Chapter 14

Equation 14.3 gives the form of the S matrix for a one-dimensional system in which a particle is scattered from an abrupt blip in the potential energy. Write down the analogous expression for scattering from a comparable dip in the potential energy. Proceed to compute the transmission probability for positive energies given that the particle is incident from the left.

Calculate the angular components of the flux density, \(J_{\theta}\) and \(J_{\varphi},\) for the scattered wave $$\psi=f_{k}(\theta, \varphi) \frac{\mathrm{e}^{\mathrm{i} k r}}{r}$$ and confirm that in the limit \(r \rightarrow \infty,\) only the radial component \(J_{r}\) given in Justification 14.3 need be retained.

The incoming Green's function is given by $$G^{(-)}\left(r, r^{\prime}\right)=\frac{\mathrm{e}^{-i k\left|r-r^{\prime}\right|}}{\left|r-r^{\prime}\right|}$$ Show that \(G^{(-)}\) is a solution of the equation $$\left(\nabla^{2}+k^{2}\right) G\left(r, r^{\prime}\right)=-4 \pi \delta\left(r-r^{\prime}\right)$$ Hint. Use an analysis similar to that given in Further information 14.1. Although the incoming Green's function does not yield the desired asymptotic form of the stationary scattering state \((\mathrm{eqn} 14.14), G^{(-)}\) appears in some of the formal expressions of scattering theory.

The differential cross-section for the Yukawa potential using the Born approximation is given in Example 14.3 Plot it as a function of the angle \(\theta\) for (a) zero energy, (b) moderate energy \((k \approx \alpha),\) and (c) high energy \((k \gg \alpha)\) For the plots, choose the range of the \(y\) -axis to be 0 to \(\left\\{\left(2 \mu V_{0}\right) /\left(\hbar^{2} \alpha^{2}\right)\right\\}^{2} .\) For moderate energy, take \(k=\alpha / 2 ;\) for high energy, take \(k=10 \alpha\)

Use the Born approximation to calculate the differential cross-section for scattering from the spherical square-well potential (Section 14.5 ). Hint. Use integration by parts to determine the scattering amplitude.

Consider the scattering of an electron by an atom of atomic number \(Z .\) The interaction potential energy can be approximated by the screened Coulomb potential energy \(V(r)=-\left(Z e^{2} / 4 \pi \varepsilon_{0} r\right) \mathrm{e}^{-r / a},\) where \(a\) is the screening length. (a) Use the Born approximation to calculate the differential cross-section for scattering from the screened Coulomb potential. (b) Proceed to evaluate the integral scattering cross-section. (c) In the limit \(a \rightarrow \infty, V(r)\) becomes exactly the Coulomb potential energy. Evaluate the differential and integral cross-sections obtained in parts (a) and (b) in this limit.

Consider the differential cross-section for elastic scattering given in eqn \(14.46 .\) At a given energy, sketch its dependence on the scattering angle \(\theta\) when the \(l=1\) partial wave dominates the scattering. Do the same for the \(l=0\) and \(l=2\) partial waves.

Show for the elastic scattering of a particle by a central potential \(V(r)\) that approaches zero more rapidly than \(1 / r\) as \(r \rightarrow \infty\) that the integral cross-section can be written as $$\sigma_{\mathrm{tot}}=\frac{4 \pi}{k} \mathrm{im} f_{k}(0)$$ where im \(f_{k}(0)\) is the imaginary part of the forward scattering amplitude \((\theta=0) .\) This is the so-called optical theorem. Hint. The Legendre polynomials are required to satisfy \(P_{l}(1)=1\) for all values of \(l\)

For elastic scattering by a central potential, it is possible to show analytically that if the potential is repulsive, with \(V(r)>0\) for all \(r,\) then the scattering phase shift \(\delta_{l}(E)\) is negative; likewise, if the potential is attractive, with \(V(r)<0\) for all \(r,\) then the phase shift \(\delta_{l}\) is positive. Explain this result qualitatively by considering the effect of a repulsive (or attractive) potential on the wavelength of the scattered particle.

Show that in the limit of low energies, the scattering phase shift for P-wave scattering by a hard sphere is proportional to \((k a)^{3}\) and therefore is negligible compared to the S-wave scattering phase shift. Hint. Use the asymptotic forms given in eqn \(14.32 \mathrm{c}\)