Dependency Graph

This page is a prerequisite map for the volume. It is not a complete site graph. Its job is to show which ideas should usually be in place before a reader studies a canonical model, and which later pages can safely assume those ideas without rederiving them.

The graph has a simple organizing principle:

$$\text{representation and dynamics} \longrightarrow \text{domains, spectra, and currents} \longrightarrow \text{canonical models} \longrightarrow \text{reusable physical mechanisms}.$$

Use this page when choosing a reading order, adding prerequisite metadata to a page, or deciding whether a derivation belongs on a model page or should be linked from a more general home.

Foundation Spine

The foundation pages are the common trunk of the volume. A canonical model may not need every detail at first reading, but it should not silently assume ideas that have not appeared in this spine.

Node	Read After	What It Enables
Coordinate Representation	basic Hilbert-space notation	wavefunctions as coordinate amplitudes
Wavefunctions and Probability Density	coordinate representation	normalization, probability densities, Born-rule interpretation
Time-Dependent Schrödinger Equation	wavefunctions and Hamiltonians	evolution of arbitrary initial states
Time-Independent Schrödinger Equation	time-dependent dynamics for time-independent Hamiltonians	stationary states and spectra
Boundary Conditions	stationary equations	domains, quantization, matching conditions
Normalization Conventions	probability density and spectra	bound-state, box, continuum, and flux normalization
Probability Current	time-dependent dynamics	scattering fluxes and conservation checks
Dimensionless Variables and Scaling	Hamiltonians and units	clean parameter counting and limiting cases

In compact form:

$$\begin{aligned} \text{coordinate amplitudes} &\longrightarrow \text{probability density} \\ &\longrightarrow \text{Schrödinger dynamics} \\ &\longrightarrow \text{stationary spectra} \\ &\longrightarrow \text{domains, normalization, and currents}. \end{aligned}$$

The last line is where many wrong solutions fail. A differential expression is not yet a quantum problem until its domain, inner product, and physical interpretation have been specified.

One-Dimensional Bound-State Chain

The bound-state chain teaches how boundary conditions and localization create discrete spectra.

Model Page	Minimum Prior Nodes	Main Dependency
Infinite Square Well	coordinate representation, TISE, boundary conditions, normalization	Dirichlet walls quantize the wavenumber
Finite Square Well	infinite well, matching conditions, dimensionless scaling	finite walls create evanescent tails and threshold behavior
Delta-Function Potential	boundary conditions, normalization, probability current	singular potentials produce derivative jumps
Double-Well Potential	finite wells, oscillator intuition	tunneling splits nearly degenerate localized states

The conceptual route is:

$$\text{hard walls} \longrightarrow \text{finite walls} \longrightarrow \text{singular contacts} \longrightarrow \text{coupled wells and tunneling splittings}.$$

This chain is the safest place to introduce node counting, parity, exponential tails, and the difference between exact solvability and qualitative spectral reasoning.

Free-Motion And Wave-Packet Chain

Free motion is the entry point to continuous spectra and localized states.

Model Page	Minimum Prior Nodes	Main Dependency
Free Particle	coordinate representation, TDSE, TISE, normalization conventions	plane-wave eigenstates and continuous energy
Gaussian Wave Packets	free particle, Fourier transforms, expectation values	localization by superposition of momentum components
Wave-Packet Spreading	Gaussian packets, group velocity	dispersion from nonlinear energy-momentum relation
Group Velocity and Phase Velocity	free particle and packets	packet motion versus carrier oscillation
Free-Particle Propagator: First Encounter	TDSE, Fourier representation, Gaussian integrals	kernels as time-evolution amplitudes

The dependency is not merely chronological. Plane waves are generalized eigenstates, not normalizable particles. Wave packets explain how continuum eigenstates combine into localized states with finite probability normalization.

Scattering And Tunneling Chain

Scattering pages depend on probability current more strongly than bound-state pages do. Transmission probabilities are current ratios, not merely squared amplitude ratios.

