Spherical Harmonics

When angular momentum is orbital — arising from motion in space — its eigenstates are concrete functions on the sphere. These functions, the spherical harmonics $Y_\ell^m(\theta, \phi)$ , are the position-space realization of the abstract states $|\ell, m\rangle$ .

The operators in spherical coordinates

Using spherical coordinates $(r, \theta, \phi)$ , the angular-momentum operators become differential operators in the angles alone. In particular

\hat{L}_z = -i\hbar\,\frac{\partial}{\partial\phi},

\hat{L}^2 = -\hbar^2\left[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta} \left(\sin\theta\,\frac{\partial}{\partial\theta}\right) + \frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\phi^2}\right].

The bracketed expression is the angular part of the Laplacian. The eigenvalue problem $\hat{L}^2 Y = \hbar^2\ell(\ell+1) Y$ and $\hat{L}_z Y = \hbar m\,Y$ is a partial differential equation on the unit sphere.

The eigenfunctions

The simultaneous solutions are the spherical harmonics

Y_\ell^m(\theta, \phi) = N_\ell^m\,P_\ell^m(\cos\theta)\,e^{im\phi},

where $P_\ell^m$ is an associated Legendre function and $N_\ell^m$ is a normalization constant. The $\phi$ -dependence $e^{im\phi}$ is fixed immediately by the simple form of $\hat{L}_z$ , and single-valuedness under $\phi \to \phi + 2\pi$ forces $m$ — and hence $\ell$ — to be an integer. This is precisely why orbital angular momentum excludes the half-integer values that the abstract algebra otherwise permits.

The simplest examples

The lowest spherical harmonics are

Y_0^0 = \frac{1}{\sqrt{4\pi}}, \qquad Y_1^0 = \sqrt{\frac{3}{4\pi}}\,\cos\theta, \qquad Y_1^{\pm 1} = \mp\sqrt{\frac{3}{8\pi}}\,\sin\theta\,e^{\pm i\phi}.

$Y_0^0$ is constant — an $s$ state is perfectly spherically symmetric. The $\ell = 1$ harmonics have the angular shapes of the familiar $p_x, p_y, p_z$ orbitals once real linear combinations are taken.

Role in the hydrogen atom

Any central-potential problem, $V = V(r)$ , separates into a radial equation and an angular equation. The angular part is always solved by $Y_\ell^m$ , independent of the specific potential. For the hydrogen atom the full wavefunction factorizes as

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

so the spherical harmonics carry all the angular information of every atomic orbital. The quantum numbers $\ell$ and $m$ you learned to enumerate are exactly the labels on these functions.

The takeaway

Spherical harmonics $Y_\ell^m(\theta, \phi)$ are the orbital-angular-momentum eigenfunctions: eigenstates of both $\hat{L}^2$ (eigenvalue $\hbar^2\ell(\ell+1)$ ) and $\hat{L}_z$ (eigenvalue $\hbar m$ ). Their single-valuedness pins $\ell$ and $m$ to integers, and they supply the universal angular factor in every central-potential wavefunction.