How This Volume Connects to the Site

Core Formalism is the canonical home for the shared grammar of quantum mechanics. Other volumes use that grammar to solve models, develop methods, organize physical domains, or provide compact lookup entries.

The guiding rule is simple: define the basic formal object here once, then link to that definition wherever the object is used.

What Core Formalism Owns

Core Formalism owns the standard language of:

• quantum states, rays, state vectors, and wavefunction representations;
• observables, operators, projectors, spectra, and eigenstates;
• probability amplitudes, the Born rule, expectation values, and variances;
• commutators, compatibility, and uncertainty;
• projective measurement and state update;
• Hamiltonians, the Schrödinger equation, and unitary evolution;
• tensor products, first encounters with entanglement, and density operators;
• compact statements of the postulates and their scope;
• the correspondence principle as the first bridge from quantum formalism to classical approximations.

It does not own every calculation using these objects. The harmonic oscillator, scattering amplitudes, angular momentum addition, path integrals, and many-body Fock space all need the formalism, but their detailed canonical homes are elsewhere.

Connection Map

Core topic	Canonical home	Used most directly by
State vectors	State Vectors	Canonical Systems, symmetry, quantum information, AMO
Wavefunctions as representations	Wavefunctions as Representations	Canonical Systems, Fourier analysis, scattering
Observables	Observables	Symmetry, Angular Momentum, and Spin, approximation methods, reference pages
Born rule	Born Rule	measurement, foundations, Reference
Commutators	Commutators	Dynamics and Formulations, symmetry, canonical systems
Hamiltonian evolution	Hamiltonians	Dynamics and Formulations, Approximation and Scattering, QFT bridge topics
Tensor products	Tensor Products	Composite Systems and Entanglement, quantum information, many-body physics
Density operators	Density Operators	Composite Systems and Entanglement, open systems, quantum information
Finite and infinite dimension	Finite vs Infinite-Dimensional Quantum Mechanics	Mathematical Toolkit, wave mechanics, rigorous quantum mechanics
Postulates	Why Postulates Matter, Minimal Postulates, Finite-Dimensional Postulates, Wave-Mechanics Postulates, Density-Matrix Formulation, Equivalent Formulations, Assumptions and Scope, and What the Postulates Do Not Say	all later volumes
Classical correspondence	Correspondence Principle, Ehrenfest Theorem Overview, Classical Limit, Semiclassical Limit Overview, Decoherence Preview, Quantization vs Classical Limit, and Common Misstatements About the Classical Limit	Dynamics and Formulations, Approximation and Scattering, canonical systems
Formalism reference aids	Symbol Map, Representation Translation Table, Common Checks and Sanity Tests, Common Mistakes in the Formalism, Glossary for Core Formalism, and Exercises and Problems	readers moving between Core pages, Reference

The table is deliberately asymmetric. A topic may be used everywhere, but it still has one preferred teaching page.

How Mathematical Toolkit Relates

Mathematical Toolkit supplies reusable mathematics. Core Formalism should not rederive linear algebra, Hilbert-space completeness, Fourier transforms, distribution theory, or group representation theory every time they appear.

For example, Core Formalism explains what a quantum state does. Mathematical Toolkit explains inner products, finite-dimensional Hilbert spaces, $L^2$ spaces, tensor products as mathematical constructions, Fourier transforms, and matrix diagonalization.

When a page needs a mathematical result as a tool, link to the toolkit. When a page needs the physical role of the object in quantum mechanics, link to Core Formalism.

How Canonical Systems Relates

Wave Mechanics and Model Systems is where the formalism becomes concrete in coordinate-space problems. It owns the particle in a box, free-particle wave packets, tunneling barriers, the harmonic oscillator, and other standard solvable models.

Core Formalism should state that the Hamiltonian generates time evolution. Canonical Systems should show how a specific Hamiltonian, boundary condition, and Hilbert space produce an actual spectrum and wavefunctions.

How Dynamics and Symmetry Relate

Dynamics and Formulations develops the Schrödinger, Heisenberg, interaction-picture, propagator, and path-integral viewpoints. Core Formalism introduces unitary evolution and Hamiltonians; Dynamics and Formulations explains how equivalent calculational languages are built from them.

Symmetry, Angular Momentum, and Spin develops unitary and antiunitary symmetries, generators, angular momentum, spinors, geometric phases, and representation-theoretic constraints. Core Formalism introduces operators, commutators, and states; the symmetry volume shows how those objects organize physical structure.

How Composite Systems Relates

Core Formalism introduces tensor products, entangled states, and density operators. Composite Systems and Entanglement owns the detailed theory of subsystems, partial trace, Schmidt decomposition, entanglement measures, identical particles, Fock space, and second quantization.

This separation is important. The first encounter with entanglement belongs in Core Formalism because every reader needs it. The mature treatment belongs in the composite-systems volume because it has its own machinery and applications.

How Reference Relates

Reference is the lookup layer. It should summarize concepts, symbols, formulas, and model cards, but it should not duplicate full explanations.

A reference entry for the Born rule should give the formula, assumptions, symbols, and a link to the Core Formalism page. A reference entry for the harmonic oscillator should point to the canonical-system page for derivation and interpretation.

Rules for Cross-Linking

Use these rules when writing or revising pages:

• Link to the canonical home the first time a major concept is needed.
• Do not rederive a result merely because another page uses it.
• If a topic is foundational vocabulary, it probably links to Core Formalism.
• If a topic is a mathematical tool, it probably links to Mathematical Toolkit.
• If a topic is a concrete solvable system, it probably links to Canonical Systems.
• If a topic is a compact lookup item, it probably links from Reference back to the teaching page.
• If a page needs a future volume that is not yet built, name the domain plainly and avoid creating broken internal links.

Common Mistakes

• Copying the Born rule derivation into every measurement, scattering, or foundations page.
• Teaching Hilbert-space theory from scratch inside a model page.
• Treating reference entries as miniature textbooks.
• Letting a downstream application redefine a core concept with incompatible conventions.
• Linking only sideways to similar applications instead of linking back to the canonical home.

References

• P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed., Oxford University Press, 1958.
• 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.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.

Exercises

1. A page on tunneling through a rectangular barrier needs to mention the Born rule, boundary conditions, and the specific barrier solution. Which volume should own each explanation?

Solution

The Born rule belongs in Core Formalism. Boundary-condition mathematics belongs in Mathematical Toolkit or the canonical-systems foundation pages, depending on the level of detail. The actual rectangular-barrier solution belongs in Wave Mechanics and Model Systems.

2. A reference entry gives the formula $U(t)=e^{-iHt/\hbar}$. What should it link to for a full explanation of the Hamiltonian’s role?

Solution

It should link to the Core Formalism pages on Hamiltonians and unitary time evolution, because those pages explain the physical and formal role of $H$ as the generator of time translations.