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Chapter 14: Problem 23

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

Short Answer

Expert verified
The phase shift \(\delta_0\) in terms of the WKB approximation can be found by integrating over the effective potential from the classical turning points. Specific formulas and the exact answer will depend on the value of the constant \(A\) and the energy \(E\).

Step by step solution

01

Write down the radial equation and the WKB approximation

The radial Schrödinger equation for S-waves (l = 0) is given by \(- \hbar^{2}/2m (d^{2} u / dr^{2}) + V(r)u = E u\), where \(u(r)\) is the wave function. The WKB approximation for the phase shift in quantum scattering is given by \(\delta_0 = - \frac{1}{\hbar} \int^{r_b}_{r_a} \sqrt{2m(E - V(r))} dr\), where \(r_a\) and \(r_b\) are the classical turning points.
02

Find the turning points

The classical turning points are the radii, \(r_a\) and \(r_b\), where the potential energy \(V(r)\) equals the total energy \(E\). Solve the equation \(E = A / r^{2}\) to find these points.
03

Integrate to find the phase shift

Substitute the potential \(V(r) = A / r^{2}\) into the WKB approximation and perform the integral to find the phase shift \(\delta\). This will require some knowledge of integral calculus, particularly dealing with powers of r.

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Most popular questions from this chapter

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 \(K_{l}=\tan \delta_{l}\)

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 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.
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