Introduction: The Power of Symmetry
A central potential depends only on distance from the origin: \(V = V(r)\). The resulting rotational symmetry means angular momentum is conserved and dictates the structure of solutions.
Conservation of Angular Momentum
For central \(V(r)\):
\[[\hat{H}, \hat{L}^2] = 0, \quad [\hat{H}, \hat{L}_z] = 0\]States can be labeled by energy, \(l\), and \(m\) simultaneously.
Structure of Solutions
\[\psi_{nlm}(r, \theta, \phi) = R_{nl}(r)Y_l^m(\theta, \phi)\]Where:
- \(R_{nl}(r)\): radial wavefunction (depends on specific \(V(r)\))
- \(Y_l^m(\theta, \phi)\): spherical harmonic (universal for all central potentials)
Examples of Central Potentials
- Free particle: \(V = 0\)
- Isotropic harmonic oscillator: \(V = \frac{1}{2}m\omega^2 r^2\)
- Coulomb potential: \(V = -\frac{e^2}{4\pi\epsilon_0 r}\) (hydrogen)
- Infinite spherical well: \(V = 0\) for \(r < a\), \(\infty\) otherwise
The Quantum Connection
All atoms have central potentials (to good approximation), so atomic physics is built on the mathematics of central forces. The \((2l + 1)\)-fold degeneracy of each \(l\) subshell comes from rotational symmetry. Electric and magnetic fields break this symmetry, splitting the levels.