Quantization of Angular Momentum

Equivalent forms

\hat{L}^2\,Y_l^m = l(l+1)\hbar^2\,Y_l^m

\hat{L}_z\,Y_l^m = m\hbar\,Y_l^m, \quad m = -l, \dots, +l

The vector is always longer than its largest projection — perfect alignment would violate the uncertainty principle between L_x and L_y.

Symbol	Name	SI unit	Dimension	Range
$L$	Angular momentum magnitudeoutput Length of the orbital angular momentum vector	J*s	ML^2T^-1	0 – 5e-34
$L_z$	z-projectionoutput Component of angular momentum along the chosen axis	J*s	ML^2T^-1	-5e-34 – 5e-34
$l$	Orbital quantum number Non-negative integer labeling the L^2 eigenvalue	dimensionless	1	0 – 4
$m$	Magnetic quantum number Integer projection label, restricted to -l <= m <= +l	dimensionless	1	-4 – 4
$hbar$	Reduced Planck constant h/(2*pi)	J*s	ML^2T^-1	1.054571817e-34 – 1.054571817e-34

Unit systems

Symbol	SI	CGS	Imperial
$L$	J*s	erg*s	ftlbfs
$L_z$	J*s	erg*s	ftlbfs
$l$	dimensionless	dimensionless	dimensionless
$m$	dimensionless	dimensionless	dimensionless
$hbar$	J*s	erg*s	ftlbfs

Where it holds

Exact for orbital angular momentum of any non-relativistic system; l integer for orbital motion, half-integer values allowed only for intrinsic spin. Spin-orbit coupling reorganizes states into total angular momentum j.

Dimensional analysis

|L| -> M*L^2*T^-1

sqrt(l(l+1))*hbar -> dimensionless * M*L^2*T^-1

Both sides carry units of action (angular momentum).

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Discovery

Arnold Sommerfeld · 1916

Sommerfeld proposed 'space quantization' to explain spectral fine structure — orbits could only tilt at discrete angles. Stern and Gerlach confirmed discrete splitting in 1922, and the full operator theory with the strange sqrt(l(l+1)) magnitude emerged from matrix mechanics in 1925-26.

Try this

Why can an electron's orbit only tilt at certain angles?

Classically a spinning body can point anywhere. Quantum mechanically, the magnitude of angular momentum and its projection on any axis come in discrete rungs — and the vector can never fully align with the axis you measure.

Research status: stable

Real-world applications

Selection rules $(\delta l = +/-1)$ governing atomic spectra and lasers
Rotational spectroscopy used to identify molecules in interstellar clouds
MRI: nuclear spin projections in a magnetic field
Quantum numbers organizing the periodic table's s, p, d, f blocks

Common misconceptions

|L| is NOT l*hbar — it is sqrt(l(l+1))*hbar, always longer than the maximum projection l*hbar
The vector-model cone is a visualization: L_x and L_y have no definite values, they are not secretly rotating
m quantization holds along ANY chosen axis — but only one axis at a time

Experimental verification

Stern-Gerlach experiment (1922): a silver beam through an inhomogeneous magnet splits into discrete spots, not a smear. Zeeman splitting of spectral lines counts exactly 2l+1 components. Rotational spectra of diatomic molecules show energies proportional to l(l+1), confirming the magnitude formula.

Derivation

The components obey [L_x,

L_y] = i

*hbar*L_z and cyclic permutations.

Define ladder operators

L_pm = L_x +/- i*L_y

.

Since L^2 commutes with L_z, simultaneous eigenstates exist; ladder operators shift m by 1 while preserving l.

Requiring the ladder to terminate at both ends (norms must stay non-negative) forces eigenvalues

L^2 = l(l+1)

*hbar^2 with m running from -l to +l in integer steps — pure algebra, no differential equations needed.

Limiting cases

l = 0

⟶

L = 0

, only

m = 0

s-states are spherically symmetric — no angular momentum at all.

m = +l

(maximum projection)⟶

\cos (\theta ) = l/\sqrt(l(l+1))

< 1Even maximal alignment leaves the vector tilted — L_x and L_y retain zero-point fluctuations.

l -> infinity⟶ sqrt(l(l+1)) -> l, spacing/L -> 0Correspondence principle: rungs blur into the classical continuum of orientations.

What if…

What if L could align exactly with the z-axis?

Then $L_x = L_y = 0$ exactly, violating [L_x, $L_y] = i$ *hbar*L_z. The sqrt(l(l+1)) excess length is the uncertainty principle in disguise.

What if l could be half-integer for orbital motion?

The wavefunction Y_l^m would change sign after a full 2*pi rotation in space — impossible for a single-valued spatial function. Half-integers belong to spin only.

What if you measure L_x right after L_z?

The state collapses onto an L_x eigenstate, randomizing L_z — successive measurements along different axes never settle.

1

Magnitude and tilt for a d-electron (l = 2)

Given ·

l:: 2
m:: 2

Find · |L| and minimum tilt angle

Steps

| $L| = \sqrt(l(l+1))$ *hbar $= \sqrt{6} * 1.054571817e-34$
$\sim 2.58e-34\,\mathrm{J}*s$
$\cos (\theta ) = m/\sqrt(l(l+1)) = 2/2.449 = 0.816$ -> $\theta \sim 35.3\,\mathrm{deg}$

Result · |

L| = \sqrt{6}

*hbar

\sim 2.58e-34\,\mathrm{J}*s

;

\cos (\theta ) = 2/\sqrt{6} = 0.816

,

\theta \sim 35.3

degrees.

2

Counting Zeeman components

Given ·

l:: 1

Find · number of m levels

Steps

m runs from -l to +l in integer steps
count $= 2l + 1 = 3$
Energy shifts: $\delta E = m * \mu _B * B$ (normal Zeeman effect)

Result ·

2l + 1 = 3

levels

(m = -1

, 0, +1) — a p-line splits into a triplet in a magnetic field.

Spin-1/2 Operators→Stern–Gerlach Deflection→Bohr Hydrogen Energy Levels→Time-Independent Schrödinger Equation→