Chapter 06
Hydrogen and central forces
The first real three-dimensional system, and the starting point of all chemistry.
- Sections
- 7
- Finalised
- 7/7
- Simulations
- 0
- Estimated time
- 3 hours
- Read first
- Chapter 05
- 6.1Reducing the central-force problemTwo preparatory moves before we storm the hydrogen atom: turn the "electron plus proton" two-body problem into a one-body problem, then use spherical symmetry to split the 3D equation into radial times angular. After these two steps, all that remains is a single one-dimensional equation.
- 6.2The radial equation and the effective potentialThe substitution u = rR restyles the radial equation into a one-dimensional Schrödinger equation, at the price of a centrifugal barrier added to the potential. A single plot of the effective potential then reads off the existence of bound states, why larger l keeps the electron farther out, and why the Coulomb potential holds infinitely many levels.
- 6.3The complete analytic solution of hydrogenThe full derivation from the radial equation to −13.6 eV/n²: nondimensionalisation, pinning down the asymptotics at both ends, the series solution, and the termination condition that decides everything. The foundation of all of chemistry, with no step skipped.
- 6.4Energy levels, degeneracy, and quantum numbersThree quantum numbers, three jobs: n runs the energy, l the magnitude of angular momentum, m the orientation. Count the n²-fold degeneracy of each level, ask what entitles the Coulomb potential to its "accidental" surplus, then use selection rules to connect the level diagram to the real spectrum.
- 6.5The spatial distribution of probability densityHow to "look at" an orbital correctly: probability density and the radial distribution are two different things — the 1s density peaks on the nucleus, yet the most probable radius is a₀; the angular shapes are orientation distributions of probability, not racetracks for the electron. The most widespread misreadings, dismantled one by one.
- 6.6Fine structureTurn the spectrometer up one notch and Hα splits in two — a 0.016 nm gap the Coulomb model cannot produce. Three corrections of order α² (relativistic kinetic energy, spin–orbit coupling, the Darwin term) join forces to reshuffle the levels by j, and 1/137 steps to centre stage.
- 6.7The Zeeman and Stark effectsSplitting levels from the outside: a magnetic field picks out a direction in space and cracks the m degeneracy open (Zeeman); an electric field mixes degenerate states of opposite parity and promotes the shift from quadratic to linear (Stark). The perturbative idea of "the bigger term calls the tune" gets its first live combat here — straight into Chapter 7.