Spherical Harmonics Quick Reference

This page is a compact reference for spherical harmonics as orbital angular-momentum states. It fixes conventions, records the formulas most often needed in calculations, and points to the canonical table and derivation pages.

For the physical interpretation, use Spherical Harmonics. For low-order explicit formulas, use Spherical Harmonics. For the mathematical basis and addition theorem details, use Spherical Harmonics.

Complex Spherical Harmonics

Use the spherical coordinates

$$\Omega=(\theta,\phi), \qquad 0\leq\theta\leq\pi, \qquad 0\leq\phi<2\pi,$$

with sphere measure

$$d\Omega=\sin\theta\,d\theta\,d\phi.$$

Use normalized complex spherical harmonics with the Condon–Shortley sign included in the associated Legendre function:

$$P_\ell^m(x)=(-1)^m(1-x^2)^{m/2}\frac{d^mP_\ell(x)}{dx^m},\qquad m\ge0.$$

Then

$$Y_\ell^m(\theta,\phi) = \sqrt{ \frac{2\ell+1}{4\pi} \frac{(\ell-m)!}{(\ell+m)!} } P_\ell^m(\cos\theta)e^{im\phi}.$$

Negative-$m$ functions are fixed by

$$Y_\ell^{-m} = (-1)^m \left(Y_\ell^m\right)^*, \qquad m>0.$$

If a reference defines $P_\ell^m$ without the factor $(-1)^m$, that factor must instead be placed outside in $Y_\ell^m$. Include it exactly once. For example, $Y_1^1=-\sqrt{3/(8\pi)}\sin\theta\,e^{i\phi}$ in the convention here.

Labels and Eigenvalue Equations

The labels are

$$\ell=0,1,2,\ldots, \qquad m=-\ell,-\ell+1,\ldots,\ell.$$

For ordinary scalar wavefunctions on the sphere, $\ell$ is an integer. Spherical harmonics diagonalize $L^2$ and $L_z$:

$$L^2Y_\ell^m = \hbar^2\ell(\ell+1)Y_\ell^m, \qquad L_zY_\ell^m = \hbar mY_\ell^m.$$

The labels $\ell$ and $m$ are dimensionless. The eigenvalues contain the factors of $\hbar$.

For a fixed $\ell$, the angular multiplet has

$$2\ell+1$$

states. This is the orbital counterpart of the usual $j$ multiplet dimension.

Orthonormality and Completeness

The normalized harmonics satisfy

$$\int_0^{2\pi} \int_0^\pi Y_\ell^m(\theta,\phi)^* Y_{\ell'}^{m'}(\theta,\phi) \sin\theta\,d\theta\,d\phi = \delta_{\ell\ell'}\delta_{mm'}.$$

The coefficients of an angular function are

$$c_{\ell m} = \int_{S^2} Y_\ell^m(\Omega)^* f(\Omega)\,d\Omega,$$

and for $f\in L^2(S^2)$ the expansion converges in the $L^2$ norm:

$$f(\Omega) = \sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{\ell} c_{\ell m}Y_\ell^m(\Omega).$$

The completeness relation is best read distributionally:

$$\sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{\ell} Y_\ell^m(\Omega) Y_\ell^m(\Omega')^* = \delta(\Omega,\Omega').$$

It is a statement about what happens under angular integration, not an ordinary pointwise equality.

Ladder Action

With

$$L_\pm=L_x\pm iL_y,$$

the ladder action is

$$L_\pm Y_\ell^m = \hbar \sqrt{ \ell(\ell+1)-m(m\pm1) } Y_\ell^{m\pm1}.$$

Endpoint functions satisfy

$$L_+Y_\ell^\ell=0, \qquad L_-Y_\ell^{-\ell}=0.$$

Use Ladder Operators for the derivation and Angular Momentum Identity Index for the broader identity map.

