91Ó°ÊÓ

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.

Prove Equation 11.33, starting with Equation 11.32. Hint: Exploit the orthogonality of the Legendre polynomials to show that the coefficients with different values of l must separately vanish.

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$\left(E≪v_0\right)$.

What are the partial wave phase shifts$\left({\mathrm{δ}}_l\right)$for hard-sphere scattering?

Find theS-wave(l=0) partial wave phase shift${\mathbf{δ}}_{\mathbf{0}}\mathbf{\left(}\mathbf{k}\mathbf{\right)}$ for scattering from a delta-function shell (Problem 11.4). Assume that the radial wave functionu(r)goes to 0 as$\mathbf{r}\mathbf{→}\mathbf{∞}$.

Check that Equation 11.65 satisfies Equation 11.52, by direct substitution. Hint:${\mathbf{∇}}^{\mathbf{2}}\left(1/r\right)\mathbf{=}\mathbf{-}\mathbf{4}\mathit{π}{\mathit{δ}}^{\mathbf{3}}\mathbf{\left(}\mathit{r}\mathbf{\right)}$

Show that the ground state of hydrogen (Equation 4.80) satisfies the integral form of the Schrödinger equation, for the appropriateV and E(note that Eis negative, so $\mathit{k}\mathbf{=}\mathit{i}\mathit{k}$, where $\mathit{k}\mathbf{=}\sqrt{\mathbf{-}\mathbf{2}\mathbf{m}\mathbf{E}\mathbf{/}\mathbf{h}}$).