1.4: Scattering in 2D and 3D
We now wish to consider scattering experiments in spatial dimension \(d \ge 2\) , which have a new and important feature. For \(d = 1\) , the particle can only scatter forward or backward, but for \(d \ge 2\) it can be scattered to the side.
Far from the scatterer, where \(V(\mathbf{r})\rightarrow 0\) , the scattered wavefunction \(\psi_s(\mathbf{r})\) satisfies
\[-\frac{\hbar^2}{2m} \nabla^2 \psi_s(\mathbf{r}) = E \psi_s(\mathbf{r}),\]
where \(\nabla^2\) denotes the \(d\) -dimensional Laplacian. Let \(E = \hbar^2 k^2 / 2m\) , where \(k \in \mathbf{R}^+\) is the wave-number in free space. Then the above equation can be written as
\[\left[\nabla^2 + k^2\right] \psi_s(\mathbf{r}) = 0,\]
which is the Helmholtz equation in \(d\) -dimensional space.
One set of elementary solutions to the Helmholtz equation are the plane waves
\[\big\{\exp(i\mathbf{k}\cdot\mathbf{r}),\;\;\mathrm{where}\;\; |\mathbf{k}| = k \big\}.\]
But we’re looking for an outgoing solution, and a plane wave can’t be said to be “outgoing”.
Therefore, we turn to curvilinear coordinates. In 2D, we use the polar coordinates \((r,\phi)\) . We will skip the mathematical details of how to solve the 2D Helmholtz equation in these coordinates; the result is that the general solution can be written as a linear combination
\[\psi(\mathbf{r})=\sum_{\pm}\sum_{m=-\infty}^\infty c_m^\pm\,\Psi_m^\pm(r,\phi), \;\;\;\mathrm{where}\;\;\,\Psi_m^\pm(r,\phi) = H_m^\pm(kr)\,e^{im\phi}.\]
This is a superposition of circular waves \(\Psi_m^\pm(r,\phi)\) , with coefficients \(c_m^\pm \in \mathbb{C}\) . Each circular wave is a solution to the 2D Helmholtz equation with angular momentum quantum number \(m \in \mathbb{Z}\) . Its \(r\) -dependence is given by \(H_m^\pm\) , called a Hankel function of the “first kind” ( \(+\) ) or “second kind” ( \(-\) ). Some Hankel functions of the first kind are plotted below:
The \(H^-_m\) functions are the complex conjugates of \(H^+_m\) . For large values of the input,
\[H_m^\pm(kr) \overset{r\rightarrow\infty}{\longrightarrow} \sqrt{\frac{2}{\pi kr}} \, \exp\left[\pm i\left(kr - \frac{(m+\frac{1}{2})\pi}{2}\right)\right] \;\sim\; r^{-1/2} e^{\pm ikr}.\]
Therefore, the \(\pm\) index specifies whether the circular wave is an outgoing wave directed outward from the origin ( \(+\) ), or an incoming wave directed toward the origin ( \(-\) ).
The 3D case is treated similarly. We use spherical coordinates \((r,\theta,\phi)\) , and the solutions of the 3D Helmholtz equation are superpositions of incoming and outgoing spherical waves:
\[\psi(\mathbf{r})=\sum_{\pm}\sum_{\ell=0}^\infty\sum_{m=-\ell}^\ell c_{\ell m}^\pm \,\Psi_{\ell m}^\pm(r,\theta,\phi)\;\;\;\mathrm{where}\;\;\Psi_{\ell m}^\pm(r,\theta,\phi) = \,h_\ell^\pm(kr)\,Y_{\ell m}(\theta,\phi).\]
The \(c_{\ell m}^\pm\) factors are complex coefficients. Each \(h_\ell^\pm\) is a spherical Hankel function , and each \(Y_{\ell m}\) is a spherical harmonic . The \(\ell\) and \(m\) indices specify the angular momentum of the spherical wave. For large inputs, the spherical Hankel functions have the limiting form
\[h_\ell^\pm(kr) \overset{r\rightarrow\infty}{\longrightarrow} \pm \frac{\exp\!\left[\pm i\!\left(kr-\frac{\ell\pi}{2}\right)\right]}{ikr}.\]
Hence, the \(\pm\) index specifies whether the spherical wave is outgoing ( \(+\) ) or incoming ( \(-\) ). More discussion about these spherical waves can be found in Appendix A.
It is now clear what we need to do to get a scattered wavefunction \(\psi_s(\mathbf{r})\) that is outgoing at infinity. We take a superposition with only outgoing ( \(+\) ) wave components:
\[\psi_s(\mathbf{r}) = \begin{cases} \displaystyle\sum_{m} c_m^+\,H_m^+(kr)\,e^{im\phi}, &d=2\\ \displaystyle\sum_{\ell m} c_{\ell m}^+\,h_\ell^+(kr)\,Y_{\ell m}(\theta,\phi),&d=3.\end{cases}\]
For large \(r\) , the outgoing wavefunction has the \(r\) -dependence
\[\psi_s(\mathbf{r}) \; \overset{r\rightarrow\infty}{\sim} \; r^{\frac{1-d}{2}} \,\exp\left(ikr\right).\]
For \(d > 1\) , the magnitude of the wavefunction decreases with distance from the origin. This is as expected, because with increasing \(r\) each outgoing wave spreads out over a wider area. The probability current density is \(\mathbf{J} = (\hbar/m) \mathrm{Im}\left[\psi_s^*\nabla\psi_s\right]\) , and its \(r\) -component is
\[\begin{align}\begin{aligned}J_r \; &\overset{r\rightarrow\infty}{\sim} \; \mathrm{Im}\left[r^{\frac{1-d}{2}} e^{-ikr} \frac{\partial}{\partial r}\left(r^{\frac{1-d}{2}} e^{ikr}\right)\right] \\ &\;\;=\;\;\;\mathrm{Im}\left[\frac{1-d}{2}\, r^{-d} + ik r^{1-d}\right]\\ &\;\;=\;\;\; k \,r^{1-d}.\end{aligned}\end{align}\]
In \(d\) dimensions, the area of a wave-front scales as \(r^{d-1}\) , so the probability flux goes as \(J_r \,r^{d-1} \sim k\) , which is positive and independent of \(r\) . This describes a constant probability flux flowing outward from the origin. Note that if we plug \(d=1\) into the above formula, we find that \(J_r\) scales as \(r^0\) (i.e., a constant), consistent with the results of the previous section: waves in 1D do not spread out with distance as there is no transverse dimension.