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Chapter 14: Problem 10

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.

Short Answer

Expert verified
A repulsive potential \(V(r)>0\) increases the kinetic energy of the scattered particle and decreases its wavelength, leading to a negative scattering phase shift. On the other hand, an attractive potential \(V(r)<0\) decreases the kinetic energy of the particle and increases its wavelength, resulting in a positive phase shift.

Step by step solution

01

Understand the Scattering Phase Shift

The scattering phase shift \(\delta_{l}(E)\) is a concept in quantum mechanics that refers to the angular change in the spherical wave function of a scattered particle resulting from an interaction with a potential. It describes how much the scattering potential changes the phase of the plane wave. Simply speaking, it's the 'shift' in the direction of the scattered wave.
02

Relation Between Repulsive Potential and Phase Shift

When the potential \(V(r)>0\), it is repulsive. This means, in the instance of scattering, it pushes the scattered particle away, leading to an increase in kinetic energy of the particle and hence, a decrease in the particle's wavelength (according to the de-Broglie relation). As phase shift is inversely proportional to wavelength, an decrease in wavelength would result in a higher phase shift — but since it is a deflection in the opposite direction, it is denoted as negative.
03

Relation Between Attractive Potential and Phase Shift

Conversely, when the potential \(V(r)<0\), it is attractive. In scattering, an attractive potential pulls the scattered particle towards it. This leads to a decrease in kinetic energy of this particle, hence an increase in the particle's wavelength. Since phase shift is inversely proportional to wavelength, an increase in wavelength would result in a lower phase shift. As this is in the same direction, it's represented with a positive sign.

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Most popular questions from this chapter

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.

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 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).

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 \(r=b\)

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 \(r=a\)
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