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Chapter 11: Q19P (page 419)

Prove the optical theorem, which relates the total cross-section to the imaginary part of the forward scattering amplitude:

蟽=4蟺kIm(f(0))

Short Answer

Expert verified

蟽=4蟺kJf0

Hence, it鈥檚 proved.

Step by step solution

01

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胃=1k鈭�l=0鈭�2l+1ei未sin未lPlcos胃鈥︹赌︹赌︹赌︹赌︹赌�.(1)

蟽=4蟺k2鈭�l=0鈭�2l+1sin2未l鈥︹赌︹赌︹赌︹赌︹赌︹赌︹赌�..(2)

If胃=0, we can use the fact that Pl1=1for alll, so we get:

f胃=1k鈭�l=0鈭�2l+1ei未sin未l

But,

ei未l=cos未l+isin未l

So,

f0=1k鈭�2l+1sin未lcos未l+isin2未ll-0鈭�

Taking the imaginary part off0we get:

Jf0=1k鈭�l-0鈭�2l+1sin2未l

So equation (2) can be written as:

蟽=4蟺kJf0

Hence, it鈥檚 proved.

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Most popular questions from this chapter

Nothing that show that the determinant of a Hermitian matrix is real, the determinant of a unitary matrix has modulus 1 (hence the name), and the determinant of an orthogonal matrix is or .

A particle of massmand energyrole="math" localid="1656064863125" Eis incident from the left on the potential

vx=0,x<-a-V0,-a鈮�x鈮�0鈭�,x>0

(a) If the incoming wave isAeikx(wherek=2mElh), find the reflected wave.

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

(c) Find the phase shift未(Equation 11.40) for a very deep wellE鈮�v0.

Rutherford scattering. An incident particle of charge q1andkinetic energy scatters off a heavy stationary particle of chargeq2 .

(a) Derive the formula relating the impact parameter to the scattering angle. 2 Answer:

b=q1q2/8蟺虁o0Ecot(胃/2) .
(b) Determine the differential scattering cross-section. Answer:

D(胃)=q1q216蟺虁o0Esin2(胃/2)2

(c) Show that the total cross-section for Rutherford scattering is infinite. We say that the1/r potential has "infinite range"; you can't escape from a Coulomb force.

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

Find the Green's function for the one-dimensional Schr枚dinger equation, and use it to construct the integral form (analogous to Equation 11.67).
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