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Mathematics for Physics

129 Orbital Angular Momentum

An angular momentum associated with spatial motion that is used to describe rotation of a particle about an origin in position space.

\(\mathbf{L}=\mathbf{r}\times\mathbf{p}\). Orbital angular momentum is built from the position and momentum operators. This principle is used to distinguish \(\mathbf{L}\) from spin.

The orbital angular momentum is

\[ \mathbf{L} = \mathbf{r}\times\mathbf{p} \]

where

  • \(\mathbf{L}\) is the orbital angular momentum.
  • \(\mathbf{r}\) is the position operator.
  • \(\mathbf{p}\) is the momentum operator.

The definition of \(L^{2}\). The squared operator is the sum of the squares of the components. This principle is used to write \(L^{2}\) in the radial Schrödinger equation.

The squared orbital angular momentum is

\[ L^{2} = L_{x}^{2} + L_{y}^{2} + L_{z}^{2} \]

where

  • \(L_{x}\), \(L_{y}\), and \(L_{z}\) are the Cartesian components.

The \((\ell,m)\) spectrum. Simultaneous eigenstates of \(L^{2}\) and \(L_{z}\) are labeled by integers \(\ell\) and \(m\). This principle is used to name \(s\), \(p\), \(d\), and \(f\) orbitals.

The orbital eigenvalue equations are

\[ L^{2}|\ell,m\rangle = \hbar^{2}\ell(\ell+1)|\ell,m\rangle \]

\[ L_{z}|\ell,m\rangle = m\hbar|\ell,m\rangle \]

where

  • \(\ell\) is the orbital quantum number.
  • \(m\) is the magnetic quantum number.
  • \(\hbar\) is the reduced Planck constant.

Note: Also denoted \(\mathbf{L}\). Also called orbital angular momentum to distinguish it from spin.

129.1 References

  1. Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics. Cambridge University Press, 2021. — \(\mathbf{L}=\mathbf{r}\times\mathbf{p}\).
  2. Hall, B. C. Quantum Theory for Mathematicians. Springer, 2013. — \(L^{2}=L_{x}^{2}+L_{y}^{2}+L_{z}^{2}\).
  3. Shankar, R. Fundamentals of Physics II. Yale University Press, 2020. — orbital versus spin angular momentum.