Editing Cross section (physics) (section)

== Quantum scattering ==
In the [[Stationary state|time-independent]] formalism of [[Quantum mechanics|quantum]] scattering, the initial [[wave function]] (before scattering) is taken to be a plane wave with definite [[momentum]] {{math|''k''}}:
: <math>\phi_-(\mathbf r) \;\stackrel{r \to \infty}{\longrightarrow}\; e^{i k z},</math>
where {{math|''z''}} and {{math|''r''}} are the ''relative'' coordinates between the projectile and the target. The arrow indicates that this only describes the ''asymptotic behavior'' of the wave function when the projectile and target are too far apart for the interaction to have any effect.

After scattering takes place it is expected that the wave function takes on the following asymptotic form:
: <math>\phi_+(\mathbf r) \;\stackrel{r \to \infty}{\longrightarrow}\; f(\theta,\phi) \frac{e^{i k r}}{r},</math>
where {{math|''f''}} is some function of the angular coordinates known as the [[scattering amplitude]]. This general form is valid for any short-ranged, energy-conserving interaction. It is not true for long-ranged interactions, so there are additional complications when dealing with electromagnetic interactions.

The full wave function of the system behaves asymptotically as the sum
: <math>\phi(\mathbf r) \;\stackrel{r \to \infty}{\longrightarrow}\; \phi_-(\mathbf r) + \phi_+(\mathbf r).</math>

The differential cross section is related to the scattering amplitude:
: <math>\frac{\mathrm d \sigma}{\mathrm d \Omega}(\theta, \phi) = \bigl|f(\theta, \phi)\bigr|^2.</math>

This has the simple interpretation as the probability density for finding the scattered projectile at a given angle.

A cross section is therefore a measure of the effective surface area seen by the impinging particles, and as such is expressed in units of area. The cross section of two [[Elementary particle|particles]] (i.e. observed when the two particles are [[Collision|colliding]] with each other) is a measure of the interaction event between the two particles. The cross section is proportional to the probability that an interaction will occur; for example in a simple scattering experiment the number of particles scattered per unit of time (current of scattered particles {{math|''I''<sub>r</sub>}}) depends only on the number of incident particles per unit of time (current of incident particles {{math|''I''<sub>i</sub>}}), the characteristics of target (for example the number of particles per unit of surface {{math|''N''}}), and the type of interaction. For {{math|''Nσ'' ≪ 1}} we have
: <math>\begin{align}
I_\text{r} &= I_\text{i}N\sigma, \\
\sigma &= \frac{I_\text{r}}{I_\text{i}} \frac{1}{N} \\
&= \text{probability of interaction} \times \frac{1}{N}.
\end{align}</math>

=== Relation to the S-matrix ===
If the [[reduced mass]]es and [[Momentum|momenta]] of the colliding system are {{math|''m''<sub>i</sub>}}, {{math|'''p'''<sub>i</sub>}} and {{math|''m''<sub>f</sub>}}, {{math|'''p'''<sub>f</sub>}} before and after the collision respectively, the differential cross section is given by{{clarify|reason=This section is an uncontextualized leap. We have jumped, without announcement, from the Schrödinger PDE for a wave in a potential to the language of QFT for two particles colliding.|date=September 2016}}
: <math>\frac{\mathrm d\sigma}{\mathrm d\Omega} = \left(2\pi\right)^4 m_\text{i} m_\text{f} \frac{p_\text{f}}{p_\text{i}} \bigl|T_{\text{f}\text{i}}\bigr|^2,</math>
where the on-shell {{math|''T''}} matrix is defined by
: <math>S_{\text{f}\text{i}} = \delta_{\text{f}\text{i}} - 2\pi i \delta\left(E_\text{f} - E_\text{i}\right) \delta\left(\mathbf{p}_\text{i} - \mathbf{p}_\text{f}\right) T_{\text{f}\text{i}}</math>
in terms of the [[S-matrix]]. Here {{math|''δ''}} is the [[Dirac delta function]]. The computation of the S-matrix is the main goal of the [[scattering theory]].