Spherical Harmonics

This table gives low-order spherical harmonics in the common physics convention.

Convention

The harmonics use the Condon–Shortley phase and are normalized by

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

They obey

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

Low-Order Harmonics

Harmonic	Formula
$Y_{00}$	$\frac{1}{\sqrt{4\pi}}$
$Y_{10}$	$\sqrt{\frac{3}{4\pi}}\cos\theta$
$Y_{1,\pm1}$	$\mp\sqrt{\frac{3}{8\pi}}\sin\theta\,e^{\pm i\phi}$
$Y_{20}$	$\sqrt{\frac{5}{16\pi}}(3\cos^2\theta-1)$
$Y_{2,\pm1}$	$\mp\sqrt{\frac{15}{8\pi}}\sin\theta\cos\theta\,e^{\pm i\phi}$
$Y_{2,\pm2}$	$\sqrt{\frac{15}{32\pi}}\sin^2\theta\,e^{\pm2i\phi}$

Associated Operators

Operator	Eigenvalue Equation
$L^2$	$L^2Y_{\ell m}=\hbar^2\ell(\ell+1)Y_{\ell m}$
$L_z$	$L_zY_{\ell m}=\hbar mY_{\ell m}$

Common Mistakes

• Dropping the Condon–Shortley phase and then comparing signs across sources.
• Forgetting the angular measure $\sin\theta\,d\theta\,d\phi$.
• Confusing $Y_{\ell m}$ with real tesseral harmonics used in chemistry.
• Treating angular normalization as radial normalization.

Canonical Links

• Spherical Harmonics Quick Reference
• Spherical Harmonics
• Angular and Radial Separation
• Rigid Rotor
• Hydrogen Atom

References

• A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, 1957.
• G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, 7th ed., Academic Press, 2013.
• D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific, 1988.