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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

Use your result in Problem 11.16 to develop the Born approximation for one-dimensional scattering (on the interval $- \infty < x < \infty$ , with no "brick wall" at the origin). That is, choose $ψ_{0} (x) = {Ae}^{ikx}$ , and assume $ψ (x_{0}) ≅ ψ_{0} (x_{0})$ to evaluate the integral. Show that the reflection coefficient takes the form:

$R ≅ {(\frac{m}{h^{2} k})}^{2} \int_{- \infty}^{\infty} e^{2 ikx} V (x) {dx}^{2}$

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

A particle of mass $m$ and energyrole="math" localid="1656064863125" $E$ is incident from the left on the potential

$v (x) = \{\begin{matrix} 0, (x < - a) \\ - V_{0}, (- a \leq x \leq 0) \\ \infty, (x > 0) \end{matrix}$

(a) If the incoming wave is $A e^{i k x}$ (where $k = \sqrt{2 m E} l h$ ), find the reflected wave.

(b) Confirm that the reflected wave has the same amplitude as the incident wave.

(c) Find the phase shift $δ$ (Equation 11.40) for a very deep well $(E ≪ v_{0})$ .

Find theS-wave(l=0) partial wave phase shift $δ_{0} (k)$ for scattering from a delta-function shell (Problem 11.4). Assume that the radial wave functionu(r)goes to 0 as $r \to \infty$ .

Consider the case of low-energy scattering from a spherical delta function shell is $V (r) = aδ (r - a)$ .Where $α$ and $a$ are constants. Calculate the scattering amplitude, $F (θ)$ , the differential cross-section, $D (θ)$ , and the total cross-section, $σ$ .

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