16. Angular Momentum

Before going into details, an introduction is likely to be required.

Spatial (orbital) angular momentum, \(\hat{\mathbf{L}}\). Spatial angular motion is the quantum mechanical counterpart of the angular momentum in classical mechanics, \(\hat{L} = \mathbf{r} \times \mathbf{p}\), due to the spatial motion of the system w.r.t. a point. The angular momentum operator in quantum mechanics can be defined promoting the classical space and momentum variables to the corresponding operators,

\[\mathbf{L} = \mathbf{r} \times \mathbf{p} \qquad \rightarrow \qquad \hat{\mathbf{L}} = \hat{\mathbf{r}} \times \hat{\mathbf{p}} \ ,\]

whose representation using spatial basis reads

\[\langle \mathbf{r} | \hat{\mathbf{L}} = -i \hbar \mathbf{r} \times \nabla_{\mathbf{r}} \langle \mathbf{r} | \ .\]

This operator has several properties that will be investigated in details in the dedicated section. This operator applies to a wave function representing the position, momentum, and physical quantities with classical analogous, here called \(| \psi \rangle\).

Spin (intrinsic) angular momentum, \(\hat{\mathbf{S}}\). Quantum systems may have intrinsic properties, with no classical counterpart. Spin momentum is one of these variables. Spin angular momentum \(\hat{\mathbf{S}}\) is defined in analogy with the spatial orbital momentum, with the same properties and acts on a spin state vector \(| s \rangle\).

Thus, the full state of the system belongs to a space state that is a tensor product of a space and a spin state space,

\[\mathcal{H} = \mathcal{H}_{space} \otimes \mathcal{H}_{spin} \ ,\]

defined as the tensor product of a spatial state vector and a spin state vector,

\[| \Psi \rangle = | \psi \rangle \otimes | s \rangle \ .\]

Total angular momentum \(\hat{\mathbf{J}}\). Total angular momentum can be defined as the sum of the spatial and the spin angular momentum,

\[\hat{\mathbf{J}} = \left( \hat{\mathbf{L}} \otimes \hat{\mathbf{1}} \right) + \left( \hat{\mathbf{1}} \otimes \hat{\mathbf{S}} \right) \ ,\]

or, remembering that \(\hat{\mathbf{L}}\) only acts on \(| \psi \rangle\) and \(\hat{\mathbf{S}}\) onlt acts on \(| s \rangle\), it’s usually written as \(\hat{\mathbf{J}} = \hat{\mathbf{L}} + \hat{\mathbf{S}}\).