where

Note that Neumann functions are allowed to appear in the previous expression, because its region of validity does not include the origin (where ). The logarithmic derivative of the th radial wavefunction, , just outside the range of the potential is given by

(10.95) |

where denotes , et cetera. The previous equation can be inverted to give

Thus, the problem of determining the phase-shift, , is equivalent to that of determining .

The most general solution to Schrödinger's equation inside the range of the potential ( ) that does not depend on the azimuthal angle, , is

where

(10.98) |

and

(See Exercise 12.) The boundary condition

ensures that the radial wavefunction is well behaved at the origin. We can launch a well-behaved solution of the previous equation from , integrate out to , and form the logarithmic derivative [of ]

(10.101) |

Because and its first derivatives are necessarily continuous for physically acceptable wavefunctions, it follows that

(10.102) |

The phase-shift, , is then obtained from Equation (10.97).