# 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 .

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:

Since the potential is a Coulomb potential , the Darwin term can also be written as

It is

- the fine structure constant
- the reduced Planck quantum of action
- the electron mass
- the speed of light
- the Laplace operator
- the atomic number
- 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 ( ).

## 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

- .

## literature

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

## Individual evidence

- ↑ 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 .