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<title>Partial-wave analysis</title>

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Partial-wave analysis

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Partial-wave analysis, in the context of <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, refers to a technique for solving <a href="Scattering" title="Scattering">scattering</a> problems by decomposing each wave into its constituent <a href="Angular_momentum" title="Angular momentum">angular-momentum</a> components and solving using <a href="Boundary_condition" class="mw-redirect" title="Boundary condition">boundary conditions</a>. Partial wave analysis is typically useful for low energy scattering where only a few angular momentum components dominate. At high energy were scattering is weak, an alternative called the <a href="Born_approximation" title="Born approximation">Born approximation</a> is used.<a href="#cite_note-Griffiths-1">[1]</a>: 507

<div class="mw-heading mw-heading2"><h2 id="Preliminary_scattering_theory">Preliminary scattering theory</h2></div>

A steady beam of particles scatters off a spherically symmetric potential <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(r)}">

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<annotation encoding="application/x-tex">{\displaystyle V(r)}</annotation>

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</math><img src="./114fdc48547ee60d02d7a2f4765d52a6ee3507d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.645ex; height:2.843ex;" alt="{\displaystyle V(r)}" loading="lazy">, which is short-ranged, so that for large distances <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\to \infty }">

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<annotation encoding="application/x-tex">{\displaystyle r\to \infty }</annotation>

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</math><img src="./dcd3a85ea2e3d6b4027434e502cace4177d7a3e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.986ex; height:1.843ex;" alt="{\displaystyle r\to \infty }" loading="lazy">, the particles behave like free particles. The incoming beam is assumed to be a collimated <a href="Plane_wave" title="Plane wave">plane wave</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(ikz)}">

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<annotation encoding="application/x-tex">{\displaystyle \exp(ikz)}</annotation>

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</math><img src="./85a9440ee17dcc882321a58a3ce031cd321272a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.464ex; height:2.843ex;" alt="{\displaystyle \exp(ikz)}" loading="lazy"> traveling along the z axis. Because the beam is switched on for times long compared to the time of interaction of the particles with the scattering potential, a steady state is assumed. This means that the stationary Schrödinger equation for the wave function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi (\mathbf {r} )}">

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<annotation encoding="application/x-tex">{\displaystyle \Psi (\mathbf {r} )}</annotation>

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</math><img src="./e478d6f260286a13b6516ffecb4787fb0b2aaae5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.72ex; height:2.843ex;" alt="{\displaystyle \Psi (\mathbf {r} )}" loading="lazy"> representing the particle beam should be solved:

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<annotation encoding="application/x-tex">{\displaystyle \left[-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(r)\right]\Psi (\mathbf {r} )=E\Psi (\mathbf {r} ).}</annotation>

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</math><img src="./ede16cf59dfb186225a8faaeb91fa2ca9346824c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.125ex; height:6.343ex;" alt="{\displaystyle \left[-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(r)\right]\Psi (\mathbf {r} )=E\Psi (\mathbf {r} ).}" loading="lazy"></dd></dl>

We make the following <a href="Ansatz" title="Ansatz">ansatz</a>:

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<annotation encoding="application/x-tex">{\displaystyle \Psi (\mathbf {r} )=\Psi _{0}(\mathbf {r} )+\Psi _{\text{s}}(\mathbf {r} ),}</annotation>

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</math><img src="./7ac47491e7abe11bab54149988dbb6c31237038f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.679ex; height:2.843ex;" alt="{\displaystyle \Psi (\mathbf {r} )=\Psi _{0}(\mathbf {r} )+\Psi _{\text{s}}(\mathbf {r} ),}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{0}(\mathbf {r} )\propto \exp(ikz)}">

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<annotation encoding="application/x-tex">{\displaystyle \Psi _{0}(\mathbf {r} )\propto \exp(ikz)}</annotation>

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</math><img src="./4c421c1bf87b19cf31603a61f6e212b45f8bac86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.336ex; height:2.843ex;" alt="{\displaystyle \Psi _{0}(\mathbf {r} )\propto \exp(ikz)}" loading="lazy"> is the incoming plane wave, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{\text{s}}(\mathbf {r} )}">

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<annotation encoding="application/x-tex">{\displaystyle \Psi _{\text{s}}(\mathbf {r} )}</annotation>

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</math><img src="./03f4c2531687a27e4d200123dd83062b865876da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.6ex; height:2.843ex;" alt="{\displaystyle \Psi _{\text{s}}(\mathbf {r} )}" loading="lazy"> is a scattered part perturbing the original wave function.

