Angular Momentum Operators

Angular momentum is one of the deepest organizing principles in quantum mechanics. It governs the structure of atoms, the selection rules of spectroscopy, and the classification of elementary particles. In this module we build the full quantum theory of angular momentum from its operators.

From classical to quantum

In classical mechanics the angular momentum of a particle about the origin is the cross product of its position and momentum,

\mathbf{L} = \mathbf{r} \times \mathbf{p}.

Writing out the three Cartesian components,

L_x = y\,p_z - z\,p_y, \qquad L_y = z\,p_x - x\,p_z, \qquad L_z = x\,p_y - y\,p_x.

To pass to quantum mechanics we promote position and momentum to operators. The position operators act by multiplication, $\hat{x}\,\psi = x\,\psi$ , and the momentum operators are the differential operators

\hat{p}_x = -i\hbar\,\frac{\partial}{\partial x}, \qquad \hat{p}_y = -i\hbar\,\frac{\partial}{\partial y}, \qquad \hat{p}_z = -i\hbar\,\frac{\partial}{\partial z}.

The three component operators

Substituting the operator forms gives the quantum orbital angular momentum operators:

\hat{L}_x = -i\hbar\left(y\,\frac{\partial}{\partial z} - z\,\frac{\partial}{\partial y}\right),

\hat{L}_y = -i\hbar\left(z\,\frac{\partial}{\partial x} - x\,\frac{\partial}{\partial z}\right),

\hat{L}_z = -i\hbar\left(x\,\frac{\partial}{\partial y} - y\,\frac{\partial}{\partial x}\right).

Each carries units of $\hbar$ (joule-seconds), the natural quantum of angular momentum.

Why they are Hermitian

Each component operator is Hermitian, $\hat{L}_i^\dagger = \hat{L}_i$ , which is why angular momentum is an observable with real measured values. Hermiticity follows because position is Hermitian, the differential momentum operators are Hermitian, and in each product (such as $y\,\hat{p}_z$ ) the two factors commute — $y$ and $\partial/\partial z$ act on independent coordinates — so there is no ordering ambiguity to spoil the symmetry.

The role of L_z

The component $\hat{L}_z$ is especially convenient. In spherical coordinates it depends only on the azimuthal angle $\phi$ ,

\hat{L}_z = -i\hbar\,\frac{\partial}{\partial \phi},

so its eigenfunctions are the simple phases $e^{im\phi}$ . This is why $\hat{L}_z$ is almost always chosen as the component whose eigenvalues we label states by. The other two components do not share a common set of eigenstates with $\hat{L}_z$ , a fact we will trace directly to their commutation relations in the next lesson.

The takeaway

Quantum angular momentum is built by promoting $\mathbf{r}\times\mathbf{p}$ to operators. The three Hermitian components $\hat{L}_x, \hat{L}_y, \hat{L}_z$ are observables, they follow a cyclic $x\to y\to z$ symmetry, and $\hat{L}_z$ takes the simplest form. Everything that follows in this module — commutators, the total $\hat{L}^2$ , ladder operators, and the $(\ell, m)$ spectrum — flows from these definitions.