Sudden Perturbations

Consider, for example, a constant perturbation that is suddenly switched on at time $t=0$ :

$$H_1(t) = 0 \mbox{\hspace{2.0cm}for $t<0$}$$

$$H_1(t) = H_1 \mbox{\hspace{1.75cm}for $t\geq 0$} ,$$ (801)

where $H_1$ is time-independent, but is generally a function of the position, momentum, and spin operators. Suppose that the system is definitely in state $\vert i\rangle$ at time $t=0$ . According to Equations (795)-(797) (with $t_0=0$ ),

$$c_n^{(0)}(t) = \delta_{i\,n},$$ (802)

$$c_n^{(1)}(t) = -\frac{{\rm i}}{\hbar}\, H_{ni} \int_0^t dt'\,\exp[\,{\rm i}\, ... ...{ni}\,(t'-t)]= \frac{H_{ni}}{E_n - E_i}\, [1- \exp(\,{\rm i}\,\omega_{ni}\,t)],$$ (803)

giving

$$P_{i\rightarrow n}(t) \simeq \vert c_n^{(1)}\vert^{\,2} = \frac{4... ...vert E_n - E_i\vert^{\,2}}\, \sin^2\left[ \frac{(E_n-E_i)\,t}{2\,\hbar}\right],$$ (804)

for $i\neq n$ . The transition probability between states $\vert i\rangle$ and $\vert n\rangle$ can be written

$$P_{i\rightarrow n}(t) = \frac{\vert H_{ni}\vert^{\,2} \,t^2}{\hbar^2} \,{\rm sinc}^2\left[ \frac{(E_n-E_i)\,t}{2\,\hbar}\right],$$ (805)

where

$${\rm sinc}(x)\equiv \frac{\sin x}{x}.$$ (806)

The sinc function is highly oscillatory, and decays like $1/\vert x\vert$ at large $\vert x\vert$ . It is a good approximation to say that ${\rm sinc}(x)$ is small except when $\vert x\vert \stackrel {_{\normalsize <}}{_{\normalsize\sim}}\pi$ . It follows that the transition probability, $P_{i\rightarrow n}$ , is small except when

$$\vert E_n - E_i\vert \stackrel {_{\normalsize <}}{_{\normalsize\sim}}\frac{2\pi\, \hbar}{t}.$$ (807)

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

$${\mit\Delta} E \,{\mit\Delta} t \stackrel {_{\normalsize <}}{_{\normalsize\sim}}h,$$ (808)

where ${\mit\Delta} E = \vert E_n - E_i\vert$ is the change in energy of the system associated with the transition, and ${\mit\Delta} t = t$ 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., $E_n=E_i$ ) is

$$P_{i\rightarrow n} (t) = \frac{\vert H_{in}\vert^{\,2}\,t^2}{\hbar^2},$$ (809)

where use has been made of ${\rm sinc}(0) = 1$ . 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, $t$ to $t+dt$ , grows linearly with $t$ , despite the fact that $H_1$ is constant for $t>0$ . In practice, there is usually a group of final states, all possessing nearly the same energy as the energy of the initial state $\vert i\rangle$ . It is helpful to define the density of states, $\rho(E)$ , where the number of final states lying in the energy range $E$ to $E+dE$ is given by $\rho(E)\,dE$ . Thus, the probability of a transition from the initial state $i$ to any of the continuum of possible final states is

$$P_{i\rightarrow} (t) = \int dE_n\,P_{i\rightarrow n}(t) \,\rho(E_n),$$ (810)

giving

$$P_{i\rightarrow} (t) = \frac{2\, t}{\hbar} \int dx\, \vert H_{ni}\vert^{\,2}\, \rho(E_n) \,{\rm sinc}^2(x),$$ (811)

where

$$x=(E_n-E_i)\,t/2\,\hbar,$$ (812)

and use has been made of Equation (805). We know that in the limit $t\rightarrow \infty$ the function ${\rm sinc}(x)$ is only non-zero in an infinitesimally narrow range of final energies centered on $E_n=E_i$ . It follows that, in this limit, we can take $\rho(E_n)$ and $\vert H_{ni}\vert^{\,2}$ out of the integral in the above formula to obtain

$$P_{i\rightarrow[n]} (t) = \left.\frac{2\pi}{\hbar}\, \overline{\vert H_{ni}\vert^{\,2}} \,\rho(E_n)\,t\, \right\vert _{E_n\simeq E_i},$$ (813)

where $P_{i\rightarrow [n]}$ denotes the transition probability between the initial state $\vert i\rangle$ and all final states $\vert n\rangle$ that have approximately the same energy as the initial state. Here, $\overline{\vert H_{ni}\vert^{\,2}}$ is the average of $\vert H_{ni}\vert^{\,2}$ 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

$$\int_{-\infty}^{\infty} dx\,\,{\rm sinc}^2(x) = \pi.$$ (814)