Model Page	Minimum Prior Nodes	Main Dependency
Potential Step	free particle, boundary conditions, probability current	reflection and transmission at a single interface
Reflection and Transmission Coefficients	probability current, free particle	flux-normalized probabilities
Rectangular Barrier Tunneling	potential step, matching conditions	evanescent waves across a finite forbidden region
Quantum Tunneling	rectangular barrier, finite wells	the general mechanism behind barrier penetration
Transfer Matrix Method	region matching, current conservation	systematic composition of piecewise-constant regions
Resonant Transmission	transfer matrices, wells, barriers	interference and quasi-bound states in open systems

The core chain is:

$$\text{free waves} \longrightarrow \text{one interface} \longrightarrow \text{two interfaces} \longrightarrow \text{barriers, wells, and resonances}.$$

Whenever a scattering page claims a reflection or transmission probability, the prerequisite should include probability current or a page that has already established the current convention.

Oscillator Chain

The harmonic oscillator is both a wave-mechanics model and a bridge to normal modes.

Model Page	Minimum Prior Nodes	Main Dependency
Quantum Harmonic Oscillator	TISE, normalization, dimensionless scaling	the quadratic Hamiltonian and zero-point energy
Differential-Equation Solution	oscillator setup, asymptotic normalizability	Hermite-polynomial eigenfunctions
Ladder-Operator Solution: First Encounter	oscillator setup, operator algebra	algebraic spectrum construction
Zero-Point Energy	oscillator spectrum, uncertainty relation	nonzero ground-state energy and its limits
Number States	ladder operators	occupation-number basis and matrix elements
Coherent States	number states, wave packets	minimum-uncertainty states with classical-like motion
Squeezed States: First Encounter	coherent states, uncertainty relation	quadrature squeezing and anti-squeezing
Displaced Oscillator	oscillator spectrum, coherent states	shifted equilibrium and unchanged spacing
Coupled Oscillators: First Encounter	oscillator spectrum, linear algebra	normal modes and the field-mode analogy
Oscillator as a Universal Local Model	oscillator spectrum, energy scales, normal modes	quadratic approximations near stable equilibria

The oscillator can be read after the one-dimensional bound-state pages, but it does not require scattering. Its main dependencies are stationary states, normalizability, and enough operator language to understand ladder methods.

Two-Level Chain

Two-level systems are the finite-dimensional canonical models in this volume. They do not require coordinate-space differential equations, but they do require basis discipline.

Model Page	Minimum Prior Nodes	Main Dependency
Two-Level Systems	state vectors, finite-dimensional Hilbert spaces	two-dimensional state space and physical examples
Two-State Hamiltonians	two-level systems, diagonalization	mixing, coherent oscillations, avoided crossings
Pauli-Matrix Hamiltonians	two-state Hamiltonians, Pauli matrices	effective-field form and two-level rotations
Bloch Sphere: Wave-Mechanics Perspective	Pauli-matrix Hamiltonians, rays and global phase	pure-state geometry, relative phase, and rotation picture
Coupled Wells and Avoided Crossings	double-well potential, two-state Hamiltonians	localized basis states, tunneling matrix elements, and avoided crossings
Tight-Binding Dimer	coupled wells, two-state Hamiltonians	two-site hopping, bonding and antibonding states, and lattice-model language
Landau-Zener Problem: First Encounter	coupled wells, Pauli-matrix Hamiltonians, TDSE	swept avoided crossings and the transition-probability statement
Rabi Oscillations: First Encounter	two-level systems, Pauli-matrix Hamiltonians, TDSE	near-resonant driven population oscillations
Spin-1/2 as a Canonical System: First Encounter	two-level systems, Pauli-matrix Hamiltonians, spin-half Hilbert space	spin magnetic-field Hamiltonian and Larmor precession

The spin page closes the introductory two-level chain while pointing the full spin formalism to the symmetry volume.

Three-Dimensional And Angular Chain

Three-dimensional models require a new layer: coordinate measures, separation of variables, and angular eigenfunctions.

