Formula Compendium

The Formula Compendium is an assumption-aware lookup layer. Each card gives a usable equation together with its mathematical setting, symbol definitions, units, validity limits, common failure modes, and a link to the canonical explanation or derivation.

A formula card is not a substitute for the page that derives and interprets the result. It is the place to confirm exactly which version of a formula applies before using it.

Formula Routes

Task	Start here
Compute an outcome probability	Born Rule
Compute a mean or spread	Expectation Values and Variance and Standard Deviation
Represent a mixed state	Density Operators
Reduce a composite state	Partial Trace
Check a quantum operation	Quantum Operations
Look up the ideal magnetic spectrum	Landau Levels
Check relativistic wave equations	Relativistic QM

The subject collections below organize additional task cards. Use the Core Formulas Index, Symbol Index, and Tables for complementary lookup. Cards differ in depth and editorial status; follow the canonical explanation when a derivation or a validity boundary matters.

Before Applying a Formula

Check these questions in order:

1. What mathematical object is being used? A ket, wavefunction, density operator, observable, superoperator, distribution, and classical function obey different rules.
2. Which assumptions are active? Look for finite versus infinite dimension, pure versus mixed state, closed versus open dynamics, discrete versus continuous spectrum, and nondegenerate versus degenerate cases.
3. Which convention fixes the signs and factors? Fourier normalization, metric signature, angular-momentum normalization, unit system, and picture choice can change the displayed form.
4. Are domains and boundary conditions relevant? Differential operators and commutators involving unbounded operators require more than formal algebra.
5. Do the units match? Natural-unit formulas may hide factors of $\hbar$, $c$, $k_{\mathrm B}$, or $4\pi\epsilon_0$.
6. Is the formula exact or approximate? For an approximation, identify the small parameter, retained order, asymptotic regime, and failure condition.
7. Does a limiting case agree with a known result? Dimensional, normalization, symmetry, and limiting-case checks catch many transcription errors.

Formula Card Contract

Every mature card should answer the following:

• Formula: the result in a convention that is stated or linked.
• Meaning: what the equation computes and what it does not compute.
• Assumptions: the hypotheses under which the expression is valid.
• Symbols: mathematical type, physical meaning, and units where applicable.
• Validity: domain, spectrum, approximation, and boundary limitations.
• Canonical derivation: one authoritative home for the derivation.
• Worked use: examples or model pages where the formula is applied.
• Common mistakes: sign, normalization, degeneracy, unit, and interpretation traps.
• Related formulas: nearby results that are often needed in the same calculation.
• References: sources that state or derive the result in compatible conventions.

If one of these items materially affects the result, silence is not a harmless shortcut. A formula without its assumptions is incomplete.

States and Probability

States and Probability connects probability formulas with their canonical explanations.

• Born Rule
• Normalization
• Expectation Value
• Variance
• Probability Current

Use these cards only after identifying whether the state is a vector, wavefunction, density operator, or generalized state. Continuous-outcome probabilities are integrals of densities, not point probabilities.

Operators and Commutators

Operators and Commutators collects algebraic identities, uncertainty bounds, projectors, and unitary transformations.

• Canonical Commutation Relations
• Commutator Identities
• Uncertainty Relations
• Baker–Campbell–Hausdorff
• Projection Identities
• Unitary Identities

For unbounded operators, a formal commutator identity may require a common invariant domain. Series formulas also require convergence conditions or an explicitly formal interpretation. In infinite dimension, distinguish an isometry from a surjective unitary and state the convergence mode of projector resolutions.

Dynamics

Dynamics separates state and observable evolution, picture changes, ordered exponentials, evolution kernels, inverse kernels, and path-integral representations.

• Schrödinger Equation
• Time-Evolution Operator
• Dyson Series
• Interaction-Picture Evolution
• Heisenberg Equation
• Ehrenfest Theorem Formulas
• Propagator Composition Law
• Green Function Equations
• Path-Integral Propagator

Always identify the picture, time dependence, basis measure, and boundary prescription. Ordinary exponentials and time-ordered exponentials are not interchangeable, and evolution kernