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 $\phi$ about the $z$-axis. Equations (1248) and (1249) 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: i.e.,

$$f(\theta, \phi) = f(\theta).$$ (1261)

It follows that neither the incident wave-function,

$$\psi_0({\bf r}) = \sqrt{n}\,\exp(\,{\rm i}\,k\,z)= \sqrt{n}\,\exp(\,{\rm i}\,k\,r\cos\theta),$$ (1262)

nor the large-$r$ form of the total wave-function,

$$\psi({\bf r}) = \sqrt{n} \left[ \exp(\,{\rm i}\,k\,r\cos\theta) + \frac{\exp(\,{\rm i}\,k\,r)\, f(\theta)} {r} \right],$$ (1263)

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

$$(\nabla^2 + k^2)\,\psi = 0.$$ (1264)

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

$$\psi(r,\theta) = \sum_l R_l(r)\, P_l(\cos\theta),$$ (1265)

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. 8, via

$$P_l(\cos\theta) = \sqrt{\frac{4\pi}{2\,l+1}}\, Y_{l,0}(\theta,\varphi).$$ (1266)

Equations (1264) and (1265) can be combined to give

$$r^2\,\frac{d^2 R_l}{dr^2} + 2\,r \,\frac{dR_l}{dr} + [k^2 \,r^2 - l\,(l+1)]R_l = 0.$$ (1267)

The two independent solutions to this equation are the spherical Bessel functions, $j_l(k\,r)$ and $y_l(k\,r)$, introduced in Sect. 9.3. Recall that

$$j_l(z) = z^l\left(-\frac{1}{z}\frac{d}{dz}\right)^l\left(\frac{\sin z}{z}\right),$$ (1268)

$$y_l(z) = -z^l\left(-\frac{1}{z}\frac{d}{dz}\right)^l\left(\frac{\cos z}{z}\right).$$ (1269)

Note that the $j_l(z)$ are well-behaved in the limit $z\rightarrow 0$ , whereas the $y_l(z)$ become singular. The asymptotic behaviour of these functions in the limit $z\rightarrow\infty$ is

$$j_l(z) \rightarrow \frac{\sin(z - l\,\pi/2)}{z},$$ (1270)

$$y_l(z) \rightarrow - \frac{\cos(z-l\,\pi/2)}{z}.$$ (1271)

We can write

$$\exp(\,{\rm i}\,k\,r \cos\theta) = \sum_l a_l\, j_l(k\,r)\, P_l(\cos\theta),$$ (1272)

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,

$$\int_{-1}^1 P_n(\mu) \,P_m(\mu)\,d\mu = \frac{\delta_{nm}}{n+1/2},$$ (1273)

so we can invert the above expansion to give

$$a_l \,j_l(k\,r) = (l+1/2)\int_{-1}^1 \exp(\,{\rm i}\,k\,r \,\mu) \,P_l(\mu) \,d\mu.$$ (1274)

It is well-known that

$$j_l(y) = \frac{(-{\rm i})^l}{2} \int_{-1}^1 \exp(\,{\rm i}\, y\,\mu) \,P_l(\mu)\,d\mu,$$ (1275)

where $l=0, 1, 2, \cdots$ [see M. Abramowitz and I.A. Stegun, Handbook of mathematical functions, (Dover, New York NY, 1965), Eq. 10.1.14]. Thus,

$$a_l = {\rm i}^{\,l} \,(2\,l+1),$$ (1276)

giving

$$\psi_0({\bf r}) = \sqrt{n}\,\exp(\,{\rm i}\,k\,r \cos\theta)... ... \sum_l {\rm i}^{\,l}\,(2\,l+1)\, j_l(k\,r)\, P_l(\cos\theta).$$ (1277)

The above 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 wave-function outside the scattering region is

$$\psi({\bf r}) = \sqrt{n}\sum_l\left[ A_l\,j_l(k\,r) + B_l\,y_l(k\,r)\right] P_l(\cos\theta),$$ (1278)

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 wave-function reduces to

$$\psi ({\bf r} ) \simeq \sqrt{n} \sum_l\left[A_l\, \frac{\sin... ..._l\,\frac{\cos(k\,r -l\,\pi/2)}{k\,r} \right] P_l(\cos\theta),$$ (1279)

where use has been made of Eqs. (1270) and (1271). The above expression can also be written

$$\psi ({\bf r} ) \simeq \sqrt{n} \sum_l C_l\, \frac{\sin(k\,r - l\,\pi/2+ \delta_l)}{k\,r}\, P_l(\cos\theta),$$ (1280)

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 (1280) yields

$$\psi({\bf r}) \simeq \sqrt{n} \sum_l C_l\left[ \frac{{\rm e}... ...\pi/2+ \delta_l)} }{2\,{\rm i}\,k\,r} \right] P_l(\cos\theta),$$ (1281)

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 from Eqs. (1270) and (1277) that

$$\psi_0({\bf r}) \simeq \sqrt{n} \sum_l {\rm i}^{\,l}\, (2l+1... ...\,(k\,r - l\,\pi/2)}}{2\,{\rm i}\,k\,r} \right]P_l(\cos\theta)$$ (1282)

in the large-$r$ limit. Now, Eqs. (1262) and (1263) give

$$\frac{\psi({\bf r} )- \psi_0({\bf r}) }{ \sqrt{n}} = \frac{\exp(\,{\rm i}\,k\,r)}{r}\, f(\theta).$$ (1283)

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 Eqs. (1281) and (1282) that

$$C_l = (2\,l+1)\,\exp[\,{\rm i}\,(\delta_l + l\,\pi/2)].$$ (1284)

Thus, Eqs. (1281)-(1283) yield

$$f(\theta) = \sum_{l=0}^\infty (2\,l+1)\,\frac{\exp(\,{\rm i}\,\delta_l)} {k} \,\sin\delta_l\,P_l(\cos\theta).$$ (1285)

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 $d\sigma/d\Omega$ is simply the modulus squared of the scattering amplitude $f(\theta)$ [see Eq. (1245)]. The total cross-section is thus given by

$$\sigma_{\rm total} = \int \vert f(\theta)\vert^2\,d\Omega$$

$$= \frac{1}{k^2} \oint d\phi \int_{-1}^{1} d\mu \sum_l \sum_{l'} (2\,l+1)\,(2\,l'+1) \exp[\,{\rm i}\,(\delta_l-\delta_{l'})]$$

$$\mbox{\hspace{1cm}}\times \sin\delta_l \,\sin\delta_{l'}\, P_l(\mu)\, P_{l'}(\mu),$$ (1286)

where $\mu = \cos\theta$. It follows that

$$\sigma_{\rm total} = \frac{4\pi}{k^2} \sum_l (2\,l+1)\,\sin^2\delta_l,$$ (1287)

where use has been made of Eq. (1273).