Past exam of the theoretical physics II course of the University of Cambridge / 2017 / 4

Outside a short-range scattering region, the stationary Schrödinger equation reduces to the two-dimensional Helmholtz equation. Separation in polar coordinates gives outgoing radial Hankel functions, whose large-$r$ dependence is $r^{-1/2}{e}^{ikr}$. The same power follows from conserved probability current: a circle has length proportional to $r$, so the squared outgoing amplitude must decrease as $1/r$.

For an incident plane wave in the $x$ direction, adopt the dimensionless amplitude conventionThe phase $e^{i\pi /4}$ and factor $k^{-1/2}$ specify the definition of $f$; they can be absorbed into a differently normalized scattering amplitude. Here they are kept exactly so that the later two-dimensional scattering cross-section follows consistently.

The incident plane wave carries probability current $j_{\mathrm{in}}=\hslash k/m$. For the separated outgoing wave from the previous part,The rate through a distant arc is $r{j}_r^{\mathrm{sc}}d\theta$. Dividing by incident flux defines the differential two-dimensional scattering cross-section,Because $f$ is dimensionless, $\lambda$ has units of length, as appropriate for scattering in a plane.

The incident plane wave is periodic in $\theta$, so expand it as a Fourier series ${\sum }_m{R}_m(r)e^{im\theta }$. Substitution into the Helmholtz equation, with $\rho =kr$, gives the radial Bessel differential equationRegularity at the origin selects the Bessel function of the first kind, rather than the singular second solution. ThusThe coefficients $a_m$ are independent of $r$; their values follow from matching near the origin in the next part. The completed identity is the Jacobi-Anger expansion.

For $m\ge 0$, expand $e^{i\rho \cos \theta }=\exp [i\rho (e^{i\theta }+e^{-i\theta })/2]$. The lowest power producing the Fourier harmonic $e^{im\theta }$ is $(i\rho /2{)}^m/m!$. Meanwhile $J_m(\rho )\sim (\rho /2{)}^m/m!$. Matching these lowest powers gives $a_m=i^m$.

For negative order use $J_{-m}=(-1{)}^m{J}_m$ and the equality of the positive and negative Fourier coefficients of this even function of $\theta$. This gives $a_{-m}=(-1{)}^m{i}^m=i^{-m}$. Therefore the Jacobi-Anger expansion isThe cosine form has multiplicity one for $m=0$ and two for every positive $m$. Those multiplicities are essential for matching the incident plane wave.

Let $w_0=1$ and $w_m=2$ for $m\ge 1$, as required by the cosine form of the Jacobi-Anger expansion. For a central real potential, the Partial-wave S-matrix is $S_m=e^{2i{\delta }_m}$. Splitting each shifted radial cosine into incoming and outgoing exponentials leaves its incoming coefficient unchanged and multiplies its outgoing coefficient by $S_m$. Subtracting the incident plane wave givesComparing with the prescribed scattering amplitude convention yields the two-dimensional partial-wave scattering formula,Orthogonality gives ${\int }_0^{2\pi }1d\theta =2\pi$ and ${\int }_0^{2\pi }{\cos }^2(m\theta )d\theta =\pi$ for $m\ge 1$, while all cross terms vanish. HenceThe multiplicities printed in this part are reversed. With those printed values and every ${\delta }_m=0$, the purported free wave would be $\frac{1}{2}{e}^{ikr\cos \theta }+\frac{3}{2}{J}_0(kr)$, rather than the required incident plane wave. The extra $J_0$ contains an incoming cylindrical wave, which cannot be absorbed into $f$. The formulas above use the corrected multiplicities and therefore satisfy the specified scattering boundary condition.