Physics:Scattering amplitude

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. At large distances from the centrally symmetric scattering center, the plane wave is described by the wavefunction

[math]\displaystyle{ \psi(\mathbf{r}) = e^{ikz} + f(\theta)\frac{e^{ikr}}{r} \;, }[/math]

where [math]\displaystyle{ \mathbf{r}\equiv(x,y,z) }[/math] is the position vector; [math]\displaystyle{ r\equiv|\mathbf{r}| }[/math]; [math]\displaystyle{ e^{ikz} }[/math] is the incoming plane wave with the wavenumber k along the z axis; [math]\displaystyle{ e^{ikr}/r }[/math] is the outgoing spherical wave; θ is the scattering angle (angle between the incident and scattered direction); and [math]\displaystyle{ 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]\displaystyle{ d\sigma = |f(\theta)|^2 \;d\Omega. }[/math]

The asymptotic form of the wave function in arbitrary external field takes the form

[math]\displaystyle{ \psi = e^{ikr\mathbf n\cdot\mathbf n'} + f(\mathbf n,\mathbf n') \frac{e^{ikr}}{r} }[/math]

where [math]\displaystyle{ \mathbf n }[/math] is the direction of incidient particles and [math]\displaystyle{ \mathbf n' }[/math] is the direction of scattered particles.

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

[math]\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'' }[/math]

Optical theorem follows from here by setting [math]\displaystyle{ \mathbf n=\mathbf n'. }[/math]

In the centrally symmetric field, the unitary condition becomes

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

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

Partial wave expansion

In the partial wave expansion the scattering amplitude is represented as a sum over the partial waves,

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

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 ([math]\displaystyle{ =e^{2i\delta_\ell} }[/math]) and the scattering phase shift δ_ℓ as

[math]\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} \;. }[/math]

Then the total cross section

[math]\displaystyle{ \sigma = \int |f(\theta)|^2d\Omega }[/math],

can be expanded as

[math]\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 }[/math]

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

For [math]\displaystyle{ \theta\neq 0 }[/math], we can write

[math]\displaystyle{ 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, 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

• Veneziano amplitude
• Plane wave expansion