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# Determination of Phase-Shifts

Let us now consider how the phase-shifts in Eq. (1306) can be evaluated. Consider a spherically symmetric potential which vanishes for , where is termed the range of the potential. In the region , the wavefunction satisfies the free-space Schrödinger equation (1285). The most general solution which is consistent with no incoming spherical-waves is
 (1309)

where
 (1310)

Note that 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
 (1311)

where denotes , etc. The above equation can be inverted to give
 (1312)

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

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

 (1313)

where
 (1314)

and
 (1315)

The boundary condition
 (1316)

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
 (1317)

Since and its first derivatives are necessarily continuous for physically acceptible wavefunctions, it follows that
 (1318)

The phase-shift is then obtainable from Eq. (1312).

Next: Hard Sphere Scattering Up: Scattering Theory Previous: Partial Waves
Richard Fitzpatrick 2010-07-20