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Physics LibreTexts

14.5: Determination of Phase-Shifts

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Let us now consider how the phase-shifts, δl, in Equation ([e17.73]) can be evaluated. Consider a spherically symmetric potential, V(r), that vanishes for r>a, where a is termed the range of the potential. In the region r>a, the wavefunction ψ(r) satisfies the free-space Schrödinger equation ([e17.54]). The most general solution that is consistent with no incoming spherical-waves is ψ(r)=nl=0,il(2l+1)Rl(r)Pl(cosθ), where Rl(r)=exp(iδl)[cosδljl(kr)sinδlyl(kr)]. Note that yl(kr) functions are allowed to appear in the previous expression because its region of validity does not include the origin (where V0). The logarithmic derivative of the lth radial wavefunction, Rl(r), just outside the range of the potential is given by βl+=ka[cosδljl(ka)sinδlyl(ka)cosδljl(ka)sinδlyl(ka)], where jl(x) denotes djl(x)/dx, et cetera. The previous equation can be inverted to give tanδl=kajl(ka)βl+jl(ka)kayl(ka)βl+yl(ka). Thus, the problem of determining the phase-shift, δl, is equivalent to that of obtaining βl+.

The most general solution to Schrödinger’s equation inside the range of the potential (r<a) that does not depend on the azimuthal angle ϕ is ψ(r)=nl=0,il(2l+1)Rl(r)Pl(cosθ), where Rl(r)=ul(r)r, and d2uldr2+[k2l(l+1)r22m2V]ul=0. The boundary condition ul(0)=0 ensures that the radial wavefunction is well behaved at the origin. We can launch a well-behaved solution of the previous equation from r=0, integrate out to r=a, and form the logarithmic derivative βl=1(ul/r)d(ul/r)dr|r=a. Because ψ(r) and its first derivatives are necessarily continuous for physically acceptable wavefunctions, it follows that βl+=βl. The phase-shift, δl, is then obtainable from Equation ([e17.82]).

Contributors and Attributions

  • Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)


This page titled 14.5: 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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