Sudden Perturbations

where is independent of time, 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 (8.58) and (8.59) (with ),

(8.62) | ||

(8.63) |

giving

(8.64) |

for . The transition probability between states and can be written

where

(8.66) |

The sinc function is highly oscillatory, and decays like at large . In fact, it is a good approximation to say that is small compared to unity except when . It follows that the transition probability, , is negligibly small unless

(8.67) |

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 that do not exactly conserve energy, provided that

(8.68) |

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 a manifestation of the well-known uncertainty relation for energy and time [53]. Incidentally, the energy-time uncertainty relation is fundamentally different from 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

(8.69) |

where use has been made of . Note that this probability grows quadratically in 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 one of the continuum of possible final states is

(8.70) |

giving

(8.71) |

where

(8.72) |

and use has been made of Equation (8.68). 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 previous formula to obtain

(8.73) |

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 previous formula, we have made use of the result [59]

(8.74) |

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,

(8.75) |

which implies that

This appealingly simple result is known as

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

Let us now calculate the second-order term in the Dyson series, using the constant perturbation (8.64). From Equation (8.60), we find that (with )

(8.78) | ||

Thus,

where use has been made of Equation (8.66). It follows, by analogy with the previous analysis, that

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 final term on the right-hand side of Equation (8.82) to average to zero during the evaluation of the transition probability. (See Exercise 4.)

According to Equation (8.83), 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 state
to state
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 previous formula clearly breaks down
if
when
. This problem can be avoided by
gradually turning on the perturbation: that is,
(where
is very small). The net result is to change the energy
denominator in Equation (8.83) from
to
.