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9.4: Partial Waves

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  • 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.,


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



    It follows that neither the incident wavefunction,


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



    nor the total wavefunction,


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


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


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


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



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


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


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


    $\displaystyle j_l(y)$ $\displaystyle \rightarrow \frac{\sin(y - l\,\pi/2)}{y},$ (964)
    $\displaystyle \eta_l(y)$ $\displaystyle \rightarrow - \frac{\cos(y-l\,\pi/2)}{y}.$ (965)



    We can write


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


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



    so we can invert the above expansion to give


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



    It is well-known that


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



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


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





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


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


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



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


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


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


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



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


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


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



    Thus, Equations (975)-(977) yield


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