Scattering theory

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In quantum mechanics the procedure has the scattering of quantum objects are generally described in contrast to the classical mechanics , since the concept of a trajectory missing. The quantum mechanical description is called scattering theory .

Math concept

In quantum mechanics, an object (e.g. an electron ) is always described by a state . This is made up of a spatial state, a spin state and many other physical quantities (e.g. isospin ): ${\displaystyle \vert {\text{Ort}}\rangle ,}$ ${\displaystyle \vert s,m_{s}\rangle }$

${\displaystyle \vert {\text{Gesamtzustand}}\rangle =|{\text{Ort}}\rangle \otimes |{\text{Spin}}\rangle \dots }$.

In the following, only the location is considered. The wave function can then be written as a component representation of the state of the location with respect to a location base . The restriction to the local state is justified under the following assumptions: ${\displaystyle |{\text{Ort}}\rangle }$ ${\displaystyle \psi (x)=\langle x|{\text{Ort}}\rangle }$ ${\displaystyle |x\rangle }$

• no internal degrees of freedom (spin, excited state , ...)
• distinguishable particles (symmetry of the wave function for bosons or fermions is not considered here), d. H. no indistinguishable particles
• no multiple scattering.

Similar to classical mechanics, the two- particle problem can initially be reduced to an equivalent one-particle problem , in which a single quantum object approaches a center of force at its origin . The starting point of the scattering theory is the description of the interaction by a potential and the Hamilton operator derived from it : ${\displaystyle V({\vec {x}})}$

${\displaystyle H={\frac {p^{2}}{2m}}+V({\vec {x}})}$

The wave function of the incoming particle is described by a wave packet at the beginning of the scattering process : ${\displaystyle (t=t_{0})}$

${\displaystyle \psi _{0}({\vec {x}},t_{0})=\int _{\mathbb {R} ^{3}}{\frac {\mathrm {d} ^{3}{\vec {k}}}{(2\pi )^{3}}}a_{\vec {k}}e^{\mathrm {i} {\vec {k}}\cdot {\vec {x}}}}$

This Fourier representation of the particle by plane waves can also take place via the stationary states (the eigen-states of the Hamilton operator):

${\displaystyle H\psi _{\vec {k}}=E_{k}\psi _{\vec {k}},}$

where the eigenvalues ​​are related to the wave vector via ${\displaystyle {\vec {k}}}$

${\displaystyle E_{k}={\frac {\hbar ^{2}k^{2}}{2m}}\geq 0}$

These states are referred to as scatter states , since a state with positive energy is unbound and has a finite probability of being outside the range of the potential . A single scattering state corresponds physically to an implausible situation, since the probability current density

${\displaystyle {\vec {j}}={\frac {\hbar }{2mi}}\left(\psi ^{*}{\vec {\nabla }}\psi -\psi {\vec {\nabla }}\psi ^{*}\right)}$

disappears, so the same amount of the particle always flows towards the scattering center as it flows away. But this is necessary because a steady state of a standing wave is analogous to how it is e.g. B. knows from acoustics . Only by superimposing one arrives at the illustrative situation of an initially arriving and then scattered wave packet. The stationary Schrödinger equation leads to the Helmholtz equation and its inhomogeneous solution to an implicit integral equation , which leads to the asymptotic form of the scattering states:

${\displaystyle \lim _{r\to \infty }\psi _{\vec {k}}({\vec {x}})=e^{\mathrm {i} {\vec {k}}\cdot {\vec {x}}}+f_{\vec {k}}(\theta ,\phi ){\frac {e^{\mathrm {i} kr}}{r}}}$

This asymptotic behavior, that at a great distance from the scattering center, the wave function is made up of an undisturbed plane wave and an outgoing spherical wave , is also known as Sommerfeld's boundary condition . The physical information about the scattering potential lies in the scattering amplitude, more precisely in its amount , the differential cross-section accessible through scattering experiments ${\displaystyle f_{\vec {k}}(\theta ,\phi ),}$

${\displaystyle {\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}=\left|f_{\vec {k}}(\theta ,\phi )\right|^{2}.}$

In the case of a central potential , the angular momentum is a conserved quantity , and the wave function is developed according to simultaneous eigenstates of and The scattering states are then called partial waves and, like the scattering amplitude and the scattering cross section, which are now only dependent on the angle , can be developed according to Legendre polynomials , what is also known as partial wave expansion . Another method for calculating the scattering amplitude is the Born approximation . ${\displaystyle H,\,L^{2}}$ ${\displaystyle L_{z}.}$ ${\displaystyle \theta }$