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 $\psi$:
$-\frac\nabla^2\psi + V\psi = E\psi$
where $\mu$ is the reduced mass of two scattering particles and 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:
$\psi(r\rightarrow \infty) \sim e^ + f(\mathbf_f,\mathbf_i)\frac$
The scattering amplitude, $f(\mathbf_f,\mathbf_i)$, represents the amplitude that the target will scatter into the direction $\mathbf_f$.
In general the scattering amplitude requires knowing the full scattering wavefunction:
$f(\mathbf_f,\mathbf_i) = -\frac\int \psi_f^* V(\mathbf) \psi_i d^3r$
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.
:$\psi(\mathbf) = e^ + f(\theta)\frac \;,$
where $\mathbf\equiv(x,y,z)$ is the position vector; $r\equiv, \mathbf,$; $e^$ is the incoming plane wave with the wavenumber along the axis; $e^/r$ is the outgoing spherical wave; 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
:$f(\mathbf,\mathbf') -f^*(\mathbf',\mathbf)= \frac \int f(\mathbf,\mathbf'')f^*(\mathbf,\mathbf'')\,d\Omega''$
Optical theorem follows from here by setting $\mathbf n=\mathbf n'.$

In the centrally symmetric field, the unitary condition becomes
:$\mathrm f(\theta)=\frac\int f(\gamma)f(\gamma')\,d\Omega''$
where $\gamma$ and $\gamma'$ are the angles between $\mathbf$ and $\mathbf'$ and some direction $\mathbf''$. 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 $f=, f, e^$ 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

In the partial wave expansion the scattering amplitude is represented as a sum over the partial waves,Michael Fowler/ 1/17/08 Plane Waves and Partial Waves
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:$f=\sum_^\infty (2\ell+1) f_\ell P_\ell(\cos \theta)$,
where is the partial scattering amplitude and are the Legendre polynomials. The partial amplitude can be expressed via the partial wave S-matrix element ($=e^$) and the scattering phase shift as
:$f_\ell = \frac = \frac = \frac = \frac \;.$

Then the total cross section
:$\sigma = \int , f(\theta), ^2d\Omega$,
can be expanded as
:$\sigma = \sum_^\infty \sigma_l, \quad \text \quad \sigma_l = 4\pi(2l+1), f_l, ^2=\frac(2l+1)\sin^2\delta_l$
is the partial cross section. The total cross section is also equal to $\sigma=(4\pi/k)\,\mathrm f(0)$ due to optical theorem.

For $\theta\neq 0$, we can write
:$f=\frac\sum_^\infty (2\ell+1) e^ P_\ell(\cos \theta).$

X-rays

The scattering length for X-rays is the Thomson scattering length or classical electron radius, ₀.

Neutrons

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

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

Neutron
X-rays
Electron
Scattering
Diffraction
Quantum mechanics