Spherical Harmonics

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

\int |Y_l^m|^2\,d\Omega = 1

One family of functions solves every rotationally symmetric problem, from hydrogen orbitals to the CMB sky map.

Symbol	Name	SI unit	Dimension	Range
$l$	Degree Orbital quantum number (number of angular nodes)	1	dimensionless	0 – 5
$m$	Order Magnetic quantum number, -l <= m <= l	1	dimensionless	-5 – 5
$Y2$	Angular densityoutput \|Y_l^m\|^2 as a function of polar angle	1	dimensionless	0 – 2

Unit systems

Symbol	SI	CGS	Imperial
$l$	1	1	1
$m$	1	1	1
$Y2$	1	1	1

Where it holds

Exact angular solutions of any central-potential Schrodinger equation; the complete orthonormal basis for functions on the sphere.

Dimensional analysis

Y_l^m on the unit sphere

N P_l^m e^{im phi}

Normalized to integral |Y|^2 dOmega

= 1

, so Y carries units of 1/sqrt(steradian

) =

dimensionless angular density.

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Discovery

Pierre-Simon Laplace · 1782

Laplace introduced spherical harmonics in 1782 while studying gravitational potentials. A century and a half later they reappeared as the exact angular solutions of the hydrogen atom, giving s, p, d and f orbitals their familiar shapes.

Try this

What shapes can a wave wrapped around a sphere make?

Whenever a problem has rotational symmetry — an atom, a vibrating planet, the cosmic microwave background — its angular part is built from spherical harmonics, the eigenfunctions of angular momentum that set the shapes of atomic orbitals.

Research status: stable

Real-world applications

Shapes of s, p, d, f atomic and molecular orbitals
Multipole expansion of the cosmic microwave background
Antenna radiation patterns and geopotential (gravity/geoid) models

Common misconceptions

Real orbitals (p_x, p_y) are real linear combinations of complex Y_l^{+/-m}
The number of angular nodes is l, independent of m's sign
|Y_l^m|^2 is axially symmetric (no phi dependence) even though Y itself carries e^{im phi}

Experimental verification

The angular shapes of atomic orbitals inferred from directional chemical bonding and from scanning-tunneling-microscope images of adsorbed molecules match |Y_l^m|^2. The multipole expansion of the cosmic microwave background is literally a spherical-harmonic decomposition of the sky.

Derivation

Separating the Schrodinger equation in spherical coordinates for a central potential factorizes

\psi = R(r) Y(\theta

, phi).

The angular equation is the eigenvalue problem for L^2:

L^2\,\mathrm{Y} = l(l+1)

hbar^2 Y with

L_z Y = m

hbar Y.

Its solutions are

Y_l^m = N_l^m P_l^m(\cos \theta ) e^{i m \varphi }

, where P_l^m are associated Legendre functions and N_l^m normalizes the integral over the sphere to one. l is a non-negative integer; m runs from -l to +l, giving 2l+1 states of a given l.

They are orthonormal: integral Y_l^m* Y_l'^m' dOmega

= \delta _{ll'} \delta _{mm'}

.

Limiting cases

l = 0

,

m = 0

⟶

Y_0^0 = 1/\sqrt{4\,\mathrm{pi}}

The s-orbital: perfectly spherical, no angular dependence.

l = 1

⟶ p-orbitals (p_x, p_y, p_z)Three dumbbell lobes along the axes.

m = +/- l

⟶ density hugs the equatorMaximum |m| concentrates the probability near the equatorial plane.

What if…

What if you add up |Y_l^m|^2 over all m at fixed l?

You get a constant (Unsold's theorem) — a filled subshell is spherically symmetric.

What if the potential is not central?

l and m stop being good quantum numbers; you must mix spherical harmonics to build the true angular states.

1

Count states for l = 3

Given ·

l:: 3

Find · number of m values

Steps

m ranges from -l to +l
count $= 2l+1$
$= 7$

Result ·

m = -3

,-2,-1,0,1,2,3 gives

2l+1 = 7

states (the f-subshell).

2

Shape of Y_1^0

Given ·

l:: 1
m:: 0

Find · angular density

Steps

$Y_1^0 \sim \cos (\theta )$
| $Y_1^0|^2 \sim \cos ^2(\theta )$
maxima at poles, node at equator

Result · |

Y_1^0|^2 \sim \cos ^2(\theta )

: two lobes along the z-axis — the p_z orbital.

hydrogen wavefunctionsAngular Momentum Operators→Quantization of Angular Momentum→Atomic Quantum Numbers→