It is the asymptotic form of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{\text{s}}(\mathbf {r} )}">

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<annotation encoding="application/x-tex">{\displaystyle \Psi _{\text{s}}(\mathbf {r} )}</annotation>

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</math><img src="./03f4c2531687a27e4d200123dd83062b865876da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.6ex; height:2.843ex;" alt="{\displaystyle \Psi _{\text{s}}(\mathbf {r} )}" loading="lazy"> that is of interest, because observations near the scattering center (e.g. an atomic nucleus) are mostly not feasible, and detection of particles takes place far away from the origin. At large distances, the particles should behave like free particles, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{\text{s}}(\mathbf {r} )}">

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<annotation encoding="application/x-tex">{\displaystyle \Psi _{\text{s}}(\mathbf {r} )}</annotation>

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</math><img src="./03f4c2531687a27e4d200123dd83062b865876da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.6ex; height:2.843ex;" alt="{\displaystyle \Psi _{\text{s}}(\mathbf {r} )}" loading="lazy"> should therefore be a solution to the free Schrödinger equation. For a spherically symmetric potential, these solutions should be outgoing spherical waves,<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{\text{s}}(\mathbf {r} )\propto \exp(ikr)/r}">

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<annotation encoding="application/x-tex">{\displaystyle \Psi _{\text{s}}(\mathbf {r} )\propto \exp(ikr)/r}</annotation>

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</math><img src="./a5905576ef04be4c087583bbb51ac5fa31d88971.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.333ex; height:2.843ex;" alt="{\displaystyle \Psi _{\text{s}}(\mathbf {r} )\propto \exp(ikr)/r}" loading="lazy"> at large distances. Thus the asymptotic form of the scattered wave is chosen as<a href="#cite_note-Messiah-1976-2">[2]</a>: 371

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<annotation encoding="application/x-tex">{\displaystyle \Psi _{\text{s}}(\mathbf {r} )\to f(\theta ,k){\frac {\exp(ikr)}{r}},}</annotation>

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</math><img src="./2fad6be75cdee7c68c3a42d4f447d940979c5a5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.544ex; height:5.676ex;" alt="{\displaystyle \Psi _{\text{s}}(\mathbf {r} )\to f(\theta ,k){\frac {\exp(ikr)}{r}},}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\theta ,k)}">

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<annotation encoding="application/x-tex">{\displaystyle f(\theta ,k)}</annotation>

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</math><img src="./2fe54082e5f44d7ae2868a6010baf635da169f90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.424ex; height:2.843ex;" alt="{\displaystyle f(\theta ,k)}" loading="lazy"> is the so-called scattering amplitude, which is in this case only dependent on the elevation angle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

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<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>

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</math><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"> and the energy.

This gives the following asymptotic expression for the entire wave function:

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<annotation encoding="application/x-tex">{\displaystyle \Psi (\mathbf {r} )\to \Psi ^{(+)}(\mathbf {r} )=\exp(ikz)+f(\theta ,k){\frac {\exp(ikr)}{r}}.}</annotation>

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</math><img src="./5cf25479d896bbcb26d59cdbfd71270491e31deb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:46.576ex; height:5.676ex;" alt="{\displaystyle \Psi (\mathbf {r} )\to \Psi ^{(+)}(\mathbf {r} )=\exp(ikz)+f(\theta ,k){\frac {\exp(ikr)}{r}}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Partial-wave_expansion">Partial-wave expansion</h2></div>

In case of a spherically symmetric potential <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(\mathbf {r} )=V(r)}">

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<annotation encoding="application/x-tex">{\displaystyle V(\mathbf {r} )=V(r)}</annotation>

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</math><img src="./1c3aab7c41eefb7b30e12c0245bcbc31e84d141c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.442ex; height:2.843ex;" alt="{\displaystyle V(\mathbf {r} )=V(r)}" loading="lazy">, the scattering wave function may be expanded in <a href="Spherical_harmonic" class="mw-redirect" title="Spherical harmonic">spherical harmonics</a>, which reduce to <a href="Legendre_polynomial" class="mw-redirect" title="Legendre polynomial">Legendre polynomials</a> because of azimuthal symmetry (no dependence on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">

