Wave Mechanics and Model Systems
Wave mechanics turns a Hamiltonian into predictions for energies, motion, probabilities, and currents. Work through familiar models—a confined particle, a spreading packet, a tunneling barrier, an oscillator—and learn how the potential, geometry, and boundary conditions shape each answer. Two-level models provide a parallel introduction using matrices.
Start with a solvable problem
Section titled “Start with a solvable problem”If this is your first encounter with the subject, read wavefunctions and probability density, then follow the worked problem-solving guide. The infinite square well is a good first model: its boundary conditions select the allowed wavelengths, and normalization fixes their amplitudes.
For a broader route, continue to free motion, Gaussian packets, the finite well, barrier tunneling, and the harmonic oscillator.
The shared background is a coordinate representation with a specified Hamiltonian, boundary conditions, and normalization. Distinguish the stationary equation, which finds energy eigenstates, from the time-dependent equation, which evolves an initial state. Learn probability current before interpreting scattering amplitudes as probabilities.
Choose a physical question
Section titled “Choose a physical question”| Question | Chapter |
|---|---|
| How do I set up and interpret a wavefunction calculation? | Foundations |
| How does a localized particle move and spread? | Free motion and wave packets |
| How does confinement select energies? | One-dimensional bound systems |
| How can a particle reflect or tunnel? | Scattering and tunneling |
| Why do small vibrations share one spectrum? | Harmonic oscillator |
| How do two coupled states exchange probability? | Two-level systems |
| What changes when motion has three coordinates? | Three-dimensional systems |
| How do central forces produce orbitals? | Central potentials and hydrogen |
| How are angular motion and rotation quantized? | Angular systems and rotors |
| How do electromagnetic fields affect motion? | Electromagnetic fields |
| Where can I compare formulas and assumptions? | Model encyclopedia |
| Where can I practice and check a calculation? | Worked examples and problem sets |
The model encyclopedia compares systems, formulas, and assumptions. The notation guide helps reconcile Fourier, radial, and current conventions.
Choose background for the calculation you want to make. Packets use Fourier transforms; coupled states use matrix diagonalization. Before hydrogen, practice radial normalization and angular and radial separation. Before Landau levels, recognize a shifted harmonic-oscillator Hamiltonian and review minimal coupling. You can return to a needed tool without restarting the volume.
Learn from the limits of each model
Section titled “Learn from the limits of each model”An infinite well idealizes impenetrable walls; a finite well allows tails outside the classically allowed region. A harmonic oscillator describes a quadratic potential exactly and small motion near many stable equilibria approximately. Ask which feature the model retains, then check what changes when its assumptions are relaxed. That habit makes an exact solution useful beyond the first problem in which it appears.
References
Section titled “References”- C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
- D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
- L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.