Model Page	Minimum Prior Nodes	Main Dependency
Schrödinger Equation in Three Dimensions	coordinate representation, TDSE, TISE	gradients, Laplacians, and three-dimensional inner products
Separation of Variables	three-dimensional equation	product solutions and separated eigenvalue problems
Three-Dimensional Box	separation, boundary conditions	separable spectra and degeneracy
Free Particle in Three Dimensions	3D Schrödinger equation, plane-wave normalization	momentum vectors, energy shells, and periodic regulators
Density of States: First Encounter	free particle in three dimensions, box normalization	large-box state counting and free-particle density of states
Spherical Coordinates	3D Schrödinger equation, separation of variables	coordinate convention, volume element, and spherical Laplacian
Angular and Radial Separation	spherical coordinates, separation of variables	angular eigenvalues and effective radial potentials
Degeneracy in Separable Systems	3D box, angular and radial separation	repeated energies, symmetry labels, and level splitting
Central Potentials	angular and radial separation, orbital angular momentum	rotationally invariant potentials and central-potential quantum numbers
Radial Schrödinger Equation	central potentials, angular and radial separation	radial half-line equation and boundary conventions
Effective Radial Potential	radial equation	centrifugal barrier, turning points, and qualitative radial spectra
Boundary Conditions for Radial Wavefunctions	radial equation, effective radial potential	origin regularity, radial measure, normalizability, and singular-potential cautions
Coulomb Potential	radial equation, radial boundary conditions	charge sign, reduced mass, Bohr scale, Rydberg scale, and bound-continuum split
Hydrogen Atom	radial equation, spherical harmonics	Coulomb spectrum and orbitals
Hydrogenic Ions	Coulomb potential, hydrogen atom	$Z$ scaling, reduced mass, ion examples, and transition scaling
Radial Wavefunctions	hydrogen atom, Laguerre polynomials	normalized $R_{n\ell}$ functions, radial probabilities, nodes, and expectation values
Atomic Orbitals	radial wavefunctions, spherical harmonics	$s,p,d,\ldots$ notation, real and complex orbital bases, nodal surfaces, and visualization cautions
Degeneracy in the Hydrogen Atom	hydrogen spectrum, symmetry	accidental degeneracy and its later breaking
Continuum States of the Coulomb Problem	Coulomb potential, radial equation	positive-energy Coulomb states, threshold behavior, delta normalization, and long-range asymptotics
Particle on a Ring	periodic boundary conditions, probability current	integer angular momentum, $\pm m$ degeneracy, ring currents, and magnetic-flux preview
Particle on a Sphere	spherical coordinates, particle on a ring	angular Laplacian, spherical harmonics, $\ell,m$ labels, and $2\ell+1$ degeneracy
Rigid Rotor	angular equations, normalization on the sphere	rotational spectra and spherical harmonics
Spherical Harmonics as Wavefunctions	particle on a sphere, spherical harmonics	angular probabilities, orthogonality, angular nodes, real and complex bases, and visualization cautions
Rotational Spectra	rigid rotor, angular-momentum selection rules	ideal molecular rotation lines, rotational constants, moment of inertia, and model limitations
Angular Probability Distributions	particle on a ring, particle on a sphere, probability densities	angular measures, $\sin\theta$ factors, marginals, equal-area sampling, and visualization cautions
Rotor in External Fields: First Encounter	rigid rotor, angular probabilities, perturbation theory	symmetry breaking by static fields, $M$ labels, electric-field mixing, and degeneracy splitting

The important change is the measure. In spherical coordinates, radial normalization and angular normalization separate only after the volume element is handled correctly:

$$d^3r=r^2\sin\theta\,dr\,d\theta\,d\phi.$$

This is why three-dimensional pages should depend on the measure convention, not only on the formal Schrödinger equation.

Electromagnetic-Field Chain

Electromagnetic fields add gauge-dependent wavefunctions and gauge-invariant observables.

