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:

Figure $\PageIndex{1}$

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.