Scattering amplitude

Template:Use American English Template:Short description 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.<ref>Quantum Mechanics: Concepts and Applications Template:Webarchive By Nouredine Zettili, 2nd edition, page 623. Template:ISBN Paperback 688 pages January 2009</ref>

Formulation

Scattering in quantum mechanics begins with a physical model based on the Schrodinger wave equation for probability amplitude <math>\psi</math>: <math display="block">-\frac{\hbar^2}{2\mu}\nabla^2\psi + V\psi = E\psi</math> where <math>\mu</math> is the reduced mass of two scattering particles and Template:Mvar 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:<ref name="Schiff-1987">Template:Cite book</ref>Template:Rp <math display="block">\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}</math> The scattering amplitude, <math>f(\mathbf{k}_f,\mathbf{k}_i)</math>, represents the amplitude that the target will scatter into the direction <math>\mathbf{k}_f</math>.<ref name=Baym-1990>Template:Cite book</ref>Template:Rp In general the scattering amplitude requires knowing the full scattering wavefunction: <math display="block">f(\mathbf{k}_f,\mathbf{k}_i) = -\frac{\mu}{2\pi\hbar^2}\int \psi_f^* V(\mathbf{r}) \psi_i d^3r</math> 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<ref name=landau>Landau, L. D., & Lifshitz, E. M. (2013). Quantum mechanics: non-relativistic theory (Vol. 3). Elsevier.</ref>

<math>

\psi(\mathbf{r}) = e^{ikz} + f(\theta)\frac{e^{ikr}}{r} \;, </math> where <math>\mathbf{r}\equiv(x,y,z)</math> is the position vector; <math>r\equiv|\mathbf{r}|</math>; <math>e^{ikz}</math> is the incoming plane wave with the wavenumber Template:Mvar along the Template:Mvar axis; <math>e^{ikr}/r</math> is the outgoing spherical wave; Template:Mvar is the scattering angle (angle between the incident and scattered direction); and <math>f(\theta)</math> 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,

<math>

d\sigma = |f(\theta)|^2 \;d\Omega. </math>

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 haveTemplate:R

<math>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</math>

Optical theorem follows from here by setting <math>\mathbf n=\mathbf n'.</math>

In the centrally symmetric field, the unitary condition becomes

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

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

Partial wave expansion

In the partial wave expansion the scattering amplitude is represented as a sum over the partial waves,<ref>Michael Fowler/ 1/17/08 Plane Waves and Partial Waves</ref>

<math>f=\sum_{\ell=0}^\infty (2\ell+1) f_\ell P_\ell(\cos \theta)</math>,

where Template:Math is the partial scattering amplitude and Template:Math are the Legendre polynomials. The partial amplitude can be expressed via the partial wave S-matrix element Template:Math (<math>=e^{2i\delta_\ell}</math>) and the scattering phase shift Template:Math as

<math>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} \;.</math>

Then the total cross section<ref name=Schiff1968>Template:Cite book</ref>

<math>\sigma = \int |f(\theta)|^2d\Omega </math>,

can be expanded asTemplate:R

<math>\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</math>

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

For <math>\theta\neq 0</math>, we can writeTemplate:R

<math>f=\frac{1}{2ik}\sum_{\ell=0}^\infty (2\ell+1) e^{2i\delta_l} P_\ell(\cos \theta).</math>

X-rays

The scattering length for X-rays is the Thomson scattering length or classical electron radius, Template:Mvar₀.

Neutrons

The nuclear neutron scattering process involves the coherent neutron scattering length, often described by Template:Mvar.

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