Introduction: The Most Important Atom
Hydrogen—one proton, one electron—is the only atom exactly solvable in quantum mechanics. Its solution is the foundation for understanding all atoms and molecules.
The Potential
\[V(r) = -\frac{e^2}{4\pi\epsilon_0 r} = -\frac{e^2}{r} \quad \text{(Gaussian units)}\]Attractive Coulomb potential between proton and electron.
Natural Units
Bohr radius: \(a_0 = \frac{4\pi\epsilon_0\hbar^2}{me^2} = 0.529\) Å
Rydberg energy: \(E_R = \frac{me^4}{2(4\pi\epsilon_0)^2\hbar^2} = 13.6\) eV
The Radial Equation
\[\left[-\frac{\hbar^2}{2m}\frac{d^2}{dr^2} + \frac{\hbar^2 l(l+1)}{2mr^2} - \frac{e^2}{4\pi\epsilon_0 r}\right]u = Eu\]Competition between centrifugal repulsion and Coulomb attraction.
The Quantum Connection
Hydrogen's spectrum was the first quantum mystery. Classical physics predicted the electron should spiral into the nucleus, radiating continuously. Instead, discrete spectral lines emerge—explained by quantized energy levels. Solving the hydrogen atom was a triumph of quantum mechanics.