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Mathematics for Physics

94 Atomic Orbitals

A spatial wavefunction that is used to describe the region around a nucleus where an electron is most likely found, where a spatial wavefunction is a function of the electron’s position.

The \((n,\ell,m)\) labels. A hydrogen orbital is labeled by the quantum numbers \(n\), \(\ell\), and \(m\). The principal quantum number \(n\) sets the energy. The orbital quantum number \(\ell\) sets the angular momentum. The magnetic quantum number \(m\) sets the \(z\) component. This principle is used to name the orbitals \(1s\), \(2p\), \(3d\).

The orbital probability density. The probability density of an orbital is \(|\psi_{n\ell m}(\mathbf{r})|^{2}\). This principle is used to draw the familiar orbital shapes as regions of high finding probability.

The orbital probability density is

\[ \rho_{n\ell m}(\mathbf{r}) = \lvert\psi_{n\ell m}(\mathbf{r})\rvert^{2} \]

where

  • \(\rho_{n\ell m}\) is the probability density.
  • \(\psi_{n\ell m}\) is the hydrogen wavefunction.

The ranges of \(\ell\) and \(m\). The orbital angular momentum quantum number obeys \(0\leq\ell\leq n-1\), and \(m\) runs from \(-\ell\) to \(\ell\). This principle is used to count the orbitals in a shell.

Subshells in many-electron atoms. In many-electron atoms, orbitals of the same \(n\) and \(\ell\) form a subshell that is filled according to the Pauli principle. A subshell is the set of orbitals sharing \(n\) and \(\ell\). This principle is used to build electron configurations.

94.1 References

  1. Griffiths, D. J. Introduction to Quantum Mechanics. Cambridge University Press, 2018. §4.2 — hydrogen wavefunctions.
  2. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — atomic orbitals.