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 $\psi$: $$-{\frac {\hbar ^{2}}{2\mu }}\nabla ^{2}\psi +V\psi =E\psi$$ 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: $$\psi (r\rightarrow \infty )\sim e^{i\mathbf {k} _{i}\cdot \mathbf {r} }+f(\mathbf {k} _{f},\mathbf {k} _{i}){\frac {e^{i\mathbf {k} _{f}\cdot \mathbf {r} }}{r}}$$ The scattering amplitude, $f(\mathbf {k} _{f},\mathbf {k} _{i})$, represents the amplitude that the target will scatter into the direction $\mathbf {k} _{f}$. In general the scattering amplitude requires knowing the full scattering wavefunction: $$f(\mathbf {k} _{f},\mathbf {k} _{i})=-{\frac {\mu }{2\pi \hbar ^{2}}}\int \psi _{f}^{*}V(\mathbf {r} )\psi _{i}d^{3}r$$ 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

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

where $\mathbf {r} \equiv (x,y,z)$ is the position vector; $r\equiv |\mathbf {r} |$; $e^{ikz}$ 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

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

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

In the centrally symmetric field, the unitary condition becomes

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

where $\gamma$ and $\gamma '$ are the angles between $\mathbf {n}$ and $\mathbf {n} '$ and some direction $\mathbf {n} ''$. 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 $f(\theta )\rightarrow -f^{*}(\theta )$.

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,

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

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

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

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

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

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

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