Orbital Angular Momentum and Spherical Harmonics

Classically the angular momentum of a particle about the origin is $\vec{L}=\vec{r}\times \vec{p}$, a vector that is conserved whenever the force is central. Promoting $\vec{r}$ and $\vec{p}$ to operators makes $\vec{L}$ an operator, and the non-commutativity of position and momentum propagates into the components of $\vec{L}$. The consequence is structural: the three components cannot be measured simultaneously, but the magnitude and one component can. Diagonalizing that compatible pair over the sphere quantizes angular momentum and yields the spherical harmonics, the functions that carry the angular dependence of every central-potential eigenstate, hydrogen included.

The orbital angular momentum operator

The definition copies the classical cross product with the canonical operators${\hat{x}}_i$ and ${\hat{p}}_i=-i\hslash {\partial }_i$:

with the Levi-Civita symbol ${\epsilon }_{ijk}$ and the summation convention. In Cartesian components,

Because $x_j$ and $p_k$ commute whenever $j\ne k$, there is no ordering ambiguity in $L_i$: each term pairs a coordinate with a momentum along a different axis. Each $L_i$ is Hermitian, being a difference of products of commuting Hermitian operators, so its eigenvalues are real and it qualifies as an observable.

Commutation relations

The single algebraic fact from which everything else follows is the commutator of two components. Using $[x_i,p_j]=i\hslash {\delta }_{ij}$ and bilinearity,

Expanding into four commutators, only the two that pair the shared axis $z$ survive, because $[y{p}_z,x{p}_z]$ and $[z{p}_y,z{p}_x]$ each contain no conjugate pair:

Adding them,

The other two follow by cyclic permutation $x\to y\to z\to x$. The compact statement is the defining relation of the rotation-group algebra:

The non-commutativity is not a computational nuisance; it is the reason a classical angular-momentum vector, with three simultaneously definite components, has no quantum counterpart. The best one can do is fix the length and a single projection.

The total square commutes with each component

Define the total-square operator

It commutes with every component. Take $L_z$ and use $[A^2,B]=A[A,B]+[A,B]A$ on each term:

and $[L_z^2,L_z]=0$. The first two cancel term by term, so

Choosing $z$ is a convention; the physics singles out no axis. What the algebra forbids is sharpening two projections at once. This lesson diagonalizes $\{L^2,L_z\}$ as differential operators on the sphere; the next lesson extracts the same spectrum from the commutators alone, without reference to coordinates, and thereby uncovers the half-integer values that orbital motion misses.

Angular momentum in spherical coordinates

Central problems are separable in spherical coordinates $(r,\theta ,\phi )$, with $\theta \in [0,\pi ]$ the polar angle from the $z$-axis and $\phi \in [0,2\pi )$ the azimuth. Writing the Cartesian derivatives in terms of $r,\theta ,\phi$ and substituting into the operators above collapses the radial dependence entirely: $\vec{L}$ acts only on the angles. The projection along $z$ is the generator of rotations about that axis,

a result that also reads off the interpretation of $L_z$ as the conjugate of the azimuthal angle. The total square becomes the angular part of the Laplacian,

The connection to the kinetic energy is direct: the Laplacian separates as

so the eigenvalue of $L^2$ is what sets the centrifugal term in the radial equation. Fixing the angular momentum fixes the rotational kinetic energy at each radius.

Angular momentum generates rotations

The identification of $L_z$ with $-i\hslash {\partial }_{\phi }$ is not a coincidence of spherical coordinates; it expresses that $L_z$ is the generator of rotations about the $z$-axis. Consider rotating a wavefunction by an infinitesimal angle $\delta \phi$ about $z$. The rotated function evaluated at $\phi$ equals the original evaluated at $\phi -\delta \phi$, so to first order

Exponentiating a sequence of infinitesimal steps builds the finite rotation operator

a unitary operator because $L_z$ is Hermitian. The same construction with $L_x$ and $L_y$ generates rotations about the other axes, and the failure of rotations about different axes to commute is encoded, at the infinitesimal level, in $[L_i,L_j]=i\hslash {\epsilon }_{ijk}{L}_k$. The commutator is the algebra of the rotation group. This is the vantage point of the symmetry-and-generators treatment, where conservation of angular momentum in a central potential follows from $[L_i,H]=0$, i.e. from the rotational invariance of $H$.¹

A single-valued wavefunction must return to itself under a full $2\pi$ rotation, $R_z(2\pi )=1$. Acting on $Y_{\ell }^m$ this reads $e^{-2\pi im}=1$, reproducing integer $m$ from the group-theoretic side. The half-integer representations of the rotation algebra satisfy $R_z(2\pi )=-1$ instead, and are excluded for orbital motion precisely because a spatial wavefunction cannot change sign under a rotation that returns every point to itself.

The eigenvalue problem on the sphere

Since $[L^2,L_z]=0$ and both are differential operators in $\theta ,\phi$ only, they share a complete set of eigenfunctions $Y(\theta ,\phi )$ on the unit sphere. Write the two eigenvalue equations with dimensionless eigenvalues $\lambda$ and $m$:

$L_z$ depends only on $\phi$ and $L^2$ mixes $\theta$ and $\phi$, so try the product $Y(\theta ,\phi )=\Theta (\theta )\Phi (\phi )$.

