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Next: Harmonic Perturbations Up: Time-Dependent Perturbation Theory Previous: Sudden Perturbations

Energy-Shifts and Decay-Widths

We have examined how a state $ \vert f\rangle$ , other than the initial state $ \vert i\rangle$ , becomes populated as a result of some time-dependent perturbation applied to the system. Let us now consider how the initial state becomes depopulated.

In this case, it is convenient to gradually turn on the perturbation from zero at $ t=-\infty$ . Thus,

$\displaystyle H_1(t) = \exp(\eta\,t)\,H_1,$ (8.81)

where $ \eta$ is small and positive, and $ H_1$ is a constant.

In the remote past, $ t\rightarrow -\infty$ , the system is assumed to be in the initial state $ \vert i\rangle$ . Thus, $ c_i(t\rightarrow-\infty) =1$ , and $ c_{f\neq i}(t\rightarrow -\infty) = 0$ . We wish to calculate the time evolution of the coefficient $ c_i(t)$ . First, however, let us check that our previous Fermi golden rule result still applies when the perturbing potential is turned on slowly, instead of very suddenly. For $ c_{f \neq i}(t)$ , it follows from Equations (8.58) and (8.59) that (with $ t_0=-\infty$ )

$\displaystyle c_f^{\,(0)}(t)$ $\displaystyle =0,$ (8.82)
$\displaystyle c_f^{\,(1)}(t)$ $\displaystyle = - \frac{\rm i}{\hbar}\, H_{fi} \int_{-\infty}^t dt'\, \exp[(\et...
..., \frac{ \exp[(\eta + {\rm i}\,\omega_{fi} )\, t]}{\eta +{\rm i}\,\omega_{fi}},$ (8.83)

where $ H_{fi} = \langle f\vert\,H_1\,\vert i\rangle$ . Thus, to first order, the transition probability from state $ \vert i\rangle$ to state $ \vert f\rangle$ is

$\displaystyle P_{i\rightarrow f}(t) = \left\vert c_f^{\,(1)}(t)\right\vert^{\,2...
...^{\,2}}{\hbar^{\,2}} \frac{\exp(2\, \eta\, t)}{\eta^{\,2} + \omega_{fi}^{\,2}}.$ (8.84)

The transition rate is given by

$\displaystyle w_{i\rightarrow f}(t) = \frac{dP_{i\rightarrow f}}{dt} = \frac{2 ...
...}}{\hbar^{\,2}} \frac{\eta \exp(2 \,\eta \,t)}{\eta^{\,2} + \omega_{fi}^{\,2}}.$ (8.85)

Consider the limit $ \eta\rightarrow 0$ . In this limit, $ \exp(\eta\, t)\rightarrow 1$ , but [91]

$\displaystyle \lim_{\eta\rightarrow 0} \frac{\eta}{\eta^{\,2}+ \omega_{fi}^{\,2}} =\pi\,\delta(\omega_{fi}) = \pi\,\hbar \,\delta(E_f - E_i).$ (8.86)

(See Exercise 19.) Thus, Equation (8.88) yields the standard Fermi golden rule result

$\displaystyle w_{i\rightarrow f} = \frac{2\pi}{\hbar} \,\vert H_{fi}\vert^{\,2} \,\delta(E_f - E_i).$ (8.87)

It is clear that the delta-function in the previous formula actually represents a function that is highly peaked when its argument is close to zero. The width of the peak is determined by how fast the perturbation is switched on.

Let us now calculate $ c_i(t)$ using Equations (8.58)-(8.60). We have

$\displaystyle c_i^{(0)} (t)$ $\displaystyle =1,$ (8.88)
$\displaystyle c_i^{(1)} (t)$ $\displaystyle = \left(\frac{-{\rm i}}{\hbar}\right)H_{ii}\, \int_{-\infty}^t \exp(\eta \,t')\,dt'= -\frac{\rm i}{\hbar}\,H_{ii} \,\frac{\exp( \eta\, t)}{\eta},$ (8.89)
$\displaystyle c_i^{(2)} (t)$ $\displaystyle = \left(\frac{-{\rm i}}{\hbar} \right)^2 \sum_f \vert H_{fi}\vert...
...xp[(\eta+ {\rm i}\,\omega_{if})\, t'] \exp[ (\eta+ {\rm i}\,\omega_{fi})\, t'']$    
  $\displaystyle = \left(\frac{-{\rm i}}{\hbar} \right)^2 \sum_f \vert H_{fi}\vert^{\,2} \frac{\exp(2 \,\eta\, t)}{2\,\eta\,(\eta + {\rm i}\,\omega_{fi})}.$ (8.90)

Thus, to second order, we get

$\displaystyle c_i(t)$ $\displaystyle \simeq 1 + \left(\frac{-{\rm i}}{\hbar}\right) H_{ii}\, \frac{\exp(\eta \,t)}{\eta}$    
  $\displaystyle \phantom{=}+ \left(\frac{-{\rm i}}{\hbar} \right)^2 \vert H_{ii}\...
...i}\vert^{\,2} \exp(2\,\eta\, t)} {2\,\eta\,(E_i -E_f + {\rm i}\,\hbar \,\eta)}.$ (8.91)

Let us now consider the ratio $ \skew{3}\dot{c}_i/c_i$ , where $ \skew{3}\dot{c}_i \equiv
d c_i/dt$ . Using Equation (8.94), we can evaluate this ratio in the limit $ \eta\rightarrow 0$ . We obtain

$\displaystyle \frac{\skew{3}\dot{c}_i}{c_i}$ $\displaystyle \simeq \left[\left( \frac{-{\rm i}}{\hbar}\right) H_{ii} + \left(...
...left[ 1+ \left(\frac{-{\rm i}}{\hbar}\right) \frac{H_{ii}}{\eta} \right]\right.$    
  $\displaystyle \simeq \left(\frac{-{\rm i}}{\hbar}\right) H_{ii} + \lim_{\eta\ri...
...sum_{f\neq i} \frac{\vert H_{fi}\vert^{\,2}}{E_i - E_f+ {\rm i}\,\hbar \,\eta}.$ (8.92)

This result is formally correct to second order in perturbed quantities. Note that the right-hand side of Equation (8.95) is independent of time. We can write

$\displaystyle \frac{\skew{3}\dot{c}_i}{c_i} = \left( \frac{-{\rm i}}{\hbar}\right) {\mit\Delta}_i,$ (8.93)


$\displaystyle {\mit\Delta}_i = H_{ii} + \lim_{\eta\rightarrow 0} \sum_{f\neq i} \frac{\vert H_{fi}\vert^{\,2}}{E_i - E_f+ {\rm i}\,\hbar \,\eta}$ (8.94)

is a constant. According to a well-known result in pure mathematics known as the Plemelj formula [84],

$\displaystyle \lim_{\epsilon\rightarrow 0} \frac{1}{x+{\rm i}\,\epsilon} = {\cal P}\left(\frac{1}{x}\right) - {\rm i}\,\pi\,\delta(x),$ (8.95)

where $ \epsilon >0$ , and $ {\cal P}$ denotes the Cauchy principal part [52]. It follows that

$\displaystyle {\mit\Delta}_i = H_{ii} + {\cal P}\sum_{f\neq i} \frac{\vert H_{f...
...E_f} - {\rm i}\,\pi \sum_{n\neq i} \vert H_{fi}\vert^{\,2}\, \delta(E_i - E_f).$ (8.96)

It is convenient to normalize the solution of Equation (8.96) such that $ c_i(0) = 1$ . Thus, we obtain

$\displaystyle c_i(t) = \exp\!\left(\frac{-{\rm i}\, {\mit\Delta}_i\, t}{\hbar}\right).$ (8.97)

According to Equation (8.6), the time evolution of the initial state ket $ \vert i\rangle$ is given by

$\displaystyle \vert i, t\rangle = \exp\left[\frac{-{\rm i}\,({\mit\Delta}_i + E_i)\,t}{\hbar}\right] \vert i\rangle.$ (8.98)

We can rewrite this result as

$\displaystyle \vert i, t\rangle = \exp\left(\frac{-{\rm i}\,[E_i + {\rm Re}({\m...
...t) \exp\left[\frac{\,{\rm Im}({\mit\Delta}_i)\,t}{\hbar}\right] \vert i\rangle.$ (8.99)

It is clear that the real part of $ {\mit\Delta}_i$ gives rise to a simple shift in energy of state $ \vert i\rangle$ , whereas the imaginary part of $ {\mit\Delta}_i$ governs the growth or decay of this state. Thus,

$\displaystyle \vert i, t\rangle = \exp\left[\frac{-{\rm i}\,(E_i + {\mit\Delta}...
...r}\right] \exp\left(\frac{ - {\mit\Gamma}_i\,t}{2\,\hbar}\right)\vert i\rangle,$ (8.100)


$\displaystyle {\mit\Delta} E_i = {\rm Re}({\mit\Delta}_i) = H_{ii} + {\cal P} \sum_{f\neq i} \frac{\vert H_{fi}\vert^{\,2}}{E_i - E_f} ,$ (8.101)


$\displaystyle \frac{{\mit\Gamma}_i}{\hbar} = - \frac{2\,{\rm Im}({\mit\Delta}_i...
...= \frac{2\pi}{\hbar} \sum_{f\neq i} \vert H_{fi}\vert^{\,2}\,\delta(E_i - E_f).$ (8.102)

Note that the energy-shift, $ {\mit\Delta} E_i$ , is the same as that predicted by standard time-independent perturbation theory. (See Section 7.3.)

The probability of observing the system in state $ \vert i\rangle$ at time $ t>0$ , given that it was definitely in state $ \vert i\rangle$ at time $ t=0$ , is given by

$\displaystyle P_{i\rightarrow i} (t) = \vert c_i(t)\vert^{\,2} = \exp\left(\frac{-{\mit\Gamma}_i\,t}{ \hbar}\right),$ (8.103)


$\displaystyle \frac{{\mit\Gamma}_i}{\hbar} = \sum_{f\neq i} w_{i\rightarrow f}.$ (8.104)

Here, use has been made of Equation (8.80). Clearly, the decay rate of the initial state, $ {\mit\Gamma}_i/\hbar$ , is equal to the sum of the transition rates to all of the other states. Note that the system conserves probability up to second order in perturbed quantities, because

$\displaystyle \vert c_i\vert^{\,2} + \sum_{f\neq i} \vert c_f\vert^{\,2} \simeq...
...{{\mit\Gamma}_i\,t}{ \hbar}\right) + \sum_{f\neq i} w_{i\rightarrow f} \,t = 1.$ (8.105)

We can write

$\displaystyle \tau_i =\frac{\hbar}{{\mit\Gamma}_i},$ (8.106)


$\displaystyle P_{i\rightarrow i}(t) = \exp\left(-\frac{t}{\tau_i}\right).$ (8.107)

The quantity $ {\mit\Gamma}_i$ , which is called the decay-width of state $ \vert i\rangle$ , is thus closely related to the mean lifetime, $ \tau_i$ , of this state. (See Exercise 5.) According to Equation (8.102), the amplitude of state $ \vert i\rangle$ both oscillates and decays as time progresses. Clearly, state $ \vert i\rangle$ is not a stationary state in the presence of the time-dependent perturbation. However, we can still represent this state as a superposition of stationary states (whose amplitudes simply oscillate in time). Thus,

$\displaystyle \exp\left[\frac{-{\rm i}\,(E_i + {\mit\Delta} E_i)\,t}{\hbar}\rig...
... \int_{-\infty}^\infty dE\,f(E) \exp\left(\frac{-{\rm i}\, E\,t}{\hbar}\right),$ (8.108)

where $ f(E)$ is the weight of the stationary state with energy $ E$ in the superposition. The Fourier inversion theorem yields [51]

$\displaystyle \vert f(E)\vert^{\,2} \propto \frac{1}{(E - [E_i +{\rm Re}({\mit\Delta}_i)])^{\,2} + {\mit\Gamma}_i^{\,2}/4}.$ (8.109)

In the absence of the perturbation, $ \vert f(E)\vert^{\,2}$ is basically a delta-function centered on the unperturbed energy, $ E_i$ , of state $ \vert i\rangle$ . In other words, state $ \vert i\rangle$ is a stationary state whose energy is completely determined. In the presence of the perturbation, the energy of state $ \vert i\rangle$ is shifted by $ {\rm Re}({\mit\Delta}_i)$ . The fact that the state is no longer stationary (i.e., it decays in time) implies that its energy cannot be exactly determined. Indeed, the effective energy of the state is smeared over some range of values of width (in energy) $ {\mit\Gamma}_i$ , centered on the shifted energy, $ E_i +{\rm Re}({\mit\Delta}_i)$ . The faster the decay of the state (i.e., the larger $ {\mit\Gamma}_i$ ), the more its energy is spread out. This phenomenon is clearly a manifestation of the energy-time uncertainty relation $ {\mit\Delta} E\, {\mit\Delta} t \sim \hbar$ . One consequence of this effect is the existence of a natural width of spectral lines associated with the decay of a given excited state of an atom to the ground state. The uncertainty in energy of the excited state, due to its propensity to decay, gives rise to a slight smearing (in wavelength) of the spectral line associated with the transition. [This follows because $ \lambda = \nu/c= (E_i-E_f)/ (h\,c)$ , where $ E_i$ and $ E_f$ are the energies of the excited and ground states, respectively--see Section 8.9. Hence, $ \delta\lambda/\lambda = \delta E_i/E_i\simeq {\mit\Gamma}_i/E_i$ .] Strong lines, which correspond to fast (i.e., large $ {\mit\Gamma}_i$ ) transitions, are smeared out more that weak lines. For this reason, spectroscopists generally favor so-called forbidden lines (which correspond to relatively slow transitions) for Doppler-shift measurements. (See Section 8.11.) Such lines are not as bright as those associated with strong transitions, but they are much sharper.

next up previous
Next: Harmonic Perturbations Up: Time-Dependent Perturbation Theory Previous: Sudden Perturbations
Richard Fitzpatrick 2016-01-22