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<annotation encoding="application/x-tex">{\displaystyle \phi }</annotation>

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</math><img src="./72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy">):

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<annotation encoding="application/x-tex">{\displaystyle \Psi (\mathbf {r} )=\sum _{\ell =0}^{\infty }{\frac {u_{\ell }(r)}{r}}P_{\ell }(\cos \theta ).}</annotation>

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</math><img src="./285a5cb14ad909b018963a5e4d4f9ca446b8e0da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.957ex; height:7.009ex;" alt="{\displaystyle \Psi (\mathbf {r} )=\sum _{\ell =0}^{\infty }{\frac {u_{\ell }(r)}{r}}P_{\ell }(\cos \theta ).}" loading="lazy"></dd></dl>

In the standard scattering problem, the incoming beam is assumed to take the form of a plane wave of wave number k, which can be decomposed into partial waves using the <a href="Plane-wave_expansion" title="Plane-wave expansion">plane-wave expansion</a> in terms of <a href="Spherical_Bessel_function" class="mw-redirect" title="Spherical Bessel function">spherical Bessel functions</a> and <a href="Legendre_polynomial" class="mw-redirect" title="Legendre polynomial">Legendre polynomials</a>:

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<annotation encoding="application/x-tex">{\displaystyle \psi _{\text{in}}(\mathbf {r} )=e^{ikz}=\sum _{\ell =0}^{\infty }(2\ell +1)i^{\ell }j_{\ell }(kr)P_{\ell }(\cos \theta ).}</annotation>

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</math><img src="./d0efc8de7600391927573b4330cf2dd1253bcaba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:44.154ex; height:7.009ex;" alt="{\displaystyle \psi _{\text{in}}(\mathbf {r} )=e^{ikz}=\sum _{\ell =0}^{\infty }(2\ell +1)i^{\ell }j_{\ell }(kr)P_{\ell }(\cos \theta ).}" loading="lazy"></dd></dl>

Here we have assumed a spherical coordinate system in which the z axis is aligned with the beam direction. The radial part of this wave function consists solely of the spherical Bessel function, which can be rewritten as a sum of two <a href="Spherical_Hankel_functions" class="mw-redirect" title="Spherical Hankel functions">spherical Hankel functions</a>:

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<annotation encoding="application/x-tex">{\displaystyle j_{\ell }(kr)={\frac {1}{2}}\left(h_{\ell }^{(1)}(kr)+h_{\ell }^{(2)}(kr)\right).}</annotation>

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</math><img src="./b84c9aee995e989450425decfe20a9dc096d67f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.027ex; width:33.589ex; height:5.176ex;" alt="{\displaystyle j_{\ell }(kr)={\frac {1}{2}}\left(h_{\ell }^{(1)}(kr)+h_{\ell }^{(2)}(kr)\right).}" loading="lazy"></dd></dl>

This has physical significance: hℓ(2) asymptotically (i.e. for large r) behaves as i−(ℓ+1)eikr/(kr) and is thus an outgoing wave, whereas hℓ(1) asymptotically behaves as iℓ+1e−ikr/(kr) and is thus an incoming wave. The incoming wave is unaffected by the scattering, while the outgoing wave is modified by a factor known as the partial-wave <a href="S-matrix" title="S-matrix">S-matrix</a> element Sℓ:

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<annotation encoding="application/x-tex">{\displaystyle {\frac {u_{\ell }(r)}{r}}{\stackrel {r\to \infty }{\longrightarrow }}{\frac {i^{\ell }k}{\sqrt {2\pi }}}\left(h_{\ell }^{(1)}(kr)+S_{\ell }h_{\ell }^{(2)}(kr)\right),}</annotation>

