---
title: Orbital Angular Momentum and Spherical Harmonics
module: Angular Momentum
moduleNumber: 6
lessonNumber: 1
order: 601
summary: >
  Orbital angular momentum is the operator triple built from position and
  momentum. Its components fail to commute, so no state carries sharp values of
  more than one of them, but each commutes with the total square. Solving the
  common eigenvalue problem in spherical coordinates quantizes both the magnitude
  and the projection and produces the spherical harmonics, the angular part of
  every central-force wavefunction.
topics: [Angular Momentum]
sources:
  - book: Griffiths & Schroeter
    ref: "§4.1.2 Angular Equation; §4.3 Angular Momentum"
  - book: Shankar
    ref: "Ch. 12 — Rotational Invariance and Angular Momentum; §12.5–§12.6"
  - book: Cohen-Tannoudji
    ref: "Ch. VI — General Properties of Angular Momentum in Quantum Mechanics"
draft: false
---

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](/quantum-mechanics/central-potentials/the-schrodinger-equation-in-three-dimensions)
eigenstate, hydrogen included.

## The orbital angular momentum operator

The definition copies the classical cross product with the
[canonical operators](/quantum-mechanics/formalism/position-momentum-and-continuous-spectra)
$\hat x_i$ and $\hat p_i = -i\hbar\,\partial_i$:

$$
\vec L = \vec r \times \vec p,
\qquad
L_i = \varepsilon_{ijk}\,x_j\,p_k,
$$

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

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

Because $x_j$ and $p_k$ commute whenever $j \neq k$, 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.

$$
% caption: The angular-momentum vector is perpendicular to the plane of the
% position and momentum vectors, with magnitude fixed by the enclosed area rate.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
% origin
\fill[black] (0,0) circle (1.6pt);
\node[anchor=north east] at (0,0) {O};
% r and p in a plane
\draw[->, very thick] (0,0) -- (3.2,0.6) node[anchor=west] {position};
\draw[->, black, very thick] (3.2,0.6) -- (4.4,2.4) node[anchor=west, text=black] {momentum};
\node[anchor=north] at (1.6,0.3) {$r$};
\node[black, anchor=west] at (3.8,1.5) {$p$};
% L out of plane (drawn upward)
\draw[->, acc, very thick] (1.2,0.6) -- (1.2,3.2) node[anchor=south, text=acc] {$L = r$ x $p$};
% little swept area
\draw[black, dashed] (0,0) -- (4.4,2.4);
\fill[black!8] (0,0) -- (3.2,0.6) -- (4.4,2.4) -- cycle;
\end{tikzpicture}
$$

## Commutation relations

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

$$
[L_x, L_y] = [\,y\,p_z - z\,p_y,\; z\,p_x - x\,p_z\,].
$$

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:

$$
[\,y\,p_z,\; z\,p_x\,] = y\,[p_z, z]\,p_x = -i\hbar\,y\,p_x,
\qquad
[\,z\,p_y,\; x\,p_z\,] = x\,p_y\,[z, p_z] = +i\hbar\,x\,p_y .
$$

Adding them,

$$
[L_x, L_y] = i\hbar\,(x\,p_y - y\,p_x) = i\hbar\,L_z .
$$

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:

> **Definition (Angular-momentum algebra).** The Hermitian operators
> $L_x, L_y, L_z$ satisfy
> $$
> [L_i, L_j] = i\hbar\,\varepsilon_{ijk}\,L_k .
> $$
> No two distinct components commute, so no state is a simultaneous eigenstate of
> two of them (except the trivial $\vec L = 0$).

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

$$
L^2 = L_x^2 + L_y^2 + L_z^2 .
$$

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

$$
[L_x^2, L_z] = L_x[L_x, L_z] + [L_x, L_z]L_x = -i\hbar\,(L_x L_y + L_y L_x),
$$
$$
[L_y^2, L_z] = L_y[L_y, L_z] + [L_y, L_z]L_y = +i\hbar\,(L_y L_x + L_x L_y),
$$

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

$$
[L^2, L_z] = 0,
\qquad\text{and by symmetry}\qquad
[L^2, L_i] = 0 \;\; \text{for every } i.
$$

> **Theorem (Compatible angular-momentum observables).** $L^2$ commutes with each
> $L_i$ while the $L_i$ do not commute with one another. The largest set of
> mutually [compatible](/quantum-mechanics/formalism/observables-hermitian-operators-and-eigenvalues)
> angular-momentum observables is therefore $\{L^2, L_z\}$ (any single axis may
> replace $z$). A complete basis of angular states is labelled by the joint
> eigenvalues of this pair.

