2.1 Quantum Theory of the Hydrogen Atom
📚 Schrödinger Equation for the Hydrogen Atom
Electron in Coulomb potential \(V(r) = -\frac{e^2}{4\pi\epsilon_0 r}\):
\[ -\frac{\hbar^2}{2m}\nabla^2\psi - \frac{e^2}{4\pi\epsilon_0 r}\psi = E\psi \]
Separation of variables in spherical coordinates: \(\psi(r,\theta,\phi) = R(r)Y_l^m(\theta,\phi)\)
Radial equation:
\[ -\frac{\hbar^2}{2m}\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{dR}{dr}\right) + \left[\frac{\hbar^2 l(l+1)}{2mr^2} - \frac{e^2}{4\pi\epsilon_0 r}\right]R = ER \]
Energy eigenvalues:
\[ E_n = -\frac{me^4}{32\pi^2\epsilon_0^2\hbar^2 n^2} = -\frac{13.6\text{ eV}}{n^2}, \quad n = 1, 2, 3, \ldots \]
Summary
In this chapter, we studied 3D quantum systems. The hydrogen atom provides exact solutions for the Coulomb potential, characterized by quantum numbers (n, l, m). We examined radial wave functions described by Laguerre polynomials and their probability distributions. The spherical harmonics describe the angular dependence and shapes of atomic orbitals. Finally, the two-particle problem demonstrates the separation of center-of-mass motion, the concept of reduced mass, and isotope effects.
In the next chapter, we will study angular momentum theory and spin.