9.4: 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 \( {\bf k}'\) 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 \( \varphi\) about the \( V({\bf x}) = V(r)\) ] the scattering amplitude is a function of \( f(\theta, \varphi) = f(\theta).\) \ref{955}

It follows that neither the incident wavefunction,

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

depend on the azimuthal angle \( \phi({\bf x})\) and \( (\nabla^2 + k^2)\,\psi = 0.\) \ref{958}

Consider the most general solution to this equation in spherical polar coordinates that does not depend on the azimuthal angle \( \psi(r,\theta) = \sum_{l=0,\infty} R_l(r)\, P_l(\cos\theta),\) \ref{959}

since the Legendre polynomials \( \theta\) -space. The Legendre polynomials are related to the spherical harmonics introduced in Chapter 4 via

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

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

\( = y^l\left(-\frac{1}{y}\frac{d}{dy}\right)^l \frac{\sin y}{y},\) \ref{962} \( = -y^l\left(-\frac{1}{y}\frac{d}{dy}\right)^l \frac{\cos y}{y}.\) \ref{963}

Note that spherical Bessel functions are well-behaved in the limit \( y\rightarrow \infty\) is

\( \rightarrow \frac{\sin(y - l\,\pi/2)}{y},\) \ref{964} \( \rightarrow - \frac{\cos(y-l\,\pi/2)}{y}.\) \ref{965}

We can write

\( a_l\) are constants. Note there are no Neumann functions in this expansion, because they are not well-behaved as \( \int_{-1}^1 d\mu\,P_n(\mu) \,P_m(\mu) = \frac{\delta_{n\,m}}{n+1/2},\) \ref{967}

so we can invert the above expansion to give

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

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

giving

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

where the \( B_l\) 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

\ref{973}

where use has been made of Equations \ref{964}-\ref{965}. The above expression can also be written

\( \delta_l\) .

Equation \ref{974} yields

\ref{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

\ref{976}

in the large-\( r\) limit. Now, Equations \ref{956} and \ref{957} give

\( r\) expansions of \( \phi({\bf x})\) must be equal. It follows from Equations \ref{975} and \ref{976} that

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

Clearly, determining the scattering amplitude \( \delta_l\) .

Contributors

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

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