As a specific example, let us consider scattering by a finite
potential well, characterized by for , and
for . Here, is a constant. The potential
is repulsive for , and attractive for .
The outside wavefunction is given by [see Eq. (1310)]

(1332) |

where use has been made of Eqs. (1289) and (1290). The inside wavefunction follows from Eq. (1315). We obtain

where use has been made of the boundary condition (1316). Here, is a constant, and

(1334) |

(1335) |

(1336) |

for , and

(1338) |

Consider an attractive potential, for which . Suppose that
(*i.e.*, the depth of the potential well is much larger than
the energy of the incident particles), so that . We can see
from Eq. (1337) that, unless becomes extremely large, the right-hand side is much less that unity, so replacing the tangent of a
small quantity with the quantity itself, we obtain

(1339) |

(1340) |

Now

so for sufficiently small values of ,

(1343) |

Note that there are values of (*e.g.*,
) at which
, and
the scattering cross-section (1341) vanishes, despite the very strong
attraction of the potential. In reality, the cross-section is not
exactly zero, because of contributions from partial waves. But,
at low incident energies, these contributions are small. It follows that
there are certain values of and which give rise to almost perfect
transmission of the incident wave. This is called the *Ramsauer-Townsend
effect*, and has been observed experimentally.