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</math><img src="./76a5a130926121ebae250e8d184d1ecfb405e4fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:40.099ex; height:6.676ex;" alt="{\displaystyle {\frac {u_{\ell }(r)}{r}}{\stackrel {r\to \infty }{\longrightarrow }}{\frac {i^{\ell }k}{\sqrt {2\pi }}}\left(h_{\ell }^{(1)}(kr)+S_{\ell }h_{\ell }^{(2)}(kr)\right),}" loading="lazy"></dd></dl>

where uℓ(r)/r is the radial component of the actual wave function. The scattering phase shift δℓ is defined as half of the phase of Sℓ:

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<annotation encoding="application/x-tex">{\displaystyle S_{\ell }=e^{2i\delta _{\ell }}.}</annotation>

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</math><img src="./a251abd68bb6d333aa66ec99aec510f5e95587ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.244ex; height:3.009ex;" alt="{\displaystyle S_{\ell }=e^{2i\delta _{\ell }}.}" loading="lazy"></dd></dl>

If flux is not lost, then |Sℓ| = 1, and thus the phase shift is real. This is typically the case, unless the potential has an imaginary absorptive component, which is often used in <a href="Phenomenological_model" title="Phenomenological model">phenomenological models</a> to simulate loss due to other reaction channels.

Therefore, the full asymptotic wave function is

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<annotation encoding="application/x-tex">{\displaystyle \psi (\mathbf {r} ){\stackrel {r\to \infty }{\longrightarrow }}\sum _{\ell =0}^{\infty }(2\ell +1)i^{\ell }{\frac {h_{\ell }^{(1)}(kr)+S_{\ell }h_{\ell }^{(2)}(kr)}{2}}P_{\ell }(\cos \theta ).}</annotation>

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</math><img src="./2d7d2a89725d67df531f1355d98287c7568b1078.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:52.817ex; height:7.843ex;" alt="{\displaystyle \psi (\mathbf {r} ){\stackrel {r\to \infty }{\longrightarrow }}\sum _{\ell =0}^{\infty }(2\ell +1)i^{\ell }{\frac {h_{\ell }^{(1)}(kr)+S_{\ell }h_{\ell }^{(2)}(kr)}{2}}P_{\ell }(\cos \theta ).}" loading="lazy"></dd></dl>

Subtracting ψin yields the asymptotic outgoing wave function:

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<annotation encoding="application/x-tex">{\displaystyle \psi _{\text{out}}(\mathbf {r} ){\stackrel {r\to \infty }{\longrightarrow }}\sum _{\ell =0}^{\infty }(2\ell +1)i^{\ell }{\frac {S_{\ell }-1}{2}}h_{\ell }^{(2)}(kr)P_{\ell }(\cos \theta ).}</annotation>

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</math><img src="./b4aea7fa1cb91d926e89f4707befa30805465cd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:48.846ex; height:7.009ex;" alt="{\displaystyle \psi _{\text{out}}(\mathbf {r} ){\stackrel {r\to \infty }{\longrightarrow }}\sum _{\ell =0}^{\infty }(2\ell +1)i^{\ell }{\frac {S_{\ell }-1}{2}}h_{\ell }^{(2)}(kr)P_{\ell }(\cos \theta ).}" loading="lazy"></dd></dl>

Making use of the asymptotic behavior of the spherical Hankel functions, one obtains

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<annotation encoding="application/x-tex">{\displaystyle \psi _{\text{out}}(\mathbf {r} ){\stackrel {r\to \infty }{\longrightarrow }}{\frac {e^{ikr}}{r}}\sum _{\ell =0}^{\infty }(2\ell +1){\frac {S_{\ell }-1}{2ik}}P_{\ell }(\cos \theta ).}</annotation>

</semantics>

</math><img src="./0c84b6a2ee6ac7b97c5691b6a861493e6b4a82f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:43.701ex; height:7.009ex;" alt="{\displaystyle \psi _{\text{out}}(\mathbf {r} ){\stackrel {r\to \infty }{\longrightarrow }}{\frac {e^{ikr}}{r}}\sum _{\ell =0}^{\infty }(2\ell +1){\frac {S_{\ell }-1}{2ik}}P_{\ell }(\cos \theta ).}" loading="lazy"></dd></dl>

Since the <a href="Scattering_amplitude" title="Scattering amplitude">scattering amplitude</a> f(θ, k) is defined from

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<annotation encoding="application/x-tex">{\displaystyle \psi _{\text{out}}(\mathbf {r} ){\stackrel {r\to \infty }{\longrightarrow }}{\frac {e^{ikr}}{r}}f(\theta ,k),}</annotation>

