Part VII: Identical Particles
Quantum mechanics of indistinguishable particles. The symmetrization postulate, fermions and bosons, Pauli exclusion principle, and its profound consequences for atomic structure and the periodic table.
45+ pages | 6 chapters
1. Symmetrization Postulate (8 pages)
Indistinguishability, symmetric and antisymmetric states
$|\psi_\pm\rangle = \frac{1}{\sqrt{2}}(|\psi_1\rangle|\psi_2\rangle \pm |\psi_2\rangle|\psi_1\rangle)$, exchange symmetry
2. Fermions & Bosons (7 pages)
Spin-statistics theorem, creation/annihilation operators
Fermions (−): half-integer spin, Bosons (+): integer spin, $[\hat{a}_i,\hat{a}_j^\dagger] = \delta_{ij}$, $\{\hat{b}_i,\hat{b}_j^\dagger\} = \delta_{ij}$
3. Pauli Exclusion Principle (6 pages)
No two fermions in same state, Slater determinants
$|\psi\rangle = \frac{1}{\sqrt{N!}}\det(|\psi_i(x_j)\rangle)$, antisymmetric wavefunctions
4. Multi-Electron Atoms (10 pages)
Helium atom, electron configurations, Hund's rules
Variational calculation, $E_{He} \approx -77.5\text{ eV}$, LS coupling, term symbols
5. Periodic Table (8 pages)
Aufbau principle, shell structure, chemical properties
Electron configurations: $1s^2 2s^2 2p^6 3s^2 3p^6\ldots$, valence electrons, periodicity
6. Exchange Interaction (6 pages)
Exchange energy, ferromagnetism, quantum statistics
Exchange integral: $J = \int\psi_1^*(r_1)\psi_2^*(r_2)\frac{e^2}{|r_1-r_2|}\psi_2(r_1)\psi_1(r_2)d^3r_1d^3r_2$