Chapter 14

A particle of mass \(m\) is scattered off a central notential \(V(r)\) of the form $$V(r)=\left\\{\begin{array}{lll} \infty & \text { if } & r=0 \\ V_{0} & \text { if } & 0V_{0},\) find an expression for the S-wave scattering phase shift \(\delta_{0} .\) Hint. Require that the wavefunction and its first derivative be continuous at \(r=a\)

A particle of mass \(m\) is scattered off a central potential \(V(r)\) of the form $$V(r)=\left\\{\begin{array}{lll} \infty & \text { if } & r=0 \\\0 & \text { if } & 0V_{0},\) find an expression for the S-wave scattering phase shift \(\delta_{0} .\) Hint. Require the wavefunction and its first derivative to be continuous at \(r=a\) and at \(r=b\)

For elastic scattering off a central potential, the scattering phase shift for partial wave \(l\) can be written \(\operatorname{as} \delta_{l}(E)=\delta_{\mathrm{bg}}(E)+\delta_{\mathrm{res}}(E),\) where the resonant part of the phase shift is given by $$\tan \delta_{\mathrm{res}}(E)=\frac{\Gamma}{2\left(E_{\mathrm{res}}-E\right)}$$ and the background phase shift is often a slowly varying function of energy. (a) Sketch the behaviour of \(\delta_{l}\) as a function of energy in the vicinity of \(E_{\text {res }}\) if \(\delta_{\mathrm{bg}}\) is taken to be independent of energy with a constant value of (i) \(0 ;\) (ii) \(\pi / 4\) (iii) \(\pi / 2 ;\) (iv) \(3 \pi / 4\) (b) The partial wave cross-section \(\sigma_{l}(E)\) is proportional to \(\sin ^{2} \delta_{l}(E) .\) Sketch the dependence of the latter on energy in the vicinity of \(E_{\mathrm{res}}\) for the four values of \(\delta_{\mathrm{bg}}\) given in part (a). Note that for \(\delta_{\mathrm{bg}}=0, \sin ^{2} \delta_{l}(E)\) has the Breit-Wigner form (eqn 14.67).

By considering flux densities, explain the appearance of the factor \(k_{\alpha} / k_{\alpha_{\rho}}\) in eqn 14.93 for the differential crosssection for scattering from an initial state \(\alpha_{0}\) to a final state \(\alpha\)

The reactance matrix, \(K\), defined in relation to the scattering matrix through \(K=\mathrm{i}(1-S)(1+S)^{-1},\) also appears in scattering theory. Show for elastic scattering by a central potential with partial wave \(l\) that \(K\) is a \(1 \times 1\) matrix with element \(K_{l}=\tan \delta_{l}\)

Find an expression for the WKB phase shift for S-wave scattering at an energy \(E\) by the potential \(V=A / r^{2}.\)