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Scattering amplitude
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{{Use American English|date = February 2019}} {{Short description|Probability amplitude in quantum scattering theory}} 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>[http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0470026790.html/ ''Quantum Mechanics: Concepts and Applications''] {{webarchive|url=https://web.archive.org/web/20101110002150/http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0470026790.html |date=2010-11-10 }} By Nouredine Zettili, 2nd edition, page 623. {{ISBN|978-0-470-02679-3}} Paperback 688 pages January 2009</ref> == Formulation == Scattering in [[quantum mechanics]] begins with a physical model based on the [[Schrodinger equation|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 {{mvar|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:<ref name="Schiff-1987">{{Cite book |last=Schiff |first=Leonard I. |title=Quantum mechanics |date=1987 |publisher=McGraw-Hill |isbn=978-0-07-085643-1 |edition=3. ed., 24. print |series=International series in pure and applied physics |location=New York}}</ref>{{rp|114}} <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>{{Cite book |last=Baym |first=Gordon |title=Lectures on quantum mechanics |date=1990 |publisher=Addison-Wesley |isbn=978-0-8053-0667-5 |edition=3 |series=Lecture notes and supplements in physics |location=Redwood City (Calif.) Menlo Park (Calif.) Reading (Mass.) [etc.]}}</ref>{{rp|194}} 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]] {{mvar|k}} along the {{mvar|z}} axis; <math>e^{ikr}/r</math> is the outgoing spherical wave; {{mvar| ΞΈ}} is the scattering angle (angle between the incident and scattered direction); and <math>f(\theta)</math> is the scattering amplitude. The [[dimensional analysis|dimension]] of the scattering amplitude is [[length]]. The scattering amplitude is a [[probability amplitude]]; the differential [[Cross section (physics)|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 have{{r|landau}} :<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>.{{r|landau}} == 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,<ref>[http://galileo.phys.virginia.edu/classes/752.mf1i.spring03/Scattering_II.htm 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 {{math|''f<sub>β</sub>''}} is the partial scattering amplitude and {{math|''P<sub>β</sub>''}} are the [[Legendre polynomial]]s. The partial amplitude can be expressed via the partial wave [[S-matrix]] element {{math|''S<sub>β</sub>''}} (<math>=e^{2i\delta_\ell}</math>) and the '''scattering phase shift''' {{math|''Ξ΄<sub>β</sub>''}} 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>{{cite book|last=Schiff|first=Leonard I.|title=Quantum Mechanics|url=https://archive.org/details/quantummechanics00schi_086|url-access=limited|date=1968|publisher=McGraw Hill|location=New York|pages=[https://archive.org/details/quantummechanics00schi_086/page/n138 119]β120}}</ref> :<math>\sigma = \int |f(\theta)|^2d\Omega </math>, can be expanded as{{r|landau}} :<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 write{{r|landau}} :<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]], {{mvar|r}}<sub>0</sub>. ==Neutrons== The nuclear [[neutron scattering]] process involves the coherent neutron scattering length, often described by {{mvar|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]] ==References== {{Reflist}} [[Category:Neutron]] [[Category:X-rays]] [[Category:Electron]] [[Category:Scattering]] [[Category:Diffraction]] [[Category:Quantum mechanics]]
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