91Ó°ÊÓ

Find the scattering amplitude, in the Born approximation, for soft sphere scattering at arbitrary energy. Show that your formula reduces to Equation 11.82 in the low-energy limit.

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 Schrödinger equation, and use it to construct the integral form (analogous to Equation 11.67).

Rutherford scattering. An incident particle of charge $q_1$andkinetic energy scatters off a heavy stationary particle of charge$q_2$ .

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

$b=\left(q_1{q}_2/8\mathrm{π}̀o_{0}E\right)cot(\mathrm{θ}/2)$ .
(b) Determine the differential scattering cross-section. Answer:

$D\left(\mathrm{θ}\right)={\left[\frac{q_1{q}_2}{16\mathrm{π}̀o_{0}Esi{n}^2(\mathrm{θ}/2)}\right]}^2$

(c) Show that the total cross-section for Rutherford scattering is infinite. We say that the$1/r$ potential has "infinite range"; you can't escape from a Coulomb force.

Use the Born approximation to determine the total cross-section for scattering from a Gaussian potential$\mathbf{V}\mathbf{\left(}\mathbf{r}\mathbf{\right)}\mathbf{=}{\mathbf{Ae}}^{\mathbf{-}{\mathbf{μ°ù}}^{\mathbf{2}}}$.

Construct the analogs to Equation 11.12 for one-dimensional and two-dimensional scattering.

Consider the case of low-energy scattering from a spherical delta function shell is$\mathrm{V}\left(\mathrm{r}\right)=\mathrm{²¹Î´}\left(\mathrm{r}-\mathrm{a}\right)$.Where$\mathrm{Î\pm }$and$a$are constants. Calculate the scattering amplitude,$F\left(\mathrm{θ}\right)$, the differential cross-section,$D\left(\mathrm{θ}\right)$, and the total cross-section,$\mathbf{σ}$.

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

$v\left(x\right)=\left\{\begin{array}{c}0,\left(x<-a\right)\\ -V_0,\left(-a≤x≤0\right)\\ ∞,\left(x>0\right)\end{array}\right.$

(a) If the incoming wave is$A{e}^{ikx}$(where$k=\sqrt{2mE}l\sout{h}$), find the reflected wave.

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

(c) Find the phase shift$\mathrm{δ}$(Equation 11.40) for a very deep well$\left(E≪v_0\right)$.