Partial waves

We can assume, without loss of generality, that the incident wave-function is characterized by a wave-vector ${\bf k}$ which is aligned parallel to the $z$-axis. The scattered wave-function is characterized by a wave-vector ${\bf k}'$ which 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 wave-vectors), and an azimuthal angle $\varphi$ about the $z$-axis. Equation (928) strongly suggests that for a spherically symmetric scattering potential [i.e., $V({\bf r}) = V(r)$] the scattering amplitude is a function of $\theta$ only:

$$f(\theta, \varphi) = f(\theta).$$ (941)

It follows that neither the incident wave-function,

$$\phi({\bf r}) = \frac{\exp( {\rm i} k z)}{(2\pi)^{3/2}}= \frac{\exp( {\rm i} k r\cos\theta)}{(2\pi)^{3/2}},$$ (942)

nor the total wave-function,

$$\psi({\bf r}) = \frac{1}{(2\pi)^{3/2}} \left[ \exp( {\rm i}... ...theta) + \frac{\exp( {\rm i} k r) f(\theta)} {r} \right],$$ (943)

depend on the azimuthal angle $\varphi$.

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

$$(\nabla^2 + k^2) \psi = 0.$$ (944)

What is the most general solution to this equation in spherical polar coordinates which does not depend on the azimuthal angle $\varphi$? Separation of variables yields

$$\psi(r,\theta) = \sum_l R_l(r) P_l(\cos\theta),$$ (945)

since 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 Sect. 5 via

$$P_l(\cos\theta) = \sqrt{\frac{4\pi}{2 l+1}} Y_l^0(\theta,\varphi).$$ (946)

Equations (944) and (945) can be combined to give

$$r^2\frac{d^2 R_l}{dr^2} + 2 r \frac{dR_l}{dr} + [k^2 r^2 - l (l+1)]R_l = 0.$$ (947)

The two independent solutions to this equation are called a spherical Bessel function, $j_l(k r)$, and a Neumann function, $\eta_l(k r)$. It is easily demonstrated that

$$j_l(y) = y^l\left(-\frac{1}{y}\frac{d}{dy}\right)^l \frac{\sin y}{y},$$ (948)

$$\eta_l(y) = -y^l\left(-\frac{1}{y}\frac{d}{dy}\right)^l \frac{\cos y}{y}.$$ (949)

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

$$j_l(y) \rightarrow \frac{\sin(y - l \pi/2)}{y},$$ (950)

$$\eta_l(y) \rightarrow - \frac{\cos(y-l \pi/2)}{y}.$$ (951)

We can write

$$\exp( {\rm i} k r \cos\theta) = \sum_l a_l j_l(k r) P_l(\cos\theta),$$ (952)

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 functions are orthonormal,

$$\int_{-1}^1 P_n(\mu) P_m(\mu) d\mu = \frac{\delta_{nm}}{n+1/2},$$ (953)

so we can invert the above expansion to give

$$a_l j_l(k r) = (l+1/2)\int_{-1}^1 \exp( {\rm i} k r \mu) P_l(\mu) d\mu.$$ (954)

It is well-known that

$$j_l(y) = \frac{(-{\rm i})^l}{2} \int_{-1}^1 \exp( {\rm i} y \mu) P_l(\mu) d\mu,$$ (955)

where $l=0, 1, 2, \cdots$ [see Abramowitz and Stegun (Dover, New York NY, 1965), Eq. 10.1.14]. Thus,

$$a_l = {\rm i}^l (2 l+1),$$ (956)

giving

$$\exp( {\rm i} k r \cos\theta) = \sum_l {\rm i}^l (2 l+1) j_l(k r) P_l(\cos\theta).$$ (957)

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 wave-function outside the scattering region is

$$\psi({\bf r}) = \frac{1}{(2\pi)^{3/2}} \sum_l\left[ A_l j_l(k r) + B_l \eta_l(k r)\right] P_l(\cos\theta),$$ (958)

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 wave-function reduces to

$$\psi ({\bf r} ) \simeq \frac{1}{(2\pi)^{3/2}} \sum_l\left[A_... ..._l \frac{\cos(k r -l \pi/2)}{k r} \right] P_l(\cos\theta),$$ (959)

where use has been made of Eqs. (950)-(951). The above expression can also be written

$$\psi ({\bf r} ) \simeq \frac{1}{(2\pi)^{3/2}} \sum_l C_l \frac{\sin(k r - l \pi/2+ \delta_l)}{k r} P_l(\cos\theta),$$ (960)

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

Equation (960) yields

$$\psi({\bf r}) \simeq \frac{1}{(2\pi)^{3/2}} \sum_l C_l \frac{\exp[ {\rm i} (k r -... .../2+ \delta_l)] -\exp[-{\rm i} (k r - l \pi/2+ \delta_l)] }{2 {\rm i} k r}$$

$$\mbox{\hspace{2cm}}\times P_l(\cos\theta),$$ (961)

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 wave-function. In fact, it is easily seen that

$$\phi({\bf r}) \simeq \frac{1}{(2\pi)^{3/2}} \sum_l {\rm i}^l (2l+1) \frac{ \exp[ {\rm i} (k r - l \pi/2)] -\exp[-{\rm i} (k r - l \pi/2)]}{2 {\rm i} k r}$$

$$\mbox{\hspace{2cm}}\times P_l(\cos\theta)$$ (962)

in the large-$r$ limit. Now, Eqs. (942) and (943) give

$$(2\pi)^{3/2}[\psi({\bf r} )- \phi({\bf r}) ] = \frac{\exp( {\rm i} k r)}{r} f(\theta).$$ (963)

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 $\psi({\bf r})$ and $\phi({\bf r})$ must be equal. It follows from Eqs. (961) and (962) that

$$C_l = (2 l+1) \exp[ {\rm i} (\delta_l + l \pi/2)].$$ (964)

Thus, Eqs. (961)-(963) yield

$$f(\theta) = \sum_{l=0}^\infty (2 l+1) \frac{\exp( {\rm i} \delta_l)} {k} \sin\delta_l P_l(\cos\theta).$$ (965)

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