9.6: Determination of Phase-Shifts
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Let us now consider how the phase-shifts \( V(r)\) that vanishes for \( a\) is termed the range of the potential. In the region \( \psi({\bf x})\) satisfies the free-space Schrödinger equation \ref{958}. The most general solution that is consistent with no incoming spherical waves is
\( A_l(r) = \exp(\,{\rm i} \,\delta_l)\, \left[ \cos\delta_l \,j_l(k\,r) -\sin\delta_l\, \eta_l(k\,r)\right].\) | \ref{986} |
Note that Neumann functions are allowed to appear in the above expression, because its region of validity does not include the origin (where \( l\) th radial wavefunction \( A_l(r)\) just outside the range of the potential is given by
| \ref{987} |
where \( dj_l(x)/dx\) , etc. The above equation can be inverted to give
\( \delta_l\) is equivalent to that of determining \( r<a\) ) that does not depend on the azimuthal angle \( \psi({\bf x}) = \frac{1}{(2\pi)^{3/2}}\sum_{l=0,\infty} {\rm i}^l \,(2\,l+1)\,R_l(r)\,P_l(\cos\theta),\) | \ref{989} |
where
\( \frac{d^2 u_l}{d r^2} +\left[k^2 - \frac{2m}{\hbar^2} \,V - \frac{l\,(l+1)} {r^2}\right] u_l = 0.\) | \ref{991} |
The boundary condition
\( r=0\), integrate out to \( \beta_{l-} = \left.\frac{1}{(u_l/r)} \frac{d(u_l/r)}{dr}\right\vert _{r=a}.\) | \ref{993} |
Because \( \beta_{l+} = \beta_{l-}.\) \ref{994} | The phase-shift \( \delta_l\) is obtainable from Equation \ref{988}.
Contributors
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)
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