Introduction: Splitting the Problem
For central potentials \(V = V(r)\), Cartesian coordinates are inefficient. Spherical coordinates \((r, \theta, \phi)\) exploit the symmetry, separating the equation into radial and angular parts.
The Laplacian in Spherical Coordinates
\[\nabla^2 = \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial}{\partial r}\right) + \frac{1}{r^2\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right) + \frac{1}{r^2\sin^2\theta}\frac{\partial^2}{\partial\phi^2}\]Separation Ansatz
For central \(V(r)\), try \(\psi(r, \theta, \phi) = R(r)Y(\theta, \phi)\):
The angular part separates further: \(Y(\theta, \phi) = \Theta(\theta)\Phi(\phi)\)
This leads to three ODEs: radial, polar, and azimuthal.
The Angular Equation
The angular part gives eigenvalue equation:
\[\hat{L}^2 Y = \hbar^2 l(l+1) Y\]Solutions are spherical harmonics \(Y_l^m(\theta, \phi)\).
The Quantum Connection
Separation in spherical coordinates reveals angular momentum as a good quantum number. The angular dependence is universal for all central potentials—only the radial part depends on \(V(r)\). This is why all atoms share the same orbital shapes (s, p, d, f...) despite having different radial structures.