# 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 ${\ displaystyle \ delta r = {\ frac {\ hbar} {m _ {\ mathrm {e}} c}} = {\ frac {\ lambda _ {\ mathrm {C}}} {2 \ pi}}}$

${\ displaystyle H _ {\ text {Darwin}} ^ {\ text {heuristic}} = {\ frac {\ hbar ^ {2}} {6m _ {\ mathrm {e}} ^ {2} c ^ {2}} } \ Delta V}$.

## 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 .