Examples by Level

Levels describe what the reader is expected to control without help. They do not measure importance: normalization and expectation values remain graduate-level failure points when conventions change.

First-Course Route

Start here if the goal is reliable calculation rather than speed.

• Normalization Examples
• Expectation-Value Examples
• Matrix-Mechanics Examples
• Scattering Examples for one-dimensional current ratios
• Exercise Sets

Expected output: stated assumptions, correct units, a normalized state, and a limiting-case check.

Advanced Undergraduate Route

At this level, examples should combine two or more methods.

• diagonalize a two-level Hamiltonian and compute time-dependent probabilities;
• normalize a radial wavefunction and interpret angular labels;
• match a barrier-scattering solution and compare with a WKB estimate;
• use nondegenerate perturbation theory for an anharmonic oscillator;
• compute spin measurement probabilities after a rotation.

Good starting pages include Pauli Matrix Hamiltonians, WKB Barrier Tunneling, and Anharmonic Oscillator Perturbation.

Graduate Route

Graduate examples should expose assumptions that first-course examples often hide.

Route	What Changes
Density-Matrix Examples	Ensembles, mixed states, partial traces, and channels replace pure-state shortcuts.
Perturbation-Theory Examples	Degeneracies, selection rules, and transition rates become central.
Scattering Examples	Born approximation, cross sections, phase shifts, and optical-theorem checks enter.
Spin Examples	Tensor products, Clebsch-Gordan coefficients, and symmetry constraints matter.

Expected output: an explicit domain of validity, a convention statement, and a comparison with an exactly solvable or limiting case.

Computational Route

Computational examples are mature only when they state what is being discretized and how convergence is checked.

• Use Numerical Notebooks Index for reproducible wave-mechanics notebooks.
• Use Sparse Eigensolvers for finite-difference Hamiltonian examples.
• Use Matrix Exponentials Numerically for finite-dimensional dynamics.
• Use Convergence Tests before trusting spectra, currents, or entropies.

Level-Selection Rule

Choose the lowest level that still contains the technical obstacle. If the obstacle is just a measure or basis convention, a first-course example is often the fastest route even for a graduate calculation.

References

• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes, 3rd ed., Cambridge University Press, 2007.