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9.6: Determination of Phase-Shifts

  • Page ID
    1243
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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

    $ \beta_{l+} = k\,a \left[\frac{ \cos\delta_l\,j_l'(k\,a) - \sin\de...
... \eta_l'(k\,a)}{\cos\delta_l \, j_l(k\,a) - \sin\delta_l\,\eta_l(k\,a)}\right],$ \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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    This page titled 9.6: Determination of Phase-Shifts is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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