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](/quantum-mechanics/angular-momentum/the-angular-momentum-algebra)
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
$(r, \theta, \varphi)$, with $\theta \in [0,\pi]$ the polar angle from the
$z$-axis and $\varphi \in [0, 2\pi)$ the azimuth. Writing the Cartesian
derivatives in terms of $r, \theta, \varphi$ 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,

$$
L_z = -i\hbar\,\frac{\partial}{\partial \varphi},
$$

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,

$$
L^2 = -\hbar^2\!\left[
  \frac{1}{\sin\theta}\,\frac{\partial}{\partial\theta}
  \!\left(\sin\theta\,\frac{\partial}{\partial\theta}\right)
  + \frac{1}{\sin^2\theta}\,\frac{\partial^2}{\partial\varphi^2}
\right].
$$

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

$$
\nabla^2 = \frac{1}{r^2}\,\frac{\partial}{\partial r}\!\left(r^2\,\frac{\partial}{\partial r}\right)
  - \frac{L^2}{\hbar^2 r^2},
$$

so the eigenvalue of $L^2$ is what sets the centrifugal term in
the [radial equation](/quantum-mechanics/central-potentials/the-schrodinger-equation-in-three-dimensions).
Fixing the angular momentum fixes the rotational kinetic energy at each radius.

### Angular momentum generates rotations

The identification of $L_z$ with $-i\hbar\,\partial_\varphi$ 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
$\delta\varphi$ about $z$. The rotated function evaluated at $\varphi$ equals the
original evaluated at $\varphi - \delta\varphi$, so to first order

$$
\psi(\varphi - \delta\varphi)
  = \psi(\varphi) - \delta\varphi\,\frac{\partial\psi}{\partial\varphi}
  = \left(1 - \frac{i}{\hbar}\,\delta\varphi\,L_z\right)\psi(\varphi).
$$

Exponentiating a sequence of infinitesimal steps builds the finite rotation
operator

$$
R_z(\phi) = \exp\!\left(-\frac{i}{\hbar}\,\phi\,L_z\right),
$$

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
$[L_i, L_j] = i\hbar\varepsilon_{ijk}L_k$. The commutator is the algebra of the
rotation group. This is the vantage point of the
[symmetry-and-generators](/quantum-mechanics/oscillator-and-symmetry/symmetries-generators-and-conservation-laws)
treatment, where conservation of angular momentum in a central potential follows
from $[L_i, H] = 0$, i.e. from the rotational invariance of $H$.[^shankar-rot]

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 $e^{-2\pi i m} = 1$, 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 $\theta, \varphi$
only, they share a complete set of eigenfunctions $Y(\theta, \varphi)$ on the
unit sphere. Write the two eigenvalue equations with dimensionless eigenvalues
$\lambda$ and $m$:

$$
L^2\,Y = \hbar^2\,\lambda\,Y,
\qquad
L_z\,Y = \hbar\,m\,Y .
$$

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

$$
% caption: The angular eigenfunction $Y(\theta,\varphi)$ factors into a polar
% amplitude $\Theta(\theta)$ that fixes the magnitude and an azimuthal phase
% $\Phi(\varphi)$ that fixes the projection.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0,
  bx/.style={draw, minimum width=30mm, minimum height=13mm, align=center}]
\definecolor{acc}{HTML}{4A6FA5}
\node[bx, thick] (Y) at (0,0) {angular state\\$Y$};
\node[bx] (T) at (5.0,1.2) {polar part\\(sets magnitude)};
\node[bx] (P) at (5.0,-1.2) {azimuthal part\\(sets projection)};
\draw[->, thick] (Y) -- (T);
\draw[->, thick] (Y) -- (P);
\node[anchor=west, font=\scriptsize] at (6.8,1.2) {$L^{2}$ eigenvalue};
\node[anchor=west, font=\scriptsize] at (6.8,-1.2) {$L_{z}$ eigenvalue};
\end{tikzpicture}
$$

