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Chapter 14: Problem 3

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.

Short Answer

Expert verified
By applying the Laplace operator to the given Green's Function and utilizing properties of the Dirac delta function, it can be shown that the Green's function \(G^{(-)}\) is indeed a solution to the given equation.

Step by step solution

01

Write down the given functions

We have the incoming Green’s function as \(G^{(-)}(r, r') = \frac{e^{-ik|r-r'|}}{|r-r'|}\) and we are required to show that this function satisfies the equation \((\nabla^2+k^2)G(r,r') = -4 \pi \delta(r-r')\).
02

Apply the Laplacian Operator to \(G^{(-)}\)

We apply the Laplace operator \(\nabla^2\) on \(G^{(-)}\). This step requires advanced calculus and the result will be a function that also depends on \(k^2\) and \(|r-r'|\).
03

Add \(k^2 G^{(-)}\) to the Result

Next, we add the term \(k^2G^{(-)}\) to the result from the previous step. This gives us the left-hand side of the equation we are working with.
04

Introduce the Dirac Delta function

Use properties of the Dirac delta function and examine the function obtained in step 3 by taking the limit as \(r \rightarrow r'\). If correctly done, this should be identical to -4π times a Dirac delta function. In this step, justify the reasoning behind the introduction of the Dirac delta function.
05

Formulate the final argument

Finally, compare the result of step 3 to the right side of the original equation to finalize the argument. If the result matches \( -4 \pi \delta(r-r')\), then it can be concluded that \(G^{(-)}\) is a solution to the given problem.

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

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 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 \(14.32 \mathrm{c}\)

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

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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