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]

Formulation

Scattering in quantum mechanics begins with a physical model based on the Schrodinger wave equation for probability amplitude ${\displaystyle \psi }$: $${\displaystyle -{\frac {\hbar ^{2}}{2\mu }}\nabla ^{2}\psi +V\psi =E\psi }$$ 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 $${\displaystyle \psi (r\rightarrow \infty )\sim e^{i\mathbf {k} _{i}\cdot \mathbf {r} }+f(\mathbf {k} _{f},\mathbf {k} _{i}){\frac {e^{i\mathbf {k} _{f}\cdot \mathbf {r} }}{r}}}$$ 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: $${\displaystyle f(\mathbf {k} _{f},\mathbf {k} _{i})=-{\frac {\mu }{2\pi \hbar ^{2}}}\int \psi _{f}^{*}V(\mathbf {r} )\psi _{i}d^{3}r}$$ 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,

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

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^[4]

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

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

In the centrally symmetric field, the unitary condition becomes

${\displaystyle \mathrm {Im} f(\theta )={\frac {k}{4\pi }}\int f(\gamma )f(\gamma ')\,d\Omega ''}$

where ${\displaystyle \gamma }$ and ${\displaystyle \gamma '}$ are the angles between ${\displaystyle \mathbf {n} }$ and ${\displaystyle \mathbf {n} '}$ 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 ${\displaystyle f(\theta )\rightarrow -f^{*}(\theta )}$.^[4]

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,^[5]

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

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]

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

can be expanded as^[4]

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

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

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

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

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
• Veneziano amplitude
• Plane wave expansion