Orbital Angular Momentum

Angular momentum controls the structure of every system with rotational symmetry — above all the hydrogen atom. Remarkably, the entire spectrum of the angular-momentum operators follows from a single commutation algebra, without solving any differential equation. That algebraic solution, carried out here for orbital angular momentum, generalizes verbatim to spin and underlies the addition of angular momenta.

1. Definition and Algebra

The orbital angular momentum operator is $\hat{\mathbf{L}}=\hat{\mathbf{r}}\times \hat{\mathbf{p}}$, with components ${\hat{L}}_i={ϵ}_{ijk}{\hat{x}}_j{\hat{p}}_k$. Using $[{\hat{x}}_i,{\hat{p}}_j]=i\hslash {\delta }_{ij}$ one finds the defining angular-momentum algebra:

$$\boxed{[{\hat{L}}_i,{\hat{L}}_j]=i\hslash {ϵ}_{ijk}{\hat{L}}_k.}$$

The three components do not commute, so they cannot be simultaneously diagonalized. But the total square ${\hat{L}}^2={\hat{L}}_x^2+{\hat{L}}_y^2+{\hat{L}}_z^2$ commutes with each:

$$[{\hat{L}}^2,{\hat{L}}_i]=0.$$

We may therefore diagonalize ${\hat{L}}^2$ together with one component, conventionally ${\hat{L}}_z$.

2. Ladder Operators and the Spectrum

Define ${\hat{L}}_{\pm }={\hat{L}}_x\pm i{\hat{L}}_y$. The algebra gives

$$[{\hat{L}}_z,{\hat{L}}_{\pm }]=\pm \hslash {\hat{L}}_{\pm },[{\hat{L}}^2,{\hat{L}}_{\pm }]=0,{\hat{L}}_{\mp }{\hat{L}}_{\pm }={\hat{L}}^2-{\hat{L}}_z^2\mp \hslash {\hat{L}}_z.$$

So ${\hat{L}}_{\pm }$ raises/lowers the ${\hat{L}}_z$-eigenvalue by $\hslash$ while preserving ${\hat{L}}^2$. Label simultaneous eigenstates $|\ell ,m⟩$:

$${\hat{L}}^2|\ell ,m⟩={\hslash }^2\ell (\ell +1)|\ell ,m⟩,{\hat{L}}_z|\ell ,m⟩=\hslash m|\ell ,m⟩.$$

Because ${\hat{L}}_z^2\le {\hat{L}}^2$, the ladder must terminate at both ends, which forces $m$ to run in integer steps between $-\ell$ and $+\ell$. Hence $2\ell$ is a non-negative integer, and:

$$\boxed{m=-\ell ,-\ell +1,\dots ,+\ell (2\ell +1\text{ values}),}$$

with action

$${\hat{L}}_{\pm }|\ell ,m⟩=\hslash \sqrt{\ell (\ell +1)-m(m\pm 1)}|\ell ,m\pm 1⟩.$$

The algebra alone permits both integer and half-integer $\ell$; orbital angular momentum realizes only integer $\ell$ (see §4). The half-integer case is physical for spin.

3. Spherical Harmonics

In the position representation with spherical coordinates, ${\hat{L}}_z=-i\hslash {\partial }_{\varphi }$ and ${\hat{L}}^2$ is (proportional to) the angular part of the Laplacian. Their simultaneous eigenfunctions are the spherical harmonics:

$$⟨\theta ,\varphi |\ell ,m⟩=Y_{\ell }^m(\theta ,\varphi )=(-1{)}^m\sqrt{\frac{(2\ell +1)}{4\pi }\frac{(\ell -m)!}{(\ell +m)!}}{P}_{\ell }^m(\cos \theta )e^{im\varphi },$$

with $P_{\ell }^m$ the associated Legendre functions. They are orthonormal on the sphere, $\int Y_{\ell }^{m\ast }{Y}_{{\ell }^{\prime }}^{m^{\prime }}d\Omega ={\delta }_{\ell {\ell }^{\prime }}{\delta }_{m{m}^{\prime }}$, and complete: any function on $S^2$ expands in them. Low cases: $Y_0^0=1/\sqrt{4\pi }$ (isotropic $s$-wave); $Y_1^0\propto \cos \theta$ (the $p_z$ orbital).

4. Why Orbital $\ell$ Is an Integer

The azimuthal dependence $e^{im\varphi }$ must be single-valued under $\varphi \to \varphi +2\pi$, forcing $m\in \mathbb{Z}$, hence $\ell \in \mathbb{Z}$. This is the topological reason orbital angular momentum excludes half-integers: it is defined on configuration space (the sphere), where a $2\pi$ rotation is the identity. Spin has no such constraint because it is not tied to a spatial wavefunction — which is exactly how half-integer angular momentum enters physics.

5. Central Potentials

When $V=V(r)$ depends only on the radius, $[\hat{H},\hat{\mathbf{L}}]=0$, so $\hat{H},{\hat{L}}^2,{\hat{L}}_z$ share eigenstates and the wavefunction separates:

$$\psi (r,\theta ,\varphi )=R_{n\ell }(r)Y_{\ell }^m(\theta ,\varphi ).$$

The angular problem is solved once and for all by the spherical harmonics; only the radial equation for $R_{n\ell }(r)$ depends on the specific potential. This reduction is what makes the hydrogen atom tractable, and the $(2\ell +1)$-fold $m$-degeneracy is the generic signature of rotational symmetry (see symmetries & conservation).

See also

• Spin — the same algebra with half-integer eigenvalues.
• Addition of angular momenta — combining $\hat{\mathbf{L}}$ and $\hat{\mathbf{S}}$.
• Hydrogen atom — the radial problem in a Coulomb potential.
• math/group-theory/00-README.md — $SO(3)$ and its representations.