8.6: Sudden Perturbations
( \newcommand{\kernel}{\mathrm{null}\,}\)
Consider, for example, a constant perturbation that is suddenly switched on at time t=0 :
H1(t) =0 \( \mbox{\hspace{2.0cm}for \) H1(t) =H1\( \mbox{\hspace{1.75cm}for \), ???where H1 is time-independent, but is generally a function of the position, momentum, and spin operators. Suppose that the system is definitely in state |i⟩ at time t=0 . According to Equations ???-??? (with t0=0 ),
c???n(t) =δin, ??? c???n(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)],$](http://farside.ph.utexas.edu/teaching/qm/lectures/img1893.png)
giving
![$ P_{i\rightarrow n}(t) \simeq \vert c_n^{\ref{1}}\vert^{\,2} = \frac{4...
...vert E_n - E_i\vert^{\,2}}\, \sin^2\left[ \frac{(E_n-E_i)\,t}{2\,\hbar}\right],$](http://farside.ph.utexas.edu/teaching/qm/lectures/img1894.png)
for i≠n . The transition probability between states |i⟩ and |n⟩ can be written
Pi→n(t)=|Hni|2t2ℏ2sinc2[(En−Ei)t2ℏ], ???where
sinc(x)≡sinxx. ???The sinc function is highly oscillatory, and decays like 1/|x| at large |x| . It is a good approximation to say that sinc(x) is small except when |x|<∼π . It follows that the transition probability, Pi→n , is small except when
|En−Ei|<∼2πℏt. ???Note that in the limit t→∞ only those transitions that conserve energy (i.e., En=Ei ) have an appreciable probability of occurrence. At finite t , is is possible to have transitions which do not exactly conserve energy, provided that
ΔEΔt<∼h, ???where ΔE=|En−Ei| is the change in energy of the system associated with the transition, and Δt=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., En=Ei ) is
Pi→n(t)=|Hin|2t2ℏ2, ???where use has been made of sinc???=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 H1 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 |i⟩ . It is helpful to define the density of states, ρ(E) , where the number of final states lying in the energy range E to E+dE is given by ρ(E)dE . Thus, the probability of a transition from the initial state i to any of the continuum of possible final states is
Pi→(t)=∫dEnPi→n(t)ρ(En), ???giving
Pi→(t)=2tℏ∫dx|Hni|2ρ(En)sinc2(x), ???where
x=(En−Ei)t/2ℏ, ???and use has been made of Equation ???. We know that in the limit t→∞ the function sinc(x) is only non-zero in an infinitesimally narrow range of final energies centered on En=Ei . It follows that, in this limit, we can take ρ(En) and |Hni|2 out of the integral in the above formula to obtain
Pi→[n](t)=2πℏ¯|Hni|2ρ(En)t|En≃Ei, ???where Pi→[n] denotes the transition probability between the initial state |i⟩ and all final states |n⟩ that have approximately the same energy as the initial state. Here, ¯|Hni|2 is the average of |Hni|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
∫∞−∞dxsinc2(x)=π. ???Note that the transition probability, Pi→[n] , is now proportional to t , instead of t2 .
It is convenient to define the transition rate, which is simply the transition probability per unit time. Thus,
wi→[n]=dPi→[n]dt, ???giving
wi→[n]=2πℏ¯|Hni|2ρ(En)|En≃Ei. ???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
wi→n=2πℏ|Hni|2δ(En−E), ???where it is understood that this formula must be integrated with ∫dEnρ(En) to obtain the actual transition rate.
Let us now calculate the second-order term in the Dyson series, using the constant perturbation ???. From Equation ??? we find that
c???n(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)$](http://farside.ph.utexas.edu/teaching/qm/lectures/img1931.png)
![$ = \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].$](http://farside.ph.utexas.edu/teaching/qm/lectures/img1932.png)
Thus,
cn(t)=c???n+c???n
where use has been made of Equation ???. It follows, by analogy with the previous analysis, that
![$ 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},$](http://farside.ph.utexas.edu/teaching/qm/lectures/img1936.png)
where the transition rate is calculated for all final states, |n⟩ , with approximately the same energy as the initial state, |i⟩ , and for intermediate states, |m⟩ whose energies differ from that of the initial state. The fact that Em≠Ei causes the last term on the right-hand side of Equation ??? to average to zero (due to the oscillatory phase-factor) during the evaluation of the transition probability.
According to Equation ???, a second-order transition takes place in two steps. First, the system makes a non-energy-conserving transition to some intermediate state |m⟩ . Subsequently, the system makes another non-energy-conserving transition to the final state |n⟩ . The net transition, from |i⟩ to |n⟩ , 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 HnmHmi≠0 when Em=Ei . This problem can be avoided by gradually turning on the perturbation: i.e., (where η is very small). The net result is to change the energy denominator in Equation ??? from Ei−Em to
.
Contributors
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)