Darwin term

The Darwin term (after Charles Galton Darwin ) is a relativistic correction term in the Hamilton operator to theoretically explain the fine structure in the hydrogen spectrum. It results from the Dirac theory . ${\displaystyle H_{\mathrm {Darwin} }}$ ${\displaystyle H}$

He describes that in a non-relativistic approximation the electrostatic interaction of the electron with the electric field of the nucleus is no longer local due to the trembling motion , but also depends on a small area of ​​the electric field around the electron:

${\displaystyle H_{\mathrm {Darwin} }={\frac {\hbar ^{2}}{8m_{\mathrm {e} }^{2}c^{2}}}\left(\Delta V\right).}$

Since the potential is a Coulomb potential , the Darwin term can also be written as ${\displaystyle V(r)=-\alpha \hbar cZ/r}$

${\displaystyle H_{\mathrm {Darwin} }=Z\alpha {\frac {\hbar ^{3}}{8m_{\mathrm {e} }^{2}c}}\cdot 4\pi \delta ^{(3)}({\vec {r}}).}$

It is

• ${\displaystyle \alpha }$the fine structure constant
• ${\displaystyle \hbar }$the reduced Planck quantum of action
• ${\displaystyle m_{\mathrm {e} }}$ the electron mass
• ${\displaystyle c}$the speed of light
• ${\displaystyle \Delta }$the Laplace operator
• ${\displaystyle Z}$the atomic number
• ${\displaystyle \delta ^{(3)}({\vec {r}})\,\!}$the delta distribution in three dimensions.

The Darwin term only plays a role for electrons with angular momentum quantum number , because only their wave functions do not disappear at the nucleus ( ). ${\displaystyle l=0}$ ${\displaystyle {\vec {r}}=0}$

Heuristic derivation

The Darwin term in the relativistic hydrogen problem can be derived formally and stringently by subtracting the relativistic correction and the spin-orbit coupling from the overall result. A heuristic derivation assumes that the electron is not exactly localized , but that its position fluctuates around the reduced Compton wavelength of the electron. Such a derivation does not lead exactly to the correct Darwin term, but only to the correct order of magnitude ${\displaystyle \delta r={\frac {\hbar }{m_{\mathrm {e} }c}}={\frac {\lambda _{\mathrm {C} }}{2\pi }}}$

${\displaystyle H_{\text{Darwin}}^{\text{heuristisch}}={\frac {\hbar ^{2}}{6m_{\mathrm {e} }^{2}c^{2}}}\Delta V}$.

literature

• Armin Wachter: Relativistic Quantum Mechanics. Springer, Berlin / Heidelberg 2005, ISBN 3-540-22922-1 , p. 167.

Individual evidence

1. ↑ Ingolf V. Hertel, Claus-Peter Schulz: Atoms, Molecules and Optical Physics 1 - Atomic Physics and Basics of Spectroscopy . Springer, Berlin / Heidelberg 2008, ISBN 978-3-540-30613-9 , pp. 221 .