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.

Formulation

Scattering in quantum mechanics begins with a physical model based on the Schrodinger wave equation for probability amplitude $\psi$: $$-\frac{{\hslash }^2}{2\mu }{\mathrm{\nabla }}^2\psi +V\psi =E\psi$$ where $\mu$ 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: $$\psi (r\to \mathrm{\infty })\sim e^{i{𝐤}_i\cdot 𝐫}+f({𝐤}_f,{𝐤}_i)\frac{e^{i{𝐤}_f\cdot 𝐫}}{r}$$ The scattering amplitude, $f({𝐤}_f,{𝐤}_i)$, represents the amplitude that the target will scatter into the direction ${𝐤}_f$. In general the scattering amplitude requires knowing the full scattering wavefunction: $$f({𝐤}_f,{𝐤}_i)=-\frac{\mu }{2\pi {\hslash }^2}\int {\psi }_f^{*}V(𝐫){\psi }_i{d}^{3}r$$ 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

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

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

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

Unitary condition

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

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

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

In the centrally symmetric field, the unitary condition becomes

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

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

Partial wave expansion

Main article: Partial wave analysis

In the partial wave expansion the scattering amplitude is represented as a sum over the partial waves,

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

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 ($=e^{2i{\delta }_{\ell }}$) and the scattering phase shift δ_ℓ as

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

Then the total cross section

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

can be expanded as

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

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

For $\theta \ne 0$, we can write

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

X-rays

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

Neutrons

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

Quantum mechanical formalism

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

Measurement

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

See also

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