Scattering (Homework 10-325)

1. Scattering theory

a) What is the scattering amplitude f(θ) for a Gaussian potential of range a, V (r) =

α exp(−r2/a2).

b) Make a plot of the differential cross-section dσ/d

c) Determine the total cross-section σ in the limit of very low energy k ! [Hint: set Q = 0 and integrate].

The scattered wave is always:

$ψ_{scattered} (r,θ)=f(θ)*(e^{i k r}/r)$

It can be show (very difficult) that for a spherically symmetrical potential (a potential where

$V(r ⃗ )=V(r)$) the function (equation [11.88])

$f(θ)=-2m/(ℏ^2 k) ∫_0^∞ r V(r) sin⁡(k r)d r$   where $k=2K*sin⁡ θ/2$

$K=\sqrt{2mE}/ℏ$   (E is energy of incident particle-wave)   and θ is the scattering angle
For the given potential

$V(r)=α*exp⁡(-r^2/a^2 )$

$f(θ)=-2m/(ℏ^2 k) α∫_0^∞ r*exp⁡(-r^2/a^2 ) sin⁡(k r)d r=$

$=α (√π/4)*a^3*exp⁡(-(a^2 k^2)/4) =(αa^3 √π)/4*exp⁡(-(a^2 K^2 sin^2⁡ θ/2)/2)$


$dσ/dΩ=|f(θ) |^2=(α^2*πa^6)/16*exp⁡ (-a^2 K^2*sin^2⁡ (θ/2)$


In the limit of weak of low energy $(k→0)$ it can be shown that

$f(θ)=-m/(2πℏ^2 )*∫ V(r ⃗ ) d^3 r=-m/(2πℏ^2 ) α ∫_0^∞ exp⁡ (-r^2/a^2 )*(r^2 d r)∬dΩ$

$f(θ)=-m/(2πℏ^2 ) α*(a^3 √π)/4*4π=-(m*α*a^3 √π)/(2ℏ^2 )$

$dσ/dΩ=|f|^2=(m^2*α^2 a^6 π)/(4ℏ^4)$


$σ=(m^2*α^2 a^6 π)/(4ℏ^4 )*∬ dΩ=(m^2*α^2 a^6 π)/(4ℏ^4 )*4π=(m^2*α^2 a^6 π^2)/ℏ^4$