### The azimuthal equation quantizes the projection

$L_z Y = \hbar m Y$ reads $-i\hbar\,\Phi' = \hbar m\,\Phi$, with solution

$$
\Phi(\varphi) = e^{i m \varphi}.
$$

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

$$
m \in \{\,0, \pm 1, \pm 2, \dots\,\} \quad (\text{integer}).
$$

This is the first quantization: the projection $L_z = m\hbar$ takes only integer
multiples of $\hbar$. 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](/quantum-mechanics/spin/spin-half-pauli-matrices-and-stern-gerlach); requiring
$\Phi$ to be single-valued is what removes them here.[^gs-azimuth]

### The polar equation quantizes the magnitude

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

$$
\frac{1}{\sin\theta}\,\frac{\d}{\d\theta}\!\left(\sin\theta\,\frac{\d\Theta}{\d\theta}\right)
  + \left[\lambda - \frac{m^2}{\sin^2\theta}\right]\Theta = 0.
$$

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

$$
\lambda = \ell(\ell + 1),
\qquad
\ell \in \{0, 1, 2, \dots\},
\qquad
|m| \le \ell .
$$

Requiring a normalizable, everywhere-finite $\Theta$ is what truncates an
otherwise-divergent power series, exactly as boundedness quantizes the
[oscillator](/quantum-mechanics/wave-mechanics-1d/operators-expectation-values-and-the-harmonic-oscillator).
The regular solutions are the associated Legendre functions
$P_\ell^{m}(\cos\theta)$, built from the Legendre polynomials $P_\ell(u)$ by

$$
P_\ell^{m}(u) = (1 - u^2)^{|m|/2}\,\frac{\d^{|m|}}{\d u^{|m|}}\,P_\ell(u),
\qquad
P_\ell(u) = \frac{1}{2^\ell\,\ell!}\,\frac{\d^\ell}{\d u^\ell}(u^2 - 1)^\ell .
$$

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.

> **Definition (Orbital quantum numbers).** The azimuthal (or orbital) quantum
> number $\ell = 0, 1, 2, \dots$ fixes the magnitude through
> $L^2 = \hbar^2 \ell(\ell+1)$; the magnetic quantum number
> $m = -\ell, -\ell+1, \dots, \ell-1, \ell$ fixes the projection $L_z = m\hbar$.
> For each $\ell$ there are exactly $2\ell + 1$ allowed values of $m$.

The magnitude is $|\vec L| = \hbar\sqrt{\ell(\ell+1)}$, strictly larger than the
maximum projection $\hbar\ell$ 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.

$$
% caption: Allowed states form a triangle: each row of fixed l holds 2l+1
% projections m running from minus l to plus l in unit steps.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=0.95]
\definecolor{acc}{HTML}{4A6FA5}
% axes
\draw[->, black] (-3.6,0) -- (3.6,0) node[anchor=west, black] {$m$};
\draw[->, black] (0,-0.4) -- (0,4.2) node[anchor=south, black] {$l$};
% rows
\foreach \l in {0,1,2,3} {
  \node[anchor=east, font=\scriptsize] at (-3.75, \l) {$l = \l$};
  \foreach \m in {-3,-2,-1,0,1,2,3} {
    \ifnum\m>\l \else \ifnum\m<-\l \else
      \fill[acc] (\m, \l) circle (2.6pt);
    \fi\fi
  }
}
% m tick labels along bottom
\foreach \m/\lab in {-3/{-3}, -2/{-2}, -1/{-1}, 0/{0}, 1/{+1}, 2/{+2}, 3/{+3}}
  \node[anchor=north, font=\scriptsize, black!70] at (\m, -0.15) {\lab};
\end{tikzpicture}
$$

## Spherical harmonics

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

$$
Y_\ell^{m}(\theta, \varphi)
  = \epsilon
    \sqrt{\frac{(2\ell+1)}{4\pi}\,\frac{(\ell - |m|)!}{(\ell + |m|)!}}\;
    P_\ell^{|m|}(\cos\theta)\,e^{i m \varphi},
$$

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

$$
L^2\,Y_\ell^{m} = \hbar^2\,\ell(\ell+1)\,Y_\ell^{m},
\qquad
L_z\,Y_\ell^{m} = \hbar\,m\,Y_\ell^{m},
$$

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

$$
\int_0^{2\pi}\!\!\int_0^{\pi} Y_{\ell'}^{m'\,\ast}(\theta,\varphi)\,
  Y_\ell^{m}(\theta,\varphi)\,\sin\theta\,\d\theta\,\d\varphi
  = \delta_{\ell'\ell}\,\delta_{m'm}.
$$

