Scattering amplitude

In quantum physics, the scattering amplitude is the probability amplitude of the outgoing spherical wave relative to the incoming plane wave in a stationary-state scattering process.^[1]

Scattering in quantum mechanics begins with a physical model based on the Schrodinger wave equation for probability amplitude ${\displaystyle \psi }$: $${\displaystyle -\frac{{\hslash }^2}{2\mu }{\mathrm{\nabla }}^2\psi +V\psi =E\psi }$$ where ${\displaystyle \mu }$ is the reduced mass of two scattering particles and E is the energy of relative motion. For scattering problems, a stationary (time-independent) wavefunction is sought with behavior at large distances (asymptotic form) in two parts. First a plane wave represents the incoming source and, second, a spherical wave emanating from the scattering center placed at the coordinate origin represents the scattered wave:^[2]: 114 $${\displaystyle \psi (r\to \mathrm{\infty })\sim e^{i{\mathbf{𝐤}}_i\cdot \mathbf{𝐫}}+f({\mathbf{𝐤}}_f,{\mathbf{𝐤}}_i)\frac{e^{i{\mathbf{𝐤}}_f\cdot \mathbf{𝐫}}}{r}}$$ The scattering amplitude, ${\displaystyle f({\mathbf{𝐤}}_f,{\mathbf{𝐤}}_i)}$, represents the amplitude that the target will scatter into the direction ${\displaystyle {\mathbf{𝐤}}_f}$.^[3]: 194 In general the scattering amplitude requires knowing the full scattering wavefunction: $${\displaystyle f({\mathbf{𝐤}}_f,{\mathbf{𝐤}}_i)=-\frac{\mu }{2\pi {\hslash }^2}\int {\psi }_f^{*}V(\mathbf{𝐫}){\psi }_i{d}^{3}r}$$ For weak interactions a perturbation series can be applied; the lowest order is called the Born approximation.

For a spherically symmetric scattering center, the plane wave is described by the wavefunction^[4]

${\displaystyle \psi (\mathbf{𝐫})=e^{ikz}+f(\theta )\frac{e^{ikr}}{r},}$

where ${\displaystyle \mathbf{𝐫}\equiv (x,y,z)}$ is the position vector; ${\displaystyle r\equiv |\mathbf{𝐫}|}$; ${\displaystyle e^{ikz}}$ is the incoming plane wave with the wavenumber k along the z axis; ${\displaystyle e^{ikr}/r}$ is the outgoing spherical wave; θ is the scattering angle (angle between the incident and scattered direction); and ${\displaystyle f(\theta )}$ is the scattering amplitude.

The dimension of the scattering amplitude is length. The scattering amplitude is a probability amplitude; the differential cross-section as a function of scattering angle is given as its modulus squared,

${\displaystyle d\sigma =|f(\theta ){|}^{2}d\mathrm{\Omega }.}$

When conservation of number of particles holds true during scattering, it leads to a unitary condition for the scattering amplitude. In the general case, we have^[4]

${\displaystyle f(\mathbf{𝐧},{\mathbf{𝐧}}^{\prime })-f^{*}({\mathbf{𝐧}}^{\prime },\mathbf{𝐧})=\frac{ik}{2\pi }\int f(\mathbf{𝐧},{\mathbf{𝐧}}^{″})f^{*}(\mathbf{𝐧},{\mathbf{𝐧}}^{″})d{\mathrm{\Omega }}^{″}}$

Optical theorem follows from here by setting ${\displaystyle \mathbf{𝐧}={\mathbf{𝐧}}^{\prime }.}$

In the centrally symmetric field, the unitary condition becomes

${\displaystyle \mathrm{I}\mathrm{m}f(\theta )=\frac{k}{4\pi }\int f(\gamma )f({\gamma }^{\prime })d{\mathrm{\Omega }}^{″}}$

where ${\displaystyle \gamma }$ and ${\displaystyle {\gamma }^{\prime }}$ are the angles between ${\displaystyle \mathbf{𝐧}}$ and ${\displaystyle {\mathbf{𝐧}}^{\prime }}$ and some direction ${\displaystyle {\mathbf{𝐧}}^{″}}$. This condition puts a constraint on the allowed form for ${\displaystyle f(\theta )}$, i.e., the real and imaginary part of the scattering amplitude are not independent in this case. For example, if ${\displaystyle |f(\theta )|}$ in ${\displaystyle f=|f|e^{2i\alpha }}$ is known (say, from the measurement of the cross section), then ${\displaystyle \alpha (\theta )}$ can be determined such that ${\displaystyle f(\theta )}$ is uniquely determined within the alternative ${\displaystyle f(\theta )\to -f^{*}(\theta )}$.^[4]

In the partial wave expansion the scattering amplitude is represented as a sum over the partial waves,^[5]

${\displaystyle f={\displaystyle \sum _{\ell =0}^{\mathrm{\infty }}}(2\ell +1)f_{\ell }{P}_{\ell }(\cos \theta )}$,

where f_ℓ is the partial scattering amplitude and P_ℓ are the Legendre polynomials. The partial amplitude can be expressed via the partial wave S-matrix element S_ℓ (${\displaystyle =e^{2i{\delta }_{\ell }}}$) and the scattering phase shift δ_ℓ as

${\displaystyle f_{\ell }=\frac{S_{\ell }-1}{2ik}=\frac{e^{2i{\delta }_{\ell }}-1}{2ik}=\frac{e^{i{\delta }_{\ell }}\sin {\delta }_{\ell }}{k}=\frac{1}{k\cot {\delta }_{\ell }-ik}.}$

Then the total cross section^[6]

${\displaystyle \sigma =\int |f(\theta ){|}^{2}d\mathrm{\Omega }}$,

can be expanded as^[4]

${\displaystyle \sigma ={\displaystyle \sum _{l=0}^{\mathrm{\infty }}}{\sigma }_l,\text{where}{\sigma }_l=4\pi (2l+1)|f_l{|}^2=\frac{4\pi }{k^2}(2l+1){\sin }^2{\delta }_l}$

is the partial cross section. The total cross section is also equal to ${\displaystyle \sigma =(4\pi /k)\mathrm{I}\mathrm{m}f(0)}$ due to optical theorem.

For ${\displaystyle \theta \ne 0}$, we can write^[4]

${\displaystyle f=\frac{1}{2ik}{\displaystyle \sum _{\ell =0}^{\mathrm{\infty }}}(2\ell +1)e^{2i{\delta }_l}{P}_{\ell }(\cos \theta ).}$

The scattering length for X-rays is the Thomson scattering length or classical electron radius, r₀.

The nuclear neutron scattering process involves the coherent neutron scattering length, often described by b.

A quantum mechanical approach is given by the S matrix formalism.

The scattering amplitude can be determined by the scattering length in the low-energy regime.

• Levinson's theorem
• Plane wave expansion
• Veneziano amplitude

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