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Introduction to Quantum Mechanics

David J. Griffiths

Physics

2nd Edition · 469 pages

ISBN: 9780131118928

Chapter 11: Q11P (page 416)

Evaluate the integral in Equation 11.91, to confirm the expression on the right.

Short Answer

Expert verified

Therefore, the integral equation was evaluated.

$f (θ) = - \frac{2 m β}{h^{2} (μ^{2} + k^{2})}$

Step by step solution

01

Given.

The amplitude of the scattering in Born approximation is given by equation11.91 as:

$f (θ) ≅ - \frac{2 m β}{h^{2} k} \int_{0}^{\infty} e^{- μ r} \sin (k r) d r f (θ) = - \frac{2 m β}{h^{2} (μ^{2} + k^{2})}$

02

To evaluate the integral equation.

We need to evaluate the integral to get the RHS of the equation, we need to convert the sine function into the complex form, that is:

$\sin (k r) = \frac{1}{2 i} (e^{i k r} - e^{- i k r})$

So the integral becomes,

$\int_{0}^{\infty} e^{- μr} \sin (k r) dr = \frac{1}{2 i} \int_{0}^{\infty} [e^{- (μ - iμ) r} - e^{- (μ + iμ) r}] dr = \frac{1}{2 i} {[\frac{e^{- (μ - iμ) r}}{- (μ - i k)} - \frac{e^{- (μ + iμ) r}}{- (μ + i k)}]}_{0}^{\infty} = \frac{1}{2 i} [\frac{1}{μ - i k} - \frac{1}{μ + i k}] = \frac{k}{μ^{2} + k^{2}}$

Thus,

$f (θ) = - \frac{2 m β}{h^{2} k} \frac{k}{μ^{2} + k^{2}} f (θ) = - \frac{2 m β}{h^{2} (μ^{2} + k^{2})}$

Therefore, the integral equation was evaluated.

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Most popular questions from this chapter

Prove Equation 11.33, starting with Equation 11.32. Hint: Exploit the orthogonality of the Legendre polynomials to show that the coefficients with different values of l must separately vanish.

Find the scattering amplitude, in the Born approximation, for soft sphere scattering at arbitrary energy. Show that your formula reduces to Equation 11.82 in the low-energy limit.

Find the scattering amplitude for low-energy soft-sphere scattering in the second Born approximation.

Use the Born approximation to determine the total cross-section for scattering from a Gaussian potential $V (r) = {Ae}^{- {μr}^{2}}$ .

Calculate the total cross-section for scattering from a Yukawa potential, in the Born approximation. Express your answer as a function of E.

See all solutions

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