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

- **Parity.** The inversion $\vec r \to -\vec r$ maps
  $(\theta, \varphi) \to (\pi - \theta, \varphi + \pi)$, under which
  $Y_\ell^{m} \to (-1)^\ell\,Y_\ell^{m}$. Angular states have definite
  [parity](/quantum-mechanics/oscillator-and-symmetry/parity-time-reversal-and-discrete-symmetries)
  $(-1)^\ell$, 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
$\ell = 0,1,2,3 \leftrightarrow s, p, d, f$:

$$
Y_0^{0} = \frac{1}{\sqrt{4\pi}},
\qquad
Y_1^{0} = \sqrt{\frac{3}{4\pi}}\,\cos\theta,
\qquad
Y_1^{\pm 1} = \mp\sqrt{\frac{3}{8\pi}}\,\sin\theta\,e^{\pm i\varphi},
$$
$$
Y_2^{0} = \sqrt{\frac{5}{16\pi}}\,(3\cos^2\theta - 1),
\qquad
Y_2^{\pm 1} = \mp\sqrt{\frac{15}{8\pi}}\,\sin\theta\,\cos\theta\,e^{\pm i\varphi},
\qquad
Y_2^{\pm 2} = \sqrt{\frac{15}{32\pi}}\,\sin^2\theta\,e^{\pm 2 i\varphi}.
$$

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

$$
% caption: Angular probability lobes: the isotropic s state, the two-lobe p
% state aligned with z, and the pinched d state, drawn as polar plots of the
% magnitude versus polar angle.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
% --- s: circle ---
\begin{scope}[xshift=0cm]
  \draw[->, black] (0,-1.6) -- (0,1.9) node[anchor=south, black, font=\scriptsize] {$z$};
  \draw[acc, very thick, fill=acc!10] (0,0) circle (1.1);
  \node[anchor=north, font=\scriptsize] at (0,-1.75) {$s$ (l=0)};
\end{scope}
% --- p_z: two lobes ---
\begin{scope}[xshift=4.4cm]
  \draw[->, black] (0,-1.9) -- (0,2.1) node[anchor=south, black, font=\scriptsize] {$z$};
  \draw[acc, very thick, fill=acc!10]
    (0,0) .. controls (1.5,0.4) and (0.9,1.7) .. (0,1.7)
          .. controls (-0.9,1.7) and (-1.5,0.4) .. (0,0);
  \draw[acc, very thick, fill=acc!10]
    (0,0) .. controls (1.5,-0.4) and (0.9,-1.7) .. (0,-1.7)
          .. controls (-0.9,-1.7) and (-1.5,-0.4) .. (0,0);
  \node[anchor=north, font=\scriptsize] at (0,-2.05) {$p$ (l=1, m=0)};
\end{scope}
% --- d_z2: pinched ---
\begin{scope}[xshift=8.8cm]
  \draw[->, black] (0,-2.0) -- (0,2.2) node[anchor=south, black, font=\scriptsize] {$z$};
  \draw[acc, very thick, fill=acc!10]
    (0,0) .. controls (1.2,0.3) and (0.7,1.8) .. (0,1.8)
          .. controls (-0.7,1.8) and (-1.2,0.3) .. (0,0);
  \draw[acc, very thick, fill=acc!10]
    (0,0) .. controls (1.2,-0.3) and (0.7,-1.8) .. (0,-1.8)
          .. controls (-0.7,-1.8) and (-1.2,-0.3) .. (0,0);
  \draw[acc, very thick, fill=acc!10] (0,0) ellipse (1.35 and 0.4);
  \node[anchor=north, font=\scriptsize] at (0,-2.15) {$d$ (l=2, m=0)};
\end{scope}
\end{tikzpicture}
$$

## The vector model and space quantization

The eigenvalues admit a semiclassical picture that predates wave mechanics and
still guides intuition. Fix $\ell$: the length $|\vec L| = \hbar\sqrt{\ell(\ell+1)}$
is sharp, and the projection $L_z = m\hbar$ 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

$$
\cos\alpha = \frac{L_z}{|\vec L|} = \frac{m}{\sqrt{\ell(\ell+1)}}.
$$

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.