</semantics>

</math><img src="./e906d85794e5ef2b03b46450f2e448047c71c5fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.448ex; height:5.676ex;" alt="{\displaystyle \psi _{\text{out}}(\mathbf {r} ){\stackrel {r\to \infty }{\longrightarrow }}{\frac {e^{ikr}}{r}}f(\theta ,k),}" loading="lazy"></dd></dl>

it follows that<a href="#cite_note-Messiah-1976-2">[2]</a>: 386

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<annotation encoding="application/x-tex">{\displaystyle f(\theta ,k)=\sum _{\ell =0}^{\infty }(2\ell +1){\frac {S_{\ell }-1}{2ik}}P_{\ell }(\cos \theta )=\sum _{\ell =0}^{\infty }(2\ell +1){\frac {e^{i\delta _{\ell }}\sin \delta _{\ell }}{k}}P_{\ell }(\cos \theta ),}</annotation>

</semantics>

</math><img src="./359adc922223aebc470eee4785cbc9a4115a1864.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:70.414ex; height:7.009ex;" alt="{\displaystyle f(\theta ,k)=\sum _{\ell =0}^{\infty }(2\ell +1){\frac {S_{\ell }-1}{2ik}}P_{\ell }(\cos \theta )=\sum _{\ell =0}^{\infty }(2\ell +1){\frac {e^{i\delta _{\ell }}\sin \delta _{\ell }}{k}}P_{\ell }(\cos \theta ),}" loading="lazy"></dd></dl>

and thus the <a href="Differential_cross_section" class="mw-redirect" title="Differential cross section">differential cross section</a> is given by

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<annotation encoding="application/x-tex">{\displaystyle {\frac {d\sigma }{d\Omega }}=|f(\theta ,k)|^{2}={\frac {1}{k^{2}}}\left|\sum _{\ell =0}^{\infty }(2\ell +1)e^{i\delta _{\ell }}\sin \delta _{\ell }P_{\ell }(\cos \theta )\right|^{2}.}</annotation>

</semantics>

</math><img src="./d21fbdd150a6838d3d1c140f223c3f923f8417d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:53.816ex; height:7.843ex;" alt="{\displaystyle {\frac {d\sigma }{d\Omega }}=|f(\theta ,k)|^{2}={\frac {1}{k^{2}}}\left|\sum _{\ell =0}^{\infty }(2\ell +1)e^{i\delta _{\ell }}\sin \delta _{\ell }P_{\ell }(\cos \theta )\right|^{2}.}" loading="lazy"></dd></dl>

This works for any short-ranged interaction. For long-ranged interactions (such as the <a href="Coulomb_interaction" class="mw-redirect" title="Coulomb interaction">Coulomb interaction</a>), the summation over ℓ may not converge. The general approach for such problems consist in treating the Coulomb interaction separately from the short-ranged interaction, as the Coulomb problem can be solved exactly in terms of <a href="Coulomb_functions" class="mw-redirect" title="Coulomb functions">Coulomb functions</a>, which take on the role of the Hankel functions in this problem.

<ul><li><a href="Levinson's_theorem" title="Levinson's theorem">Levinson's theorem</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFGriffiths,_J._D.1995" class="citation book cs1">Griffiths, J. D. (1995). Introduction to Quantum Mechanics. Pearson Prentice Hall. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-13-111892-7</bdi>.</cite>

</li>

<li id="cite_note-Messiah-1976-2">^ <a href="#cite_ref-Messiah-1976_2-0">a</a> <a href="#cite_ref-Messiah-1976_2-1">b</a> <cite id="CITEREFMessiah1976" class="citation book cs1">Messiah, Albert (1976). Quantum mechanics. 1 (22. print ed.). Amsterdam: North-Holland. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-59766-7</bdi>.</cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120425231714/http://homepages.rpi.edu/~napolj/Talks/PWALunch9Sep03.pdf">Partial Wave Analysis for Dummies</a></li>

<li><a rel="nofollow" class="external text" href="http://quantummechanics.ucsd.edu/ph130a/130_notes/node228.html">Partial Wave Analysis of Scattering</a></li></ul>

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