Model Page	Minimum Prior Nodes	Main Dependency
Minimal Coupling in Wave Mechanics	Hamiltonians, probability current, gauge phase intuition	replacing canonical momentum by kinetic momentum
Gauge Transformations: First Encounter	minimal coupling, probability current, electromagnetism prerequisites	potential redundancy, wavefunction phase, and gauge-covariant momentum
Particle in a Uniform Magnetic Field	minimal coupling, gauge transformations, oscillator algebra	cyclotron motion, kinetic-momentum algebra, and oscillator reduction
Landau Levels	uniform-field setup, oscillator structure, degeneracy	magnetic quantization and macroscopic degeneracy
Landau Gauge and Symmetric Gauge	gauge transformations, uniform-field setup, Landau levels	equivalent basis choices, gauge-dependent labels, and geometry-adapted wavefunctions
Degeneracy of Landau Levels	Landau levels, gauge choices, density-of-states intuition	finite-area guiding-center counting and flux-quantum degeneracy
Aharonov–Bohm Effect: First Encounter	minimal coupling, gauge transformations, particle on a ring	gauge-invariant phase around excluded magnetic flux
Charged Harmonic Oscillator in a Magnetic Field	oscillator spectrum, uniform-field setup, orbital angular momentum	Fock–Darwin levels and confinement-lifted Landau degeneracy

The route is:

$$\text{canonical momentum} \longrightarrow \text{minimal coupling} \longrightarrow \text{gauge covariance} \longrightarrow \text{magnetic oscillator} \longrightarrow \text{Landau degeneracy}.$$

Because gauge choices can change the appearance of wavefunctions, these pages should state which quantities are gauge-dependent and which are physical.

A Practical Reading Algorithm

When adding or reading a page, assign its prerequisites by asking four questions.

1. Which representation is being used?
2. Which dynamical equation or eigenvalue problem is assumed?
3. Which domain, measure, normalization, or current convention is needed?
4. Which earlier canonical model gives the simplest analogy?

For example, a rectangular-barrier calculation depends on the free particle, boundary matching, and current conservation. It does not need the harmonic oscillator. A coherent-state page depends on oscillator number states and wave-packet intuition, but not on central potentials.

This keeps the dependency graph sparse. Sparse prerequisite lists are easier to maintain and more useful than long lists that name every earlier page.

Common Mistakes

• Listing a whole chapter as a prerequisite when only one convention page is needed.
• Making a later model the canonical home for a general tool, such as boundary conditions or Fourier transforms.
• Linking a table as if it were a derivation; tables are summaries and checks.
• Treating plane waves as ordinary normalized states before the continuum convention has been stated.
• Repeating the same proof on multiple model pages instead of linking to the canonical derivation.
• Forgetting that three-dimensional pages depend on the measure as much as on the Hamiltonian.

Where This Is Used

• Overview uses this page to make the section reading order explicit.
• Map of Canonical Systems explains what each model teaches; this page explains what each model assumes.
• How to Solve a Wave-Mechanics Problem gives the calculation workflow that the graph organizes.
• Problem-Solving Patterns turns graph edges into reusable habits such as matching, scaling, and checking currents.

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.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.

Exercises

1. A new page derives resonant transmission through a double barrier. List the smallest useful prerequisite chain.

Solution

A compact chain is: free particle, probability current, potential step or region matching, rectangular barrier tunneling, transfer matrices, then resonant transmission. The page should not require oscillator or hydrogen material unless it uses those analogies explicitly.

2. A draft page on Landau levels begins by rederiving the harmonic-oscillator spectrum in full. What should be changed?

Solution

The Landau-level page should link to the oscillator page or the ladder-operator page for the oscillator spectrum, then explain how minimal coupling in a uniform magnetic field reduces part of the Hamiltonian to oscillator form. The canonical home for the oscillator derivation remains the harmonic-oscillator chapter.

3. Why should a three-dimensional central-potential page depend on normalization and measure conventions?

Solution

The inner product uses $d^3r=r^2\sin\theta\,dr\,d\theta\,d\phi$. If the measure is omitted, radial normalization, angular normalization, and expectation values are wrong. The radial wavefunction $R(r)$ and reduced radial wavefunction $u(r)=rR(r)$ also have different normalization conventions.