

where the transition rate is calculated for all final states, , with approximately the same energy as the initial state, , and for intermediate states, whose energies differ from that of the initial state. The fact that causes the last term on the right-hand side of Equation (819) to average to zero (due to the oscillatory phase-factor) during the evaluation of the transition probability.

According to Equation (820), a second-order transition takes place in two steps. First, the system makes a non-energy-conserving transition to some intermediate state . Subsequently, the system makes another non-energy-conserving transition to the final state . The net transition, from to , conserves energy. The non-energy-conserving transitions are generally termed virtual transitions, whereas the energy conserving first-order transition is termed a real transition. The above formula clearly breaks down if when . This problem can be avoided by gradually turning on the perturbation: i.e., (where is very small). The net result is to change the energy denominator in Equation (820) from to .