Scattering amplitude

Scattering in quantum mechanics begins with a physical model based on the Schrodinger wave equation for probability amplitude $\backslash psi$:

$$-\backslash frac\{\backslash hbar^2\}\{2\backslash mu\}\backslash nabla^2\backslash psi\; +\; V\backslash psi\; =\; E\backslash psi$$

where $\backslash mu$ 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:{{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}}{{rp|114}}

$$\backslash psi(r\backslash rightarrow\; \backslash infty)\; \backslash sim\; e^\{i\backslash mathbf\{k\}\_i\backslash cdot\backslash mathbf\{r\}\}\; +\; f(\backslash mathbf\{k\}\_f,\backslash mathbf\{k\}\_i)\backslash frac\{e^\{i\backslash mathbf\{k\}\_f\backslash cdot\backslash mathbf\{r\}\}\}\{r\}$$

The scattering amplitude, $f(\backslash mathbf\{k\}\_f,\backslash mathbf\{k\}\_i)$, represents the amplitude that the target will scatter into the direction $\backslash mathbf\{k\}\_f$.{{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.]}}{{rp|194}}

In general the scattering amplitude requires knowing the full scattering wavefunction:

$$f(\backslash mathbf\{k\}\_f,\backslash mathbf\{k\}\_i)\; =\; -\backslash frac\{\backslash mu\}\{2\backslash pi\backslash hbar^2\}\backslash int\; \backslash psi\_f^*\; V(\backslash mathbf\{r\})\; \backslash psi\_i\; d^3r$$

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 wavefunctionLandau, L. D., & Lifshitz, E. M. (2013). Quantum mechanics: non-relativistic theory (Vol. 3). Elsevier.

:$$

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

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

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}}

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

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

In the centrally symmetric field, the unitary condition becomes

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

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

{{Main article|Partial wave analysis}}

In the partial wave expansion the scattering amplitude is represented as a sum over the partial waves,[http://galileo.phys.virginia.edu/classes/752.mf1i.spring03/Scattering_II.htm Michael Fowler/ 1/17/08 Plane Waves and Partial Waves]

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

where {{math|f_ℓ}} is the partial scattering amplitude and {{math|P_ℓ}} are the Legendre polynomials. The partial amplitude can be expressed via the partial wave S-matrix element {{math|S_ℓ}} ($=e^\{2i\backslash delta\_\backslash ell\}$) and the scattering phase shift {{math|δ_ℓ}} as

:$f\_\backslash ell\; =\; \backslash frac\{S\_\backslash ell-1\}\{2ik\}\; =\; \backslash frac\{e^\{2i\backslash delta\_\backslash ell\}-1\}\{2ik\}\; =\; \backslash frac\{e^\{i\backslash delta\_\backslash ell\}\; \backslash sin\backslash delta\_\backslash ell\}\{k\}\; =\; \backslash frac\{1\}\{k\backslash cot\backslash delta\_\backslash ell-ik\}\; \backslash ;.$

Then the total cross section{{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]–120}}

:$\backslash sigma\; =\; \backslash int\; |f(\backslash theta)|^2d\backslash Omega$,

can be expanded as{{r|landau}}

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

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

For $\backslash theta\backslash neq\; 0$, we can write{{r|landau}}

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

The scattering length for X-rays is the Thomson scattering length or classical electron radius, {{mvar|r}}₀.

The nuclear neutron scattering process involves the coherent neutron scattering length, often described by {{mvar|b}}.

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

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

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

{{Reflist}}

Category:Neutron

Category:X-rays

Category:Electron

Category:Scattering

Category:Diffraction

Category:Quantum mechanics