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Atomic Units, Fine Structure, and Singlet-Triplet Splitting in Hydrogen and Helium | MIT OCW Physics Lecture
Introduction to Atomic Units
The lecture from MIT OpenCourseWare explains how the atom is described using natural or atomic units constructed from the electron charge, electron mass, and Planck's constant. The speed of light is not included in this non-relativistic formulation of the Schrödinger equation, which is appropriate for hydrogenic problems at low nuclear charges.
Fine Structure Constant and Relativity
The fine structure constant alpha, a dimensionless number, is discussed as a measure of electromagnetic interaction strength. Because alpha is small, electromagnetic effects are relatively weak in atomic physics within the non-relativistic framework, though relativistic corrections become significant at higher Z.
Atomic Scales and Units
The speaker derives typical scales: the Bohr radius for lengths, the Hartree energy for energies, typical orbital velocities v ≈ alpha c, and the electric field experienced by the electron in the 1s state. The Compton wavelength is introduced for comparison with atomic scales, illustrating how relativistic concepts enter at fundamental limits.
Hydrogenic Energies and Core Corrections
Hydrogenic energy levels scale as 1/n^2, but in multi-electron atoms, the ionic core modifies the potential felt by outer electrons. The leading correction is captured by a quantum defect Delta L, which shifts energies away from the pure hydrogenic formula. The defect is largely determined by the core region and L, and can be understood via perturbation theory and semi-classical pictures of phase shifts near the core.
Two-Electron Atoms: Helium and Exchange
Moving to helium, the simplest two-electron system shows how electron–electron repulsion raises energy levels beyond the hydrogenic estimate. Using a hydrogenic product wavefunction as a starting point, the Coulomb interaction between the two electrons yields a calculable energy correction. Variational methods with an effective nuclear charge further improve the agreement with experiment, reducing the discrepancy for the ground state energy.
Singlet and Triplet States and Exchange Energy
In excited configurations like 1s2s or 1s2p, the Coulomb and exchange interactions split degeneracies into singlet and triplet terms. The exchange term favors the triplet configuration because the spatial part of the wavefunction can be antisymmetric in space while the spin part is symmetric, reducing repulsion. This exchange energy underpins magnetic phenomena and is central to understanding ferromagnetism and spin-dependent energy splittings in atoms.
Spectroscopic Notation and Coupling Schemes
The notation for states in atoms emerges from LS coupling, with term symbols written as, for example, 3S1/2 or 3P0, reflecting total orbital angular momentum L, spin multiplicity, and total angular momentum J. In alkali atoms, the inner shells are largely closed and the outer electron dominates the angular momentum, though real atoms are superpositions of configurations. Hydrogenic labeling remains a useful guide for understanding the spectrum.
Transitions and Selection Rules
Finally, the lecture discusses possible transitions between singlet and triplet states. In the simple Coulomb-only model, dipole transitions do not couple singlet and triplet configurations because the dipole operator acts on spatial, not spin, degrees of freedom, and spin flips are not induced by transverse magnetic fields. This leads to selection-rule restrictions and an explanation for why intercombination lines are not easily driven within this approximation, a topic to be revisited with more complete Hamiltonians.
