PU-306 · Atomic & Molecular Physics
Starting from the exact solution of the one-electron atom, the unit builds outward through the hierarchy of corrections — relativistic, spin, nuclear, and external-field — that turn idealized levels into the real spectrum, then confronts electron-electron interaction to explain many-electron atoms and the periodic table. It closes by coupling atoms to the radiation field and to each other, deriving transition rates and selection rules and extending the machinery via Born-Oppenheimer to the rotational, vibrational, and electronic structure of molecules.
Lectures
| L01 | The Atomic Scale and the Shape of the Problem — |
| L02 | Central Potentials: Separating Radial and Angular Motion |
| L03 | The Hydrogen Atom Solved |
| L04 | Hydrogen Wavefunctions, Degeneracy, and Quantum Numbers |
| L05 | Why Hydrogen Is Not the Whole Story — |
| L06 | Relativistic Corrections to the Kinetic Energy |
| L07 | Spin-Orbit Coupling |
| L08 | The Fine Structure of Hydrogen |
| L09 | The Darwin Term and a Glimpse of the Lamb Shift |
| L10 | Hyperfine Structure and the 21 cm Line |
| L11 | Atoms in Magnetic Fields: The Zeeman Effect |
| L12 | Atoms in Electric Fields: The Stark Effect |
| L13 | Identical Particles and Exchange Symmetry |
| L14 | Helium: Ortho, Para, and the Exchange Splitting |
| L15 | Variational Estimates of Atomic Energies |
| L16 | Many-Electron Atoms: The Central-Field Approximation — |
| L17 | Hartree-Fock and Self-Consistent Fields |
| L18 | Building the Periodic Table: Term Symbols and Hund's Rules |
| L19 | Radiation and Atoms: Time-Dependent Perturbation Theory — |
| L20 | Selection Rules for Dipole Radiation |
| L21 | Einstein Coefficients, Spontaneous Emission, and Lasers |
| L22 | Lineshapes, Widths, and Lifetimes |
| L23 | From Atoms to Molecules: The Born-Oppenheimer Approximation |
| L24 | The Hydrogen Molecular Ion and the Chemical Bond |
| L25 | Molecular Rotation and Vibration |
| L26 | Rovibrational Spectra and the Franck-Condon Principle |
| L27 | Modern Atomic Physics: Laser Cooling and Trapping — |
Derivations homed in this unit
Separation of the Central-Potential Schrodinger Equation
Separating a spherically symmetric Hamiltonian into spherical-harmonic angular factors and a one-dimensional radial equation with an effective centrifugal potential.
Bound-State Spectrum of the Hydrogen Atom
Solving the Coulomb radial equation to obtain E_n = -13.6 eV / n^2, the associated Laguerre wavefunctions, and the n^2 degeneracy.
Relativistic Kinetic-Energy Correction
Deriving the first-order p^4 correction to hydrogen levels by expanding the relativistic kinetic energy and applying non-degenerate perturbation theory.
Spin-Orbit Coupling from the Rest Frame
Deriving the L.S interaction from the magnetic field seen in the electron rest frame, including the Thomas precession factor of one half.
Fine-Structure Energy Formula
Combining the p^4, spin-orbit, and Darwin corrections via degenerate perturbation theory to give levels depending only on n and j.
Hyperfine Splitting and the 21 cm Line
Deriving the magnetic-dipole coupling between electron and nuclear spins that splits the hydrogen ground state and produces the 21 cm transition.
Anomalous Zeeman Effect and the Lande g-Factor
Deriving the weak-field Zeeman splitting proportional to the Lande g-factor and its crossover to the Paschen-Back regime.
Linear Stark Effect in Hydrogen
Using degenerate perturbation theory on the n=2 manifold to obtain a splitting linear in the applied electric field, absent in non-hydrogenic atoms.
Exchange Symmetry and the Ortho/Para Splitting
Antisymmetrizing the two-electron wavefunction to produce the exchange integral and the singlet-triplet energy splitting of helium.
Variational Ground-State Energy of Helium
Estimating the helium ground-state energy with a screened-charge trial wavefunction, optimizing the effective nuclear charge.
The Hartree-Fock Self-Consistent-Field Equations
Deriving the coupled one-electron equations with direct and exchange potentials by minimizing the energy of a Slater determinant.
LS Coupling, Term Symbols, and Hund's Rules
Ordering many-electron terms by residual electrostatic and spin-orbit energies to justify term symbols and Hund's rules.
Electric-Dipole Selection Rules
Deriving Delta-l=+/-1 and Delta-m=0,+/-1 and the parity rule from the angular matrix elements of the dipole operator via the Wigner-Eckart theorem.
Einstein Coefficients and Spontaneous Emission
Relating the A and B coefficients through detailed balance with the Planck spectrum and computing the stimulated rate from Fermi's golden rule.
The Born-Oppenheimer Approximation
Separating fast electronic from slow nuclear motion using the nuclear-to-electron mass ratio to define potential-energy surfaces.
The Hydrogen Molecular Ion and the Chemical Bond
Building bonding and antibonding states of H2+ by the LCAO method and showing how electron sharing lowers the energy to bind the nuclei.
Molecular Rotational and Vibrational Spectra
Deriving rigid-rotor levels B J(J+1) and harmonic plus Morse-anharmonic vibrational levels, and combining them into rovibrational P and R branches.
The Franck-Condon Principle
Deriving vibronic transition intensities as squared overlaps of vibrational wavefunctions under the assumption of instantaneous electronic transitions.