Q11P Evaluate the integral in Equatio... [FREE SOLUTION]

91影视

Introduction to Quantum Mechanics

David J. Griffiths

Physics

2nd Edition 路 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.

$f (胃) = - \frac{2 m 尾}{h^{2} (渭^{2} + k^{2})}$

Step by step solution

Given.

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

$f (胃) 鈮� - \frac{2 m 尾}{h^{2} k} {鈭�}_{0}^{鈭�} e^{- 渭 r} \sin (k r) d r f (胃) = - \frac{2 m 尾}{h^{2} (渭^{2} + k^{2})}$

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,

${鈭�}_{0}^{鈭�} e^{- 渭谤} \sin (k r) dr = \frac{1}{2 i} {鈭�}_{0}^{鈭�} [e^{- (渭 - 颈渭) r} - e^{- (渭 + 颈渭) r}] dr = \frac{1}{2 i} {[\frac{e^{- (渭 - 颈渭) r}}{- (渭 - i k)} - \frac{e^{- (渭 + 颈渭) r}}{- (渭 + i k)}]}_{0}^{鈭�} = \frac{1}{2 i} [\frac{1}{渭 - i k} - \frac{1}{渭 + i k}] = \frac{k}{渭^{2} + k^{2}}$

Thus,

$f (胃) = - \frac{2 m 尾}{h^{2} k} \frac{k}{渭^{2} + k^{2}} f (胃) = - \frac{2 m 尾}{h^{2} (渭^{2} + k^{2})}$

Therefore, the integral equation was evaluated.

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91影视

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

Short Answer

Step by step solution

Given.

To evaluate the integral equation.

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