Angular Momentum Eigenvalues

Equivalent forms

m=-l,-l+1,\dots,+l

The commutators of Lx, Ly, Lz alone force the spectrum: a vector that can never point straight along its own axis, quantized in half-integer steps.

Symbol	Name	SI unit	Dimension	Range
$\hat{L}^2$	Total angular momentum²output Eigenvalue ℏ²l(l+1)	J^2*s^2	M^2L^4T^-2	—
$l$	Orbital quantum number 0, 1, 2, ... (or half-integers for spin)	1	1	0 – 4
$m$	Magnetic quantum number Projection index, −l ≤ m ≤ l	1	1	-4 – 4
$\hat{L}_z$	Projection Eigenvalue ℏm along z	J*s	ML^2T^-1	—
$\hbar$	Reduced Planck constant h/2π, the quantum of action	J*s	ML^2T^-1	—

Where it holds

General for any angular momentum (orbital l integer; spin s can be half-integer). Orbital angular momentum forbids half-integers because wavefunctions must be single-valued.

Dimensional analysis

L^{2} \to [M L^2\,\mathrm{T}^-1]^{2} = [M^2*L^4*T^-2]

\hbar ^{2} l(l+1) \to [M L^2\,\mathrm{T}^-1]^{2}\times

(number)

Both sides are (angular momentum

)^{2}

.

\checkmark

Discovery

Wolfgang Pauli / Otto Stern & Walther Gerlach · 1922

The Stern–Gerlach experiment (1922) split a silver-atom beam into discrete spots, proving space quantization of angular momentum. Pauli, Born, and others soon derived the ℏ²l(l+1) and ℏm spectrum purely from the commutation relations [L_i,L_j]=iℏε_ijk L_k.

Research status: stable

Real-world applications

Atomic orbital shapes and the periodic table (s,p,d,f)
Zeeman/Stark spectroscopy and magnetic resonance
Selection rules for atomic transitions

Common misconceptions

| $L| = \hbar \surd (l(l+1))$ , $not \hbar l$ — the vector is longer than its maximum projection
Orbital l cannot be half-integer; only spin can
m is the projection quantum number, unrelated to mass

Experimental verification

Stern–Gerlach beam splitting into 2l+1 spots; Zeeman splitting of spectral lines into equally spaced

\hbar m

sublevels; ESR and NMR transitions between m states.

Derivation

From [L_x,

L_y]=i\hbar L_z

(and cyclic) build ladder operators

L_\pm = L_x \pm iL_y

with [L_z,

L_\pm ]=\pm \hbar L_\pm

.

L_\pm

raise/lower the L_z eigenvalue

by \hbar

.

Requiring the ladder to terminate top and bottom (finite ‖

L_\pm |l

,m⟩‖

^{2}\ge 0)

forces the top value

m=l

and bottom

m=-l

with 2l an integer, and gives

L^{2}=\hbar ^{2}l(l+1)

.

Hence m runs in integer steps from

-l

to +l, 2l+1 values.

What if…

What if

l = 1/2

(an electron's spin)?

Only $m = \pm 1/2$ — two states, and | $S| = \hbar \sqrt{3}/2$ > $\hbar /2$ , so spin can never point exactly along z. This is the Stern–Gerlach two-spot result.

What if l is very large?

$\surd (l(l+1)) \approx l+\tfrac{1}{2} \to the$ projection can nearly equal the magnitude, and the classical continuous angular momentum re-emerges (correspondence principle).

1

l = 2 multiplet

Given ·

l:: 2

Find · |L|, allowed m, and number of states

Steps

$L^{2} = \hbar ^{2}\cdot 2\cdot 3 = 6\hbar ^{2} \to |L| = \sqrt{6} \hbar$
m from $-l$ to +l: $-2$ , $-1$ ,0,1,2
$2l+1 = 5$ orientations

Result · |

L| = \hbar \sqrt{6}

;

m = -2

,

-1

,0,1,2; 5 states

Angular Momentum Operators→hydrogen wavefunctionsstern gerlach