Partial wave

Partial waves , literally part waves, a concept of quantum mechanics , are stationary solutions of a scattering problem and simultaneously eigenfunctions of angular momentum . The decomposition of a scattering amplitude into partial waves , i. H. a series expansion according to angular momentum is particularly useful for interactions with a short range, such as B. the strong interaction . Because of the short range, only small angular impulses contribute to the scattering for low energies.

Explanation in the particle picture

If a moving particle is in the field of a scattering center - e.g. B. of an atomic nucleus - deflected from its orbit, an orbital angular momentum belongs to this movement . This can only assume discrete values ​​described by a quantum number ; in individual cases it depends on the impact parameter for a given particle velocity . The contributions of the individual processes are called partial waves in the wave pattern and each have a characteristic effect, e.g. B. in the distribution of the total scattered particles in the scattering directions, the angular distribution . ${\displaystyle l}$ ${\displaystyle l=0,1,2,\dots }$

The code letters for the values ​​of are used in the same way as for the bound electron in the atom , so one speaks of the s-wave ( ), p-wave ( ), d-wave ( ) etc. ${\displaystyle l}$ ${\displaystyle l=0}$ ${\displaystyle l=1}$ ${\displaystyle l=2}$

Derivation

The aim is to find a solution to the Schrödinger equation for a spherically symmetrical potential such as B. to find the Coulomb potential . ${\displaystyle V({\vec {r}})=V(r)}$

The wave function is used for asymptotic distances as the superposition of an incoming plane wave and a spherical wave modified by the scattering amplitude : ${\displaystyle \psi _{\vec {k}}({\vec {r}})}$ ${\displaystyle r\rightarrow \infty }$ ${\displaystyle f(\theta ,\phi )}$

${\displaystyle \psi _{\vec {k}}({\vec {r}}){\xrightarrow {r\rightarrow \infty }}[e^{i{\vec {k}}{\vec {r}}}+f(\theta ,\phi ){\frac {e^{ikr}}{r}}]}$

In this case the scattering amplitude is independent of the angle due to the spherical symmetry : ${\displaystyle \phi }$

${\displaystyle f(\theta ,\phi )=f(\theta )}$

After a few transformations, the solution wave function of the scattering problem for asymptotic distances results as:

${\displaystyle \psi _{\vec {k}}({\vec {r}})=\sum _{l=0}^{\infty }i^{l}(2l+1)\;R_{lk}(r)\;P_{l}(\cos \theta )}$

where are the Legendre polynomials . ${\displaystyle P_{l}(\cos \theta )}$

${\displaystyle R_{lk}}$is the solution of the radial Schrödinger equation, which consists of a linear combination of the spherical Bessel functions and the Von Neumann function : ${\displaystyle j_{l}(\rho )}$ ${\displaystyle n_{l}(\rho )}$

${\displaystyle R_{lk}{\stackrel {\mathrm {r\rightarrow \infty } }{=}}A_{l}\;j_{l}(kr)+B_{l}\;n_{l}(kr)}$

In the next step, the scattering phase is defined as follows: ${\displaystyle \delta _{l}}$

${\displaystyle A_{l}=+a_{l}\cos \delta _{l}}$

${\displaystyle B_{l}=-a_{l}\sin \delta _{l}}$

The phase of the outgoing spherical wave is shifted by the potential : with elastic scattering , the scattered wave differs from the undisturbed wave of the free particle only by a phase factor ${\displaystyle V(r)}$ ${\displaystyle e^{i\cdot \delta _{l}}.}$

By inserting the spherical Bessel and Von-Neumann functions and comparing them with the approach for the wave function for asymptotic distances, after a few transformations one comes to the following relationship between the scattering amplitude and the scattering phase : ${\displaystyle f(\theta )}$ ${\displaystyle \delta _{l}}$

${\displaystyle f(\theta )=\sum _{l=0}^{\infty }(2l+1)\;f_{l}(\delta _{l})\;P_{l}(\cos \theta )}$

in which

${\displaystyle f_{l}(\delta _{l})={\frac {1}{k}}e^{i\cdot \delta _{l}}\sin \delta _{l}}$

represents the contribution of the  l th partial wave.

Spread length

Another important parameter for the analysis of scattering problems, which can be derived from the scattering amplitude, is the scattering length  a . It results from the total scattering cross-section when the energy of the scattered particle approaches 0:

${\displaystyle {\begin{aligned}\lim _{k\to 0}\sigma _{\text{total}}&=4\pi \cdot a^{2}\\\Leftrightarrow a&=\pm {\sqrt {\frac {\lim _{k\to 0}\sigma _{\text{total}}}{4\pi }}}\end{aligned}}}$

The scattering length therefore corresponds to an effective cross-sectional area, which shows both the strength and the type of a potential.

With the following definition for the total cross section:

${\displaystyle \sigma _{\text{total}}=\int _{4\pi }{\frac {d\sigma }{d\Omega }}\cdot d\Omega =\int _{4\pi }|f(\theta )|^{2}\cdot d\Omega ={\frac {4\pi }{k^{2}}}\sum _{l=0}^{\infty }(2l+1)\sin ^{2}\delta _{l}}$

the scattering length for s-waves ( ) becomes: ${\displaystyle l=0}$

${\displaystyle a=\pm \lim _{k\to 0}{\frac {\sin \delta _{0}}{k}}\;.}$

literature

• Cohen-Tannoudji: Quantum Mechanics - Vol 2, Wiley-Interscience, 2006.

Web links

• TU Munich: Properties of the nucleon-nucleon interaction ( Memento from March 7, 2014 in the Internet Archive )