Chapter 11: Q19P (page 419)

Prove the optical theorem, which relates the total cross-section to the imaginary part of the forward scattering amplitude:
$σ = \frac{4 π}{k} Im (f (0))$

Short Answer

Expert verified

$σ = \frac{4 π}{k} J [f (0)]$

Hence, it’s proved.

Step by step solution

To prove the optical theorem.

The relationship between the total cross section and the scattering amplitude in three dimensional scattering is the optical theorem. Equations 11.47 and 11.48 are given by:

$f (θ) = \frac{1}{k} \sum_{l = 0}^{\infty} (2 l + 1) e^{i δ} \sin (δ_{l}) P_{l} (\cos θ)$ ……………….(1)

$σ = \frac{4 π}{k^{2}} \sum_{l = 0}^{\infty} (2 l + 1) \sin^{2} (δ_{l})$ ……………………..(2)

If $θ = 0$ , we can use the fact that $P_{l} (1) = 1$ for all $l$ , so we get:

$f (θ) = \frac{1}{k} \sum_{l = 0}^{\infty} (2 l + 1) e^{i δ} \sin (δ_{l})$

But,

$e^{i δ_{l}} = \cos (δ_{l}) + i \sin (δ_{l})$

So,

$f (0) = \frac{1}{k} {\sum (2 l + 1) [\sin (δ_{l}) \cos (δ_{l}) + i \sin^{2} (δ_{l})]}_{l - 0}^{\infty}$

Taking the imaginary part of $f (0)$ we get:

$J [f (0)] = \frac{1}{k} \sum_{l - 0}^{\infty} (2 l + 1) \sin^{2} (δ_{l})$

So equation (2) can be written as:

$σ = \frac{4 π}{k} J [f (0)]$

Hence, it’s proved.

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Prove the optical theorem, which relates the total cross-section to the imaginary part of the forward scattering amplitude:
$σ = \frac{4 π}{k} Im (f (0))$

Short Answer

Step by step solution

To prove the optical theorem.

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Prove the optical theorem, which relates the total cross-section to the imaginary part of the forward scattering amplitude:σ=4πkIm(f(0))

Short Answer

Step by step solution

To prove the optical theorem.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Turning Points in Physics

Electricity and Magnetism

Linear Momentum

Energy Physics

Particle Model of Matter

Electric Charge Field and Potential

Study anywhere. Anytime. Across all devices.

Prove the optical theorem, which relates the total cross-section to the imaginary part of the forward scattering amplitude:
$σ = \frac{4 π}{k} Im (f (0))$