The azimuthal equation quantizes the projection

$L_{z}Y=\hslash mY$ reads $-i\hslash {\Phi }^{\prime }=\hslash m\Phi$, with solution

Single-valuedness on the circle, $\Phi (\phi +2\pi )=\Phi (\phi )$, forces $e^{2\pi im}=1$, hence

This is the first quantization: the projection $L_z=m\hslash$ takes only integer multiples of $\hslash$. The integrality is specific to orbital angular momentum, where $\Phi$ is a genuine function of a spatial angle. The purely algebraic treatment permits half-integers as well, and those describe spin; requiring $\Phi$ to be single-valued is what removes them here.²

The polar equation quantizes the magnitude

Substituting $Y=\Theta e^{im\phi }$ into $L^{2}Y={\hslash }^2\lambda Y$ and dividing out the phase gives an ordinary differential equation for $\Theta$:

With $u=\cos \theta$ this is the associated Legendre equation. Its solutions are finite on the closed interval $\theta \in [0,\pi ]$ (equivalently $u\in [-1,1]$) only when

Requiring a normalizable, everywhere-finite $\Theta$ is what truncates an otherwise-divergent power series, exactly as boundedness quantizes the oscillator. The regular solutions are the associated Legendre functions $P_{\ell }^m(\cos \theta )$, built from the Legendre polynomials $P_{\ell }(u)$ by

The differentiation caps the order at $|m|\le \ell$: applying more than $\ell$ derivatives to a degree-$\ell$ polynomial annihilates it. That single fact links the two quantum numbers.

The magnitude is $|\vec{L}|=\hslash \sqrt{\ell (\ell +1)}$, strictly larger than the maximum projection $\hslash \ell$ for every $\ell \ge 1$. The vector can never align with the measurement axis. This is the geometric face of the uncertainty relation among the components, and the source of the vector-model picture below.

Spherical harmonics

Assembling the normalized product gives the spherical harmonics, the joint eigenfunctions of $L^2$ and $L_z$:

where the Condon–Shortley phase is $ϵ=(-1{)}^m$ for $m\ge 0$ and $ϵ=1$ for $m<0$. They satisfy

and form an orthonormal basis for square-integrable functions on the sphere:

Completeness means any angular function expands as $f(\theta ,\phi )={\sum }_{\ell ,m}{c}_{\ell m}{Y}_{\ell }^m$ with $c_{\ell m}=\int Y_{\ell }^{m\ast }f\mathbf{d}\Omega$, the angular analog of a Fourier series. Two structural properties recur throughout atomic physics:

• Parity. The inversion $\vec{r}\to -\vec{r}$ maps $(\theta ,\phi )\to (\pi -\theta ,\phi +\pi )$, under which $Y_{\ell }^m\to (-1{)}^{\ell }{Y}_{\ell }^m$. Angular states have definite parity$(-1{)}^{\ell }$, which drives the dipole selection rule $\Delta \ell =\pm 1$.
• Complex conjugation. $Y_{\ell }^{m\ast }=(-1{)}^m{Y}_{\ell }^{-m}$, so the $\pm m$ pair carries the same polar amplitude and opposite azimuthal winding.

The lowest harmonics, written in the spectroscopic naming $\ell =0,1,2,3\leftrightarrow s,p,d,f$:

The quantity with physical meaning is the angular probability density $|Y_{\ell }^m{|}^2$, independent of $\phi$ because $|e^{im\phi }|=1$. Its polar shape gives the familiar orbital lobes.

The vector model and space quantization

The eigenvalues admit a semiclassical picture that predates wave mechanics and still guides intuition. Fix $\ell$: the length $|\vec{L}|=\hslash \sqrt{\ell (\ell +1)}$ is sharp, and the projection $L_z=m\hslash$ takes $2\ell +1$ discrete values. Because the transverse components $L_x,L_y$ have zero mean but nonzero spread in an $L_z$ eigenstate, one visualizes $\vec{L}$ as a vector of fixed length lying on a cone about the $z$-axis, its azimuth completely undetermined. The permitted cones are the ones whose half-angle $\alpha$ satisfies

The name space quantization records the surprising content: the orientation of the angular-momentum vector relative to a chosen axis is restricted to discrete values, the effect first isolated by Stern and Gerlach.

The gap between $\sqrt{\ell (\ell +1)}=\sqrt{6}\approx 2.449$ and the largest projection $m=2$ is the quantitative statement that the topmost cone still opens at a finite angle $\alpha =\arccos (2/\sqrt{6})\approx 35.3^{\circ }$; the vector never stands straight up. As $\ell$ grows the ratio $\ell /\sqrt{\ell (\ell +1)}\to 1$, the top cone closes onto the axis, and the discrete orientations crowd into the continuum of classical directions, an instance of the correspondence principle.

The whole construction here rests on $\vec{L}$ being built from spatial position and momentum, which forced integer $\ell$ through single-valuedness. The same eigenvalues ${\hslash }^{2}j(j+1)$ and $\hslash m$ emerge from the commutation relations alone once the coordinate crutch is dropped, and that abstraction is what the angular-momentum algebra develops next, admitting the half-integer multiplets that spin requires.