Energy-Shifts and Decay-Widths

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

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

where

(8.94) |

is a constant. According to a well-known result in pure mathematics known as the

(8.95) |

where , and denotes the

(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

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

(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