$$
% caption: Vector model for l = 2: the angular momentum of fixed length
% $|\vec L| = \hbar\sqrt{6}$ lies on one of five cones, one per allowed
% projection, with the transverse direction uniformly uncertain.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
% z axis
\draw[->, black] (0,-3.1) -- (0,3.4) node[anchor=south, black] {$L_{z}$};
% five projection levels for l=2, |L| scaled to 2.75 (represents sqrt 6):
% m=2  height 2.246 radius 1.587 rim 0.49
% m=1  height 1.123 radius 2.510 rim 0.75
% m=0  height 0.000 radius 2.750 rim 0.82
% m=-1 height -1.123 radius 2.510 rim 0.75
% m=-2 height -2.246 radius 1.587 rim 0.49
\draw[black, dashed] (-1.587,2.246) -- (1.587,2.246);
\draw[black, dashed] (-2.510,1.123) -- (2.510,1.123);
\draw[black, dashed] (-2.750,0) -- (2.750,0);
\draw[black, dashed] (-2.510,-1.123) -- (2.510,-1.123);
\draw[black, dashed] (-1.587,-2.246) -- (1.587,-2.246);
\draw[->, acc, very thick] (0,0) -- (1.587,2.246);
\draw[->, acc, very thick] (0,0) -- (2.510,1.123);
\draw[->, acc, very thick] (0,0) -- (2.750,0);
\draw[->, acc, very thick] (0,0) -- (2.510,-1.123);
\draw[->, acc, very thick] (0,0) -- (1.587,-2.246);
\draw[black!55] (0,2.246) ellipse (1.587 and 0.49);
\draw[black!55] (0,1.123) ellipse (2.510 and 0.75);
\draw[black!55] (0,0) ellipse (2.750 and 0.82);
\draw[black!55] (0,-1.123) ellipse (2.510 and 0.75);
\draw[black!55] (0,-2.246) ellipse (1.587 and 0.49);
\node[anchor=west, font=\scriptsize, black!75] at (2.95,2.246) {m = +2};
\node[anchor=west, font=\scriptsize, black!75] at (2.95,1.123) {m = +1};
\node[anchor=west, font=\scriptsize, black!75] at (2.95,0) {m = 0};
\node[anchor=west, font=\scriptsize, black!75] at (2.95,-1.123) {m = -1};
\node[anchor=west, font=\scriptsize, black!75] at (2.95,-2.246) {m = -2};
\end{tikzpicture}
$$

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.

> **Worked example.** Take $\ell = 1$. The magnitude is
> $|\vec L| = \hbar\sqrt 2$ and the three projections are $L_z \in \{+\hbar, 0, -\hbar\}$.
> The largest cone half-angle is
> $$
> \alpha = \arccos\!\frac{1}{\sqrt 2} = 45^\circ,
> $$
> so even the most-aligned $\ell = 1$ state tips a full $45^\circ$ off the axis.
> The transverse spread follows from
> $\langle L_x^2 \rangle + \langle L_y^2 \rangle
> = \langle L^2 \rangle - \langle L_z^2 \rangle
> = 2\hbar^2 - \hbar^2 = \hbar^2$ in the $m = 1$ state, split evenly as
> $\langle L_x^2 \rangle = \langle L_y^2 \rangle = \tfrac{1}{2}\hbar^2$ by
> symmetry about $z$.

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 $\hbar^2 j(j+1)$ and $\hbar m$ emerge from the commutation relations
alone once the coordinate crutch is dropped, and that abstraction is what the
[angular-momentum algebra](/quantum-mechanics/angular-momentum/the-angular-momentum-algebra)
develops next, admitting the half-integer multiplets that spin requires.

[^gs-azimuth]: Griffiths & Schroeter, _Introduction to Quantum Mechanics_, 3rd ed. (Cambridge, 2018), §4.1.2 — the azimuthal equation and the single-valuedness condition $\Phi(\varphi + 2\pi) = \Phi(\varphi)$ restricting $m$ to integers. Publisher: https://doi.org/10.1017/9781316995433
[^shankar-rot]: Shankar, _Principles of Quantum Mechanics_, 2nd ed. (Springer, 1994), Ch. 12 §12.2–§12.3 — rotational invariance, angular momentum as the generator of rotations, and the derivation of $[L_i, L_j] = i\hbar\varepsilon_{ijk}L_k$ from the non-commutativity of finite rotations. Publisher: https://doi.org/10.1007/978-1-4757-0576-8