Conjugation, Parity, and Real Bases

Complex conjugation gives

$$\left(Y_\ell^m\right)^* = (-1)^mY_\ell^{-m}.$$

Under spatial inversion,

$$\mathbf r\mapsto-\mathbf r \quad \Longleftrightarrow \quad (\theta,\phi)\mapsto(\pi-\theta,\phi+\pi),$$

and

$$Y_\ell^m(\pi-\theta,\phi+\pi) = (-1)^\ell Y_\ell^m(\theta,\phi).$$

Thus orbital parity is $(-1)^\ell$ for scalar central-potential wavefunctions.

Chemistry and visualization pages often use real linear combinations of $Y_\ell^m$ instead of the complex $L_z$ eigenbasis. Those real tesseral harmonics are useful for drawing orbitals, but they are usually not eigenfunctions of $L_z$ when $m\ne0$. The complex convention is the default for angular-momentum algebra.

Addition Theorem

For two directions $\Omega$ and $\Omega'$, let $\gamma$ be the angle between them. Then

$$\sum_{m=-\ell}^{\ell} Y_\ell^m(\Omega)^* Y_\ell^m(\Omega') = \frac{2\ell+1}{4\pi} P_\ell(\cos\gamma).$$

This theorem explains why Legendre polynomials appear when an angular problem is rotationally invariant after summing over $m$. It is used in partial waves, multipole expansions, central Green functions, and rotationally invariant kernels.

Setting $\Omega'=\Omega$ gives the useful identity

$$\sum_{m=-\ell}^{\ell} \left|Y_\ell^m(\Omega)\right|^2 = \frac{2\ell+1}{4\pi}.$$

Central-Potential Dictionary

For a spinless central-potential problem,

$$V(\mathbf r)=V(r),$$

one may separate

$$\psi(r,\theta,\phi) = R_{n\ell}(r)Y_\ell^m(\theta,\phi)$$

when the radial problem has a discrete radial label. The spherical harmonic carries:

• the $L^2$ eigenvalue $\hbar^2\ell(\ell+1)$;
• the $L_z$ eigenvalue $\hbar m$;
• the angular normalization;
• the parity $(-1)^\ell$;
• the $2\ell+1$ rotational multiplet structure.

The radial function carries the dynamics of the specific potential. For the derivation, use Angular and Radial Separation. For the symmetry interpretation, use Central Potentials and Rotational Symmetry.

Selection-Rule Pointers

Spherical harmonics are scalar-function realizations of irreducible angular-momentum multiplets. Angular integrals with products of spherical harmonics inherit angular-momentum selection rules:

• azimuthal phases impose magnetic quantum-number conservation;
• triangle inequalities appear when products are decomposed into coupled angular momenta;
• parity imposes even-or-odd constraints on integrals over the sphere.

For the general symmetry logic, use Selection Rules. For tensor-operator matrix elements, use Wigner–Eckart Theorem. For recoupling conventions, use Wigner Symbols.

References

• G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, 7th ed., Academic Press, 2013.

• A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, 1957.

• NIST Digital Library of Mathematical Functions, Section 14.30, Spherical harmonics.

• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.

• D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific, 1988.

Quick Checks

1. How many spherical harmonics exist for $\ell=3$?

Solution

For fixed $\ell$, the magnetic label runs from $-\ell$ to $\ell$. Thus

$$2\ell+1=7$$

spherical harmonics exist for $\ell=3$.

2. What is the parity of $Y_4^m$?

Solution

Orbital parity is $(-1)^\ell$. For $\ell=4$,

$$(-1)^4=+1,$$

so every $Y_4^m$ has even parity.

3. Evaluate $\sum_{m=-1}^{1}\left|Y_1^m(\Omega)\right|^2$.

Solution

Use the addition theorem with $\Omega'=\Omega$:

$$\sum_{m=-\ell}^{\ell} \left|Y_\ell^m(\Omega)\right|^2 = \frac{2\ell+1}{4\pi}.$$

For $\ell=1$,

$$\sum_{m=-1}^{1} \left|Y_1^m(\Omega)\right|^2 = \frac{3}{4\pi}.$$