Angular Momentum Operators

Equivalent forms

\hat{\vec{L}} = \hat{\vec{r}} \times \hat{\vec{p}}

[\hat{L}^2, \hat{L}_z] = 0

[\hat{L}_i,\hat{L}_j] = i\hbar\,\epsilon_{ijk}\hat{L}_k

The entire ladder of angular-momentum states follows purely from three commutators — no differential equations required.

Symbol	Name	SI unit	Dimension	Range
$Lz$	z-componentoutput Eigenvalue of L_z in units of hbar	J*s	M L^2 T^-1	-4 – 4
$l$	Total quantum number Orbital quantum number l (sets L^2 = l(l+1) hbar^2)	1	dimensionless	0 – 5
$Lmag$	Magnitudeoutput \|L\| = sqrt(l(l+1)) hbar	J*s	M L^2 T^-1	0 – 6

Unit systems

Symbol	SI	CGS	Imperial
$Lz$	J*s	erg*s	J*s
$l$	1	1	1
$Lmag$	J*s	erg*s	J*s

Where it holds

Defines both orbital angular momentum (integer l) and, by the same algebra, spin (half-integer allowed); relativistic corrections add spin-orbit coupling.

Dimensional analysis

[L_x, L_y] has units (J s)^2/... actually J s times J s

hbar

L_z \sim (J s)(J s)

Both sides carry (angular momentum

)^2 = (J s)^2

.

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Discovery

Born, Heisenberg & Jordan · 1925

The angular-momentum commutation relations emerged from the 1925 matrix mechanics of Born, Heisenberg and Jordan. They encode rotational symmetry and became the template for all quantum spin — orbital and intrinsic alike.

Try this

Why can't you know all three components of a spin at once?

Angular momentum in quantum mechanics is built from operators that refuse to commute. Their commutation relations — not any picture of a spinning ball — dictate that only one component and the total magnitude can be known together.

Research status: stable

Real-world applications

Selection rules for atomic and molecular spectra
Angular-momentum coupling in NMR and ESR spectroscopy
Rotational band structure in molecular and nuclear physics

Common misconceptions

L is not a classical spinning arrow — its components genuinely cannot be simultaneously sharp
The magnitude is sqrt(l(l+1)) hbar, not l*hbar, so L_z is always strictly less than |L|
Spin obeys the same algebra but allows half-integer values, which orbital L cannot

Experimental verification

Stern-Gerlach beams split into exactly 2l+1 components, directly counting the allowed m values. Rotational spectra of molecules show energy levels proportional to l(l+1), confirming the L^2 eigenvalue.

Derivation

From

L = r x p

and [x_i,

p_j] = i

hbar delta_{ij} one computes [L_x,

L_y] = i

hbar L_z and cyclic permutations, i.e. [L_i,

L_j] = i

hbar epsilon_{ijk} L_k.

Because the three components don't commute, they share no common eigenbasis.

But

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

commutes with every component, so {L^2, L_z} is a maximal commuting set.

Their simultaneous eigenstates |l,m> satisfy L^2|l,m>

= l(l+1)

hbar^2 |l,m> and L_z|l,m>

= m

hbar |l,m> with

m = -l

,...,+l.

The ladder operators

L_pm = L_x +/- i L_y

step m by one.

Limiting cases

l = 0

⟶

L = 0

, spherically symmetricAn s-state has no angular momentum and no preferred axis.

commuting pair (L^2, L_z)⟶ simultaneous eigenstates |l, m>Magnitude and one projection are compatible observables.

large l⟶ L_z/|L| can approach 1The vector can point almost along z — the classical limit of a well-defined direction.

What if…

What if the components did commute?

You could fix all three at once and angular momentum would point in a definite direction — the classical, non-quantum case.

What if you measure L_x after L_z?

Measuring L_x randomizes L_z, because the two don't share eigenstates — successive projections disturb each other.

1

Magnitude for l = 2

Given ·

l:: 2

Find · |L| and allowed m

Steps

| $L| = \sqrt(l(l+1))$ hbar
$= \sqrt{6}$ hbar
m ranges -l..l -> 5 values

Result · |

L| = \sqrt{2*3}

hbar

= \sqrt{6}

hbar

\sim 2.45

hbar;

m = -2

,-1,0,1,2 (five states).

2

Do L_x and L^2 commute?

Given ·

operators:: L_x, L^2

Find · compatibility

Steps

$L^2 = L_x^2+L_y^2+L_z^2$
[L^2, $L_i] = 0$ for all i
L^2 compatible with any single component

Result · [L^2,

L_x] = 0

, so L^2 commutes with every component; you may know |L| and L_x together (just not L_x and L_z).

Quantization of Angular Momentum→Canonical Commutation Relation→Spin-1/2 Operators→Stern–Gerlach Deflection→