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given a radial potential in 3 dimension and its Schroedinguer equation

$ -D^{2}U(r) + \frac{l(l+1)}{r^{2}}+V(r) $ here D means derivative with respect to 'r'

then if we apply quantum scattering how can we calculate the PHASE SHIFT ?? $\delta $ , for a general potential ?? .. for example with the condition $ V(r) \to \infty$ as $ r\to \infty$

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  • $\begingroup$ Related: physics.stackexchange.com/q/8132/2451 $\endgroup$
    – Qmechanic ♦
    Commented Nov 17, 2011 at 15:19
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    $\begingroup$ Do you mean V(r) goes to zero at large r? If V(r) goes to infinity, there are no scattering states, the motion is bound. Also V(r) needs to go to zero fast enough for the phase shift to approach a limit, generally this means V(r) goes to zero faster than 1/r. Electrostatic potential scattering with a 1/r potential has a logarithmically diverging phase shift, making the S-matrix undefined without a small photon mass. This is a form of infrared divergence. $\endgroup$
    – Ron Maimon
    Commented Nov 17, 2011 at 15:20
  • $\begingroup$ thanks RON i forgot :) , yes the potential is $ V(r) \to 0 $ as $ r\to \infty $, whenever you say SCATTERING STATES you are referring to states with CONTINUUM energy in the form $ k^{2}+a^{2} $ here 'k' is the wave number and 'a' is a constant :) $\endgroup$
    – Jose Javier Garcia
    Commented Nov 17, 2011 at 15:50

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For a general potential $V(r)$ you can use the variable-phase method.

I think you will find the following book very useful: Calogero, F. (1967), The Variable Phase Approach to Potential Scattering. Academic Press, New York.

Interesting to note is that Calogero emphasizes in his 1967 book that the numerical calculation of phase shifts using the variable-phase equation is well within the power of a simple desk calculator!

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  • $\begingroup$ Is there any numerical implementation available of his approach in Python, C++ etc.? $\endgroup$
    – rainman
    Commented Dec 2, 2018 at 6:41
  • $\begingroup$ @omehoque I know of no dedicated public codes, though I'm sure they are there as part of larger nuclear/chemical physics routines. For the case in the original question, the differential equation is just a simple initial value problem which you can just code up yourself in Mathematica or Matlab. $\endgroup$
    – QuantumDot
    Commented Dec 2, 2018 at 7:19

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