The Yukawa potential between two particles separated by is V(r)=A

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Computational Physics · Scientific programming for physicists

Scientific Computing for Physicists, Ch Letting the computer do advanced algebra: operators, series and differentiated programs

Problem

The Yukawa potential between two particles separated by r>0r>0 is V(r)=A e−r/λr,V(r)=A\,\frac{e^{-r/\lambda}}{r}, with AA a positive energy-length constant and λ\lambda a positive screening length. A symbolic derivative gives the force magnitude F(r)=−dVdr=A e−r/λ(1λr+1r2),F(r)=-\frac{\mathrm dV}{\mathrm dr}=A\,e^{-r/\lambda}\left(\frac{1}{\lambda r}+\frac{1}{r^2}\right), from which a numerical function is generated (e.g. by lambdify) and then evaluated at randomly chosen values of rr as a test against an independently hand-derived formula. ((a)) Evaluate the generated function at r=1.2 nmr=1.2\,\mathrm{nm} for A=2.0 eV nmA=2.0\,\mathrm{eV\,nm}, λ=0.5 nm\lambda=0.5\,\mathrm{nm}, giving your answer to four significant figures. ((b)) Given binary64 arithmetic with ϵmach=2−52≈2.2×10−16\epsilon_{\rm mach}=2^{-52}\approx2.2\times10^{-16}, state the order of relative agreement expected between this generated function and an independently hand-derived evaluation of the same formula at the same point, and explain briefly why a single hand-chosen test point such as this one is not sufficient on its own to certify the generated function.

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