Map of Canonical Systems

Canonical systems are the reusable laboratories of quantum mechanics. Each one is simple enough to solve cleanly, but rich enough to teach a general principle. The infinite square well teaches quantization from boundary conditions. The free particle teaches continuous spectra and wave packets. Barriers teach tunneling. The harmonic oscillator teaches zero-point energy and ladder structure. Hydrogen teaches central potentials and degeneracy.

This map organizes the models by the lesson they teach rather than by difficulty.

Systems At A Glance

System	Main lesson	Later uses
Free particle	Continuous spectrum, plane waves, wave packets	Scattering, propagators, semiclassical motion
Gaussian wave packet	Localization, spreading, group velocity	Wave-packet dynamics, measurement models, semiclassical limits
Infinite square well	Boundary quantization, nodes, discrete spectra	Boxes, basis expansions, numerical benchmarks
Finite square well	Bound-state thresholds and evanescent tails	Molecular binding, heterostructures, tunneling devices
Potential step	Reflection from discontinuities	Scattering intuition, interface physics
Rectangular barrier	Tunneling and exponential suppression	Alpha decay, scanning tunneling microscopy, Josephson physics
Delta potential	Singular matching conditions	Contact interactions, model scattering, bound-state tests
Harmonic oscillator	Zero-point energy, Hermite functions, ladder structure	Phonons, field modes, quantum optics, molecular vibrations
Two-level system	Finite-dimensional dynamics, pure-state geometry, and avoided crossings	Qubits, spin resonance, driven atoms
Three-dimensional box	Separability and degeneracy	Density of states, finite-volume methods
Hydrogen atom	Central potentials, orbitals, accidental degeneracy	Atomic physics, chemistry, spectroscopy
Rigid rotor	Angular wavefunctions and rotational spectra	Molecular rotations, angular momentum coupling
Landau levels	Magnetic quantization and degeneracy	Quantum Hall physics, topological matter, cyclotron motion

The purpose of the table is not to replace the detailed pages. It gives the shortest route from a physical idea to the model where that idea first becomes explicit.

The Main Chains

Several conceptual chains run through the volume.

Boundary conditions lead to quantization:

$$\text{domain and boundary conditions} \longrightarrow \text{allowed eigenfunctions} \longrightarrow \text{discrete spectrum}.$$

Fourier analysis links free motion to localization:

$$\text{plane waves} \longrightarrow \text{momentum amplitudes} \longrightarrow \text{wave packets} \longrightarrow \text{spreading and group velocity}.$$

Barriers reveal that quantum motion is controlled by amplitudes and currents, not classical pass-or-fail trajectories:

$$\text{potential discontinuity} \longrightarrow \text{matching conditions} \longrightarrow \text{reflection, transmission, tunneling}.$$

Quadratic potentials lead to oscillator structure:

$$\text{stable equilibrium} \longrightarrow \text{quadratic approximation} \longrightarrow \text{harmonic oscillator} \longrightarrow \text{normal modes}.$$

Spherical symmetry leads to angular momentum:

$$\text{central potential} \longrightarrow \text{angular-radial separation} \longrightarrow \text{spherical harmonics} \longrightarrow \text{hydrogenic orbitals}.$$

Magnetic fields lead to gauge-dependent wavefunctions but gauge-invariant physics:

$$\text{minimal coupling} \longrightarrow \text{Landau levels} \longrightarrow \text{degeneracy} \longrightarrow \text{quantum Hall physics}.$$

First Route Through The Volume

For a first undergraduate pass, read in this order:

1. Coordinate Representation
2. Wavefunctions and Probability Density
3. Time-Dependent Schrödinger Equation
4. Time-Independent Schrödinger Equation
5. Boundary Conditions
6. Free Particle
7. Gaussian Wave Packets
8. Infinite Square Well
9. Finite Square Well
10. Rectangular Barrier Tunneling
11. Quantum Harmonic Oscillator

The order is not sacred, but it is pedagogically efficient. It moves from representation, to dynamics, to simple spectra, to tunneling, to the oscillator.

Research-Level Reuse

Canonical systems remain useful long after the first course.

• The free particle and Gaussian wave packet are the local building blocks for scattering, semiclassical propagation, and path-integral intuition.
• The infinite well and harmonic oscillator are benchmark problems for numerical diagonalization.
• The finite well and barrier are minimal models for tunneling, resonances, and interface states.
• The harmonic oscillator is the template for small oscillations, phonons, photons, and field modes.
• The hydrogen atom is the starting point for atomic structure, spectroscopy, degeneracy breaking, and central-force approximation schemes.
• Landau levels are the entry point to magnetic quantization, macroscopic degeneracy, edge physics, and topological quantum matter.

The same few models appear repeatedly because they isolate mechanisms that more complicated systems combine.

Common Mistakes

• Treating canonical systems as isolated textbook exercises rather than reusable models.
• Memorizing spectra without remembering which boundary conditions produced them.
• Assuming every exact model has only one method of solution; many can be solved by differential equations, algebra, symmetry, or numerical diagonalization.
• Treating finite-dimensional two-level systems as less important because they are not differential-equation models.
• Forgetting that a simple model may be exact in one context and only an approximation in another.

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.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.

Exercises

1. Choose the canonical system that best isolates each idea: tunneling, boundary quantization, central potentials, magnetic degeneracy, and localization with spreading.

Solution

Rectangular barriers isolate tunneling. The infinite square well isolates boundary quantization. The hydrogen atom is the standard central-potential system. Landau levels isolate magnetic degeneracy. Gaussian wave packets isolate localization and spreading.

2. Explain why the harmonic oscillator is more than a single-particle model in a parabolic potential.

Solution

Near a stable equilibrium, many smooth potentials are approximately quadratic. After diagonalizing small oscillations, many systems decompose into oscillator-like normal modes. This is why oscillator mathematics reappears in molecular vibrations, phonons, quantum optics, and field quantization.