Quantum Mechanics — A Refresher

Author

Daniel Fischer

Scope of This Section

Atomic, molecular, and optical physics is built on quantum mechanics. This section collects the quantum-mechanical background needed throughout the course.

It is not necessary to read all pages in this section before continuing with atomic physics. The first group provides the required fundamentals; the remaining pages introduce particular systems, approximation methods, or more advanced concepts that can be studied when they become relevant.

Some of the notes include derivations and mathematical details that go beyond what you are expected to retain. Where appropriate, the Core Knowledge — Know and Working Knowledge — Understand boxes identify the main takeaways and help distinguish them from additional detail.

I. Required Fundamentals

These pages should be read first and provide the basic language used throughout the remainder of the course.

  1. Wave Mechanics — Matter waves, interference, the Born rule, de Broglie relations, and the Schrödinger equation.
  2. States and Operators in Quantum Mechanics — Hilbert space, Dirac notation, operators, eigenstates, and observables.
  3. Quantum Measurement and the Measurement Problem — Measurement probabilities, state reduction, and the operational consequences of quantum measurement.
  4. Commutators and the Heisenberg Uncertainty Principle — Compatible observables, commutators, and quantum uncertainty.

II. Important Quantum Systems

These chapters apply the general formalism to systems that occur repeatedly in AMO physics.

  1. Angular Momenta and Their Couplings — Angular-momentum eigenvalues, quantum numbers, and coupling.
  2. The One-Dimensional Quantum Harmonic Oscillator — Ladder operators, oscillator states, and coherent states. This chapter can be used as a reference when the harmonic oscillator appears later.

III. Approximation Methods

These methods are important when a quantum problem cannot be solved exactly and can be studied when they are first needed.

  1. Time-Independent Perturbation Theory — Corrections to known energies and states caused by weak interactions.
  2. Time-Dependent Perturbation Theory and Fermi’s Golden Rule — Transitions driven by weak time-dependent interactions, resonance, and transition rates.
  3. The Variational Method — Approximate energies and states obtained from trial wave functions.

IV. Advanced Concepts and Reference Material

These pages introduce concepts needed for particular later topics but are not prerequisites for the first atomic-physics chapters.

  1. Exchange Interaction: Fermions and Bosons — Indistinguishable particles, exchange symmetry, and the Pauli principle; especially relevant for many-electron atoms.
  2. Relativistic Wave Equations and the Dirac Equation — Relativistic quantum mechanics and the Dirac equation; mainly background for spin and relativistic effects in atoms.