Given a wave packet which in the remote past represented a particle approaching a potential, what does that wave look like in the remote future?

We will first consider the nonrelativistic propagator. Huygens' principle can be written as

(6.5) |

where the integral extends over all space. is the total wave arriving at the point at time and is the original wave amplitude. is the Green's function or propagator. In general one does not consider stationary eigenstates of energy (i.e. stationary waves). Knowledge of enables us to construct the physical state which develops in time from any given initial state. This is equivalent to a complete solution to Schrödinger equation.

Consider a free-particle solution and its Green function . We introduce a potential which is ``turned on'' for a brief interval of time about ( outside of time ). acts as a source of new waves and we can write

where right-hand side is zero outside the interval .

The new wave function can be written as

(6.7) |

Substituting this solution into equation 6.6 and using the free particle Schrödinger equation gives

The second terms on both sides of the equation are smaller than the first terms on both sides of the equation. This is clear for the righthand side of the equation. To see it for the lefthand side, we drop the second term on the righthand side and write

(6.9) | |||

(6.10) |

After dropping the second two terms on both sides of equation 6.8, we have

(6.11) |

To first order

(6.12) |

The integration shows that the potential produces an additional change in during in addition to that taking place in the absence of . Since the potential vanishes after the time interval , the scattered wave propagates according to the free propagator too. This added wave at a future time leads to a new contribution to ,

(6.13) |

Here we have replaced by which is justified in the limit.

Thus the wave developing from an arbitary wave packet in the remote past is

Therefore the Green's function here is

(6.16) |

The first term represents the propagation for to as a free particle. The second term represents propagation from to , a scattering at , and free propagation from to .

If we turn on another potential for an interval at time , the additional contribution to for is

where we have used the obvious notation: and . The first term represents a single scattering at time . The second term is a double scattering.

The total wave is obtained by inserting (6.14) for into (6.17) and adding it to (6.14)

(6.19) |

The four terms of the above equation are depicted in figure 6.1.

If there are such time intervals when the potential is turned on

(6.20) |

The corresponding Green's function is

(6.21) |

is the probability amplitude for a particle wave originating at to propagate to , as depicted in figure 6.2. This amplitude is a sum of amplitudes, the th such term begin a product of factors corresponding to the figure below. Each line in the figure represents the amplitude that a particle wave originating at propagates freely to . At the point it is scattered with probability amplitude per unit space-time volume to a new wave propagating forward in time with amplitude to the next interaction. This amplitude is then summed over all space-time points in which the interactions can occur.

We may lift the time-ordering restriction , etc, if we define

(6.22) | |||

(6.23) |

is a retarded propagator since it only propagates waves forward in time. Physically this just means that no Huygens wavelets from the 'th iteration (at time ) appear until after .

In the limit of a continuous interaction and , so that we can write

(6.24) |

and obtain

(6.25) |

This multiple scattering series is assumed to converge and may be summed formally to yield

This is the Lippmann-Schwinger equation. We have ignore the possibility of bound states in the potential .

Similarly the series for the wave function can be summed, resulting in

This is the integral equation for , where the second term is the scattered wave.

2004-03-18