14.3: Partial Waves

We can assume, without loss of generality, that the incident wavefunction is characterized by a wavevector ${\bf k}$ that is aligned parallel to the $z$-axis. The scattered wavefunction is characterized by a wavevector ${\bf k}'$ that has the same magnitude as ${\bf k}$, but, in general, points in a different direction. The direction of ${\bf k}'$ is specified by the polar angle $\theta$ (i.e., the angle subtended between the two wavevectors), and an azimuthal angle $\phi$ about the $z$-axis. Equations ([e17.38]) and ([e17.39]) strongly suggest that for a spherically symmetric scattering potential [i.e., $V({\bf r}) = V(r)$] the scattering amplitude is a function of $\theta$ only: that is, $$f(\theta, \phi) = f(\theta).$$ It follows that neither the incident wavefunction,

$$\label{e17.52} \psi_0({\bf r}) = \sqrt{n}\,\exp(\,{\rm i}\,k\,z)= \sqrt{n}\,\exp(\,{\rm i}\,k\,r\cos\theta),$$ nor the large-$r$ form of the total wavefunction,

$$\label{e17.53} \psi({\bf r}) = \sqrt{n} \left[ \exp(\,{\rm i}\,k\,r\cos\theta) + \frac{\exp(\,{\rm i}\,k\,r)\, f(\theta)} {r} \right],$$ depend on the azimuthal angle $\phi$.

Outside the range of the scattering potential, both $\psi_0({\bf r})$ and $\psi({\bf r})$ satisfy the free-space Schrödinger equation,

$$\label{e17.54} (\nabla^{\,2} + k^{\,2})\,\psi = 0.$$ What is the most general solution to this equation in spherical polar coordinates that does not depend on the azimuthal angle $\phi$? Separation of variables yields

$$\label{e17.55} \psi(r,\theta) = \sum_l R_l(r)\, P_l(\cos\theta),$$ because the Legendre functions, $P_l(\cos\theta)$, form a complete set in $\theta$-space. The Legendre functions are related to the spherical harmonics, introduced in Chapter [sorb], via $$P_l(\cos\theta) = \sqrt{\frac{4\pi}{2\,l+1}}\, Y_{l,0}(\theta,\varphi).$$ Equations ([e17.54]) and ([e17.55]) can be combined to give

$$r^{\,2}\,\frac{d^{\,2} R_l}{dr^{\,2}} + 2\,r \,\frac{dR_l}{dr} + [k^{\,2} \,r^{\,2} - l\,(l+1)]\,R_l = 0.$$ The two independent solutions to this equation are the spherical Bessel functions, $j_l(k\,r)$ and $y_l(k\,r)$, introduced in Section [rwell]. Recall that

$$\begin{aligned} \label{e17.58a} j_l(z) &= z^{\,l}\left(-\frac{1}{z}\frac{d}{dz}\right)^l\left( \frac{\sin z}{z}\right), \\[0.5ex]\label{e17.58b} y_l(z) &= -z^{\,l}\left(-\frac{1}{z}\frac{d}{dz}\right)^l \left(\frac{\cos z}{z}\right).\end{aligned}$$ Note that the $j_l(z)$ are well behaved in the limit $z\rightarrow 0$, whereas the $y_l(z)$ become singular. The asymptotic behavior of these functions in the limit $z\rightarrow \infty$ is

$$\begin{aligned} \label{e17.59a} j_l(z) &\rightarrow \frac{\sin(z - l\,\pi/2)}{z},\\[0.5ex] y_l(z) &\rightarrow - \frac{\cos(z-l\,\pi/2)}{z}.\label{e17.59b}\end{aligned}$$

We can write $$\exp(\,{\rm i}\,k\,r \cos\theta) = \sum_l a_l\, j_l(k\,r)\, P_l(\cos\theta),$$ where the $a_l$ are constants. Note there are no $y_l(k\,r)$ functions in this expression because they are not well-behaved as $r \rightarrow 0$. The Legendre functions are orthonormal ,

$$\label{e17.61} \int_{-1}^1 P_n(\mu) \,P_m(\mu)\,d\mu = \frac{\delta_{nm}}{n+1/2},$$ so we can invert the previous expansion to give $$a_l \,j_l(k\,r) = (l+1/2)\int_{-1}^1 \exp(\,{\rm i}\,k\,r \,\mu) \,P_l(\mu) \,d\mu.$$ It is well known that $$j_l(y) = \frac{(-{\rm i})^{\,l}}{2} \int_{-1}^1 \exp(\,{\rm i}\, y\,\mu) \,P_l(\mu)\,d\mu,$$ where $l=0, 1, 2, \cdots$. Thus, $$a_l = {\rm i}^{\,l} \,(2\,l+1),$$ giving

$$\label{e15.49} \psi_0({\bf r}) = \sqrt{n}\,\exp(\,{\rm i}\,k\,r \cos\theta) =\sqrt{n}\, \sum_l {\rm i}^{\,l}\,(2\,l+1)\, j_l(k\,r)\, P_l(\cos\theta).$$ The previous expression tells us how to decompose the incident plane-wave into a series of spherical waves. These waves are usually termed “partial waves”.

The most general expression for the total wavefunction outside the scattering region is $$\psi({\bf r}) = \sqrt{n}\sum_l\left[ A_l\,j_l(k\,r) + B_l\,y_l(k\,r)\right] P_l(\cos\theta),$$ where the $A_l$ and $B_l$ are constants. Note that the $y_l(k\,r)$ functions are allowed to appear in this expansion because its region of validity does not include the origin. In the large-$r$ limit, the total wavefunction reduces to $$\psi ({\bf r} ) \simeq \sqrt{n} \sum_l\left[A_l\, \frac{\sin(k\,r - l\,\pi/2)}{k\,r} - B_l\,\frac{\cos(k\,r -l\,\pi/2)}{k\,r} \right] P_l(\cos\theta),$$ where use has been made of Equations ([e17.59a]) and ([e17.59b]). The previous expression can also be written

$$\label{e17.68} \psi ({\bf r} ) \simeq \sqrt{n} \sum_l C_l\, \frac{\sin(k\,r - l\,\pi/2+ \delta_l)}{k\,r}\, P_l(\cos\theta),$$ where the sine and cosine functions have been combined to give a sine function which is phase-shifted by $\delta_l$. Note that $A_l=C_l\,\cos\delta_l$ and $B_l=-C_l\,\sin\delta_l$.

Equation ([e17.68]) yields $$\psi({\bf r}) \simeq \sqrt{n} \sum_l C_l\left[ \frac{{\rm e}^{\,{\rm i}\,(k\,r - l\,\pi/2+ \delta_l)} -{\rm e}^{-{\rm i}\,(k\,r - l\,\pi/2+ \delta_l)} }{2\,{\rm i}\,k\,r} \right] P_l(\cos\theta),\label{e17.69}$$ which contains both incoming and outgoing spherical waves. What is the source of the incoming waves? Obviously, they must be part of the large-$r$ asymptotic expansion of the incident wavefunction. In fact, it is easily seen from Equations ([e17.59a]) and ([e15.49]) that

$$\psi_0({\bf r}) \simeq \sqrt{n} \sum_l {\rm i}^{\,l}\, (2l+1)\left[\frac{ {\rm e}^{\,{\rm i}\,(k\,r - l\,\pi/2)} -{\rm e}^{-{\rm i}\,(k\,r - l\,\pi/2)}}{2\,{\rm i}\,k\,r} \right]P_l(\cos\theta)\label{e17.70}$$ in the large-$r$ limit. Now, Equations ([e17.52]) and ([e17.53]) give

$$\label{e17.71} \frac{\psi({\bf r} )- \psi_0({\bf r}) }{ \sqrt{n}} = \frac{\exp(\,{\rm i}\,k\,r)}{r}\, f(\theta).$$ Note that the right-hand side consists of an outgoing spherical wave only. This implies that the coefficients of the incoming spherical waves in the large-$r$ expansions of $\psi({\bf r})$ and $\psi_0({\bf r})$ must be the same. It follows from Equations ([e17.69]) and ([e17.70]) that $$C_l = (2\,l+1)\,\exp[\,{\rm i}\,(\delta_l + l\,\pi/2)].$$ Thus, Equations ([e17.69])–([e17.71]) yield

$$\label{e17.73} f(\theta) = \sum_{l=0,\infty} (2\,l+1)\,\frac{\exp(\,{\rm i}\,\delta_l)} {k} \,\sin\delta_l\,P_l(\cos\theta).$$ Clearly, determining the scattering amplitude, $f(\theta)$, via a decomposition into partial waves (i.e., spherical waves) is equivalent to determining the phase-shifts, $\delta_l$.

Now, the differential scattering cross-section, $d\sigma/d{\mit\Omega}$, is simply the modulus squared of the scattering amplitude, $f(\theta)$. [See Equation ([e15.17]).] The total cross-section is thus given by $$\begin{aligned} \sigma_{\rm total}& = \int |f(\theta)|^{\,2}\,d{\mit\Omega}\\[0.5ex] &= \frac{1}{k^{\,2}} \oint d\phi \int_{-1}^{1} d\mu \sum_l \sum_{l'} (2\,l+1)\,(2\,l'+1) \exp[\,{\rm i}\,(\delta_l-\delta_{l'})]\, \sin\delta_l \,\sin\delta_{l'}\, P_l(\mu)\, P_{l'}(\mu),\nonumber\end{aligned}$$ where $\mu = \cos\theta$. It follows that $$\label{e17.75} \sigma_{\rm total} = \frac{4\pi}{k^{\,2}} \sum_l (2\,l+1)\,\sin^2\delta_l,$$ where use has been made of Equation ([e17.61]).

Contributors and Attributions

• Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

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