Note that the transition probability, $P_{i\rightarrow [n]}$ , is now proportional to $t$ , instead of $t^2$ .

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

$$w_{i\rightarrow [n]} = \frac{d P_{i\rightarrow [n]}}{dt},$$ (815)

giving

$$w_{i\rightarrow [n]} = \left.\frac{2\pi}{\hbar}\, \overline{\vert H_{ni}\vert^{\,2}} \,\rho(E_n) \right\vert _{E_n\simeq E_i}.$$ (816)

This appealingly simple result is known as Fermi's golden rule. Note that the transition rate is constant in time (for $t>0$ ): i.e., the probability of a transition occurring in the time interval $t$ to $t+dt$ is independent of $t$ for fixed $dt$ . Fermi's golden rule is sometimes written

$$w_{i\rightarrow n} = \frac{2\pi}{\hbar} \,\vert H_{ni}\vert^{\,2}\, \delta(E_n - E),$$ (817)

where it is understood that this formula must be integrated with $\int dE_n\,\rho(E_n)$ 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

$$c_n^{(2)}(t) = \left(\frac{-{\rm i}}{\hbar}\right)^2 \sum_m H_{nm} H_{mi} \int... ...i}\,\omega_{nm}\,t'\,) \int_0^{t'} \, dt'' \,\exp(\,{\rm i} \,\omega_{mi}\,t\,)$$

$$=\frac{\rm i}{\hbar} \sum_m \frac{H_{nm} \,H_{mi}}{E_m - E_i} \in... ...[\exp(\,{\rm i}\,\omega_{ni}\,t'\,) - \exp(\,{\rm i}\,\omega_{nm}\,t']\,\right)$$

$$= \frac{{\rm i}\,t}{\hbar} \sum_m \frac{H_{nm} H_{mi}}{E_m - E_i}... ...t/2)- \exp(\,{\rm i}\,\omega_{nm} \,t/2) \,{\rm sinc}(\omega_{nm}\,t/2)\right].$$ (818)

Thus,

$$c_n(t) = c_n^{(1)}+ c_n^{(2)} = \frac{{\rm i}\,t}{\hbar} \exp(\,{\rm i}\,\omega_{ni}\,t/2)\, \l... ... \frac{H_{nm}\,H_{mi}}{E_m - E_i}\right)\, {\rm sinc} (\omega_{ni}\,t/2)\right.$$

$$\left. - \sum_m\frac{H_{nm}\,H_{mi}}{E_m - E_i} \exp(\,{\rm i}\,\omega_{im}\,t/2)\,{\rm sinc}(\omega_{nm}\,t/2)\right],$$ (819)

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

$$w_{i\rightarrow [n]} =\left. \frac{2\pi}{\hbar}\, \overline{ \lef... ...\,H_{mi}}{E_m - E_i}\right\vert^{\,2}} \rho(E_n) \right\vert _{E_n \simeq E_i},$$ (820)

where the transition rate is calculated for all final states, $\vert n\rangle$ , with approximately the same energy as the initial state, $\vert i\rangle$ , and for intermediate states, $\vert m\rangle$ whose energies differ from that of the initial state. The fact that $E_m\neq E_i$ 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 $\vert m\rangle$ . Subsequently, the system makes another non-energy-conserving transition to the final state $\vert n\rangle$ . The net transition, from $\vert i\rangle$ to $\vert n\rangle$ , 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 $H_{nm}\,H_{mi}\neq 0$ when $E_m = E_i$ . This problem can be avoided by gradually turning on the perturbation: i.e., $H_1\rightarrow \exp(\eta\,t)\, H_1$ (where $\eta$ is very small). The net result is to change the energy denominator in Equation (820) from $E_i-E_m$ to $E_i - E_m +{\rm i}\,\hbar\,\eta$ .