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# Energy-Shifts and Decay-Widths

We have examined how a state , other than the initial state , 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 . Thus,

 (8.81)

where is small and positive, and is a constant.

In the remote past, , the system is assumed to be in the initial state . Thus, , and . We wish to calculate the time evolution of the coefficient . 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 , it follows from Equations (8.58) and (8.59) that (with )

 (8.82) (8.83)

where . Thus, to first order, the transition probability from state to state is

 (8.84)

The transition rate is given by

 (8.85)

Consider the limit . In this limit, , but [91]

 (8.86)

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

 (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 using Equations (8.58)-(8.60). We have

 (8.88) (8.89) (8.90)

Thus, to second order, we get

Let us now consider the ratio , where . Using Equation (8.94), we can evaluate this ratio in the limit . We obtain

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

 (8.93)

where

 (8.94)

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

 (8.95)

where , and denotes the Cauchy principal part [52]. It follows that

 (8.96)

It is convenient to normalize the solution of Equation (8.96) such that . Thus, we obtain

 (8.97)

According to Equation (8.6), the time evolution of the initial state ket is given by

 (8.98)

We can rewrite this result as

 (8.99)

It is clear that the real part of gives rise to a simple shift in energy of state , whereas the imaginary part of governs the growth or decay of this state. Thus,

 (8.100)

where

 (8.101)

and

 (8.102)

Note that the energy-shift, , is the same as that predicted by standard time-independent perturbation theory. (See Section 7.3.)

The probability of observing the system in state at time , given that it was definitely in state at time , is given by

 (8.103)

where

 (8.104)

Here, use has been made of Equation (8.80). Clearly, the decay rate of the initial state, , 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

 (8.105)

We can write

 (8.106)

where

 (8.107)

The quantity , which is called the decay-width of state , is thus closely related to the mean lifetime, , of this state. (See Exercise 5.) According to Equation (8.102), the amplitude of state both oscillates and decays as time progresses. Clearly, state 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,

 (8.108)

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

 (8.109)

In the absence of the perturbation, is basically a delta-function centered on the unperturbed energy, , of state . In other words, state is a stationary state whose energy is completely determined. In the presence of the perturbation, the energy of state is shifted by . 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) , centered on the shifted energy, . The faster the decay of the state (i.e., the larger ), the more its energy is spread out. This phenomenon is clearly a manifestation of the energy-time uncertainty relation . 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 , where and are the energies of the excited and ground states, respectively--see Section 8.9. Hence, .] Strong lines, which correspond to fast (i.e., large ) 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: Harmonic Perturbations Up: Time-Dependent Perturbation Theory Previous: Sudden Perturbations
Richard Fitzpatrick 2016-01-22