Energy-Shifts and Decay-Widths

In this case, it is convenient to gradually turn on the perturbation from zero at . Thus,

(821) |

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 . Basically, we want 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 we have from Equations (795)-(796) that

(822) | ||

(823) |

where . It follows that, to first order, the transition probability from state to state is

(824) |

The transition rate is given by

Consider the limit . In this limit, , but

(826) |

Thus, Equation (825) yields the standard Fermi golden rule result

(827) |

It is clear that the delta-function in the above formula actually represents a function that is highly peaked at some particular energy. The width of the peak is determined by how fast the perturbation is switched on.

Let us now calculate using Equations (795)-(797). We have

(828) | ||

(829) | ||

(830) |

Thus, to second order we have

Let us now consider the ratio , where . Using Equation (831), 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 (832) is independent of time. We can write

where

(834) |

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

(835) |

where , and denotes the principal part. It follows that

(836) |

It is convenient to normalize the solution of Equation (833) such that . Thus, we obtain

(837) |

According to Equation (744), the time evolution of the initial state ket is given by

(838) |

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,

(840) |

where

(841) |

and

(842) |

Note that the energy-shift is the same as that predicted by standard time-independent perturbation theory.

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

(843) |

where

(844) |

Here, use has been made of Equation (817). Clearly, the rate of decay of the initial state is a simple function of the transition rates to the other states. Note that the system conserves probability up to second order in perturbed quantities, because

(845) |

The quantity
is called the *decay-width* of state
.
It is
closely related to the mean lifetime of this state,

(846) |

where

(847) |

According to Equation (839), 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 it as a superposition of stationary states (whose amplitudes simply oscillate in time). Thus,

(848) |

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

(849) |

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