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Chapter 14: Problem 13

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 \(r=a\)

Short Answer

Expert verified
The S-wave scattering phase shift can be obtained by solving the wave function boundary conditions and its derivative at the potential barrier ending point (\( r = a \)). It is given by argument of the reflection amplitude, which implies it's phase of the ratio of the outgoing to incoming spherical waves at large \( r \). The detailed formula depends on the solutions of the wave equations and boundary conditions.

Step by step solution

01

Identify the function forms in the different regions

We know that the wave function of a particle in a potential \( V(r) \) satisfies the time-independent Schrödinger equation. In region 2, \( 0a \), we get \(\frac{d^2 y}{dx^2} = -k'^2 y\) with \( k'^2 = 2mE/\hbar^2 \). Solution here is \( y = H e^{i k' r} + G e^{-i k' r} \). Note that for \( r > a \), the decreasing exponent term represents an incoming spherical wave and the increasing exponent term represents a scattered spherical wave.
02

Assume wave approaches from far away

Typically in scattering problems, one assumes the wave approaches from far away with no spherical wave moving towards the origin. Therefore, one can set \( G = 0 \). So the solution in region 3 simplifies to \( y = H e^{i k' r} \).
03

Apply boundary conditions

Now we solve for the undetermined constants by applying continuity of the wave function and its first derivative at \( r = a \). Setting the wave functions in region 2 and 3 equal at \( r = a \) yields \( A sin(k a) + B cos(k a) = H e^{i k' a} \). This gives one equation. The other comes from setting the derivatives equal at \( r = a \) giving another equation. Solve these 2 equations to find A and B in terms of H.
04

Determine the S-wave scattering phase shift

The S-wave phase shift \( \delta_0 \) is given by argument of the reflection amplitude. Which is the phase of the ratio of the outgoing to incoming spherical waves at large \( r \), i.e, \( \delta_0 = arg(A/H) \). Determine \( \delta_0 \) by using the equations from step 3, and find the expression for the scattering phase shift.

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Most popular questions from this chapter

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

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.

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.

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.

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