Let us now consider how the phase-shifts can be evaluated. Consider a spherically symmetric potential that vanishes for , where is termed the range of the potential. In the region , the wavefunction satisfies the free-space Schrödinger equation (958). The most general solution that is consistent with no incoming spherical waves is
Note that Neumann functions are allowed to appear in the above 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
where denotes , etc. The above 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
The boundary condition
ensures that the radial wavefunction is well-behaved at the origin. We can launch a well-behaved solution of the above equation from , integrate out to , and form the logarithmic derivative
Because and its first derivatives are necessarily continuous for physically acceptible wavefunctions, it follows that
The phase-shift is obtainable from Equation (988).
- Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)