(1020) |

Note that the cross-section now depends on the energy. Furthermore, the magnitude of the cross-section is much larger than that given in Equation (1017) for (since ).

The origin of this rather strange behaviour is quite simple. The condition

(1021) |

is equivalent to the condition that a spherical well of depth possesses a bound state at zero energy. Thus, for a potential well that satisfies the above equation, the energy of the scattering system is essentially the same as the energy of the bound state. In this situation, an incident particle would like to form a bound state in the potential well. However, the bound state is not stable, because the system has a small positive energy. Nevertheless, this sort of

We have seen that there is a resonant effect when the phase-shift of the -wave takes the value . There is nothing special about the partial wave, so it is reasonable to assume that there is a similar resonance when the phase-shift of the th partial wave is . Suppose that attains the value at the incident energy , so that

(1022) |

Let us expand in the vicinity of the resonant energy:

(1023) |

Defining

(1024) |

we obtain

(1025) |

Recall, from Equation (984), that the contribution of the th partial wave to the scattering cross-section is

(1026) |

Thus,

(1027) |

This is the famous