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 $\varphi$ about the $z$ -axis. Equation (943) strongly suggests that for a spherically symmetric scattering potential [i.e., $V({\bf x}) = V(r)$ ] the scattering amplitude is a function of $\theta$ only: i.e.,

$$f(\theta, \varphi) = f(\theta).$$ (955)

It follows that neither the incident wavefunction,

$$\phi({\bf x}) = \frac{\exp(\,{\rm i}\,k\,z)}{(2\pi)^{3/2}}= \frac{\exp(\,{\rm i}\,k\,r\cos\theta)}{(2\pi)^{3/2}},$$ (956)

nor the total wavefunction,

$$\psi({\bf x}) = \frac{1}{(2\pi)^{3/2}} \left[ \exp(\,{\rm i}\,k\,r\cos\theta) + \frac{\exp(\,{\rm i}\,k\,r)\, f(\theta)} {r} \right],$$ (957)

depend on the azimuthal angle $\varphi$ .

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

$$(\nabla^2 + k^2)\,\psi = 0.$$ (958)

Consider the most general solution to this equation in spherical polar coordinates that does not depend on the azimuthal angle $\varphi$ . Separation of variables yields

$$\psi(r,\theta) = \sum_{l=0,\infty} R_l(r)\, P_l(\cos\theta),$$ (959)

since the Legendre polynomials $P_l(\cos\theta)$ form a complete set in $\theta$ -space. The Legendre polynomials are related to the spherical harmonics introduced in Chapter 4 via

$$P_l(\cos\theta) = \sqrt{\frac{4\pi}{2\,l+1}}\, Y_{l\,0}(\theta,\varphi).$$ (960)

Equations (958) and (959) can be combined to give

$$r^2\frac{d^2 R_l}{dr^2} + 2\,r \frac{dR_l}{dr} + [k^2 \,r^2 - l\,(l+1)]\,R_l = 0.$$ (961)

The two independent solutions to this equation are the spherical Bessel function, $j_l(k\,r)$ , and the Neumann function, $\eta_l(k\,r)$ , where

$$j_l(y) = y^l\left(-\frac{1}{y}\frac{d}{dy}\right)^l \frac{\sin y}{y},$$ (962)

$$\eta_l(y) = -y^l\left(-\frac{1}{y}\frac{d}{dy}\right)^l \frac{\cos y}{y}.$$ (963)

Note that spherical Bessel functions are well-behaved in the limit $y\rightarrow 0$ , whereas Neumann functions become singular. The asymptotic behaviour of these functions in the limit $y\rightarrow \infty$ is

$$j_l(y) \rightarrow \frac{\sin(y - l\,\pi/2)}{y},$$ (964)

$$\eta_l(y) \rightarrow - \frac{\cos(y-l\,\pi/2)}{y}.$$ (965)

We can write

$$\exp(\,{\rm i}\,k\,r \cos\theta) = \sum_{l=0,\infty} a_l\, j_l(k\,r)\, P_l(\cos\theta),$$ (966)

where the $a_l$ are constants. Note there are no Neumann functions in this expansion, because they are not well-behaved as $r\rightarrow 0$ . The Legendre polynomials are orthogonal,

$$\int_{-1}^1 d\mu\,P_n(\mu) \,P_m(\mu) = \frac{\delta_{n\,m}}{n+1/2},$$ (967)

so we can invert the above expansion to give

$$a_l \,j_l(k\,r) = (l+1/2)\int_{-1}^1 d\mu\,\exp(\,{\rm i}\,k\,r \,\mu) \,P_l(\mu).$$ (968)

It is well-known that

$$j_l(y) = \frac{(-{\rm i})^l}{2} \int_{-1}^1 d\mu\, \exp(\,{\rm i}\, y\,\mu) \,P_l(\mu),$$ (969)

where $l=0, \infty$ . Thus,

$$a_l = {\rm i}^l \,(2\,l+1),$$ (970)

giving

$$\exp(\,{\rm i}\,k\,r \cos\theta) = \sum_{l=0,\infty} {\rm i}^l\, (2\,l+1)\, j_l(k\,r)\, P_l(\cos\theta).$$ (971)

The above expression tells us how to decompose a plane wave into a series of spherical waves (or ``partial waves'').

The most general solution for the total wavefunction outside the scattering region is

$$\psi({\bf x}) = \frac{1}{(2\pi)^{3/2}} \sum_{l=0,\infty}\left[ A_l\,j_l(k\,r) + B_l\,\eta_l(k\,r)\right] P_l(\cos\theta),$$ (972)

where the $A_l$ and $B_l$ are constants. Note that the Neumann 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

$$\psi ({\bf x} ) \simeq \frac{1}{(2\pi)^{3/2}} \sum_{l=0,\infty}\l... ...\pi/2)}{k\,r} - B_l\,\frac{\cos(k\,r -l\,\pi/2)}{k\,r} \right] P_l(\cos\theta),$$ (973)

where use has been made of Equations (964)-(965). The above expression can also be written

$$\psi ({\bf x} ) \simeq \frac{1}{(2\pi)^{3/2}} \sum_l C_l\, \frac{\sin(k\,r - l\,\pi/2+ \delta_l)}{k\,r}\, P_l(\cos\theta),$$ (974)

where the sine and cosine functions have been combined to give a sine function that is phase-shifted by $\delta_l$ .

Equation (974) yields

$$\psi({\bf x}) \simeq \frac{1}{(2\pi)^{3/2}} \sum_l C_l\, \frac{\e... ...p[-{\rm i}\,(k\,r - l\,\pi/2+ \delta_l)] }{2\,{\rm i}\,k\,r}\, P_l(\cos\theta),$$ (975)

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 that

$$\phi({\bf x}) \simeq \frac{1}{(2\pi)^{3/2}} \sum_{l=0,\infty} {\r... ...i/2)] -\exp[-{\rm i}\,(k\,r - l\,\pi/2)]}{2\,{\rm i}\,k\,r} \, P_l(\cos\theta),$$ (976)

in the large-$r$ limit. Now, Equations (956) and (957) give

$$(2\pi)^{3/2}[\psi({\bf x} )- \phi({\bf x}) ] = \frac{\exp(\,{\rm i}\,k\,r)}{r}\, f(\theta).$$ (977)

Note that the right-hand side consists only of an outgoing spherical wave. This implies that the coefficients of the incoming spherical waves in the large-$r$ expansions of $\psi({\bf x})$ and $\phi({\bf x})$ must be equal. It follows from Equations (975) and (976) that

$$C_l = (2\,l+1)\,\exp[\,{\rm i}\,(\delta_l + l\,\pi/2)].$$ (978)

Thus, Equations (975)-(977) yield

$$f(\theta) = \sum_{l=0,\infty} (2\,l+1)\,\frac{\exp(\,{\rm i}\,\delta_l)} {k} \,\sin\delta_l\,P_l(\cos\theta).$$ (979)

Clearly, determining the scattering amplitude $f(\theta)$ via a decomposition into partial waves (i.e., spherical waves) is equivalent to determining the phase-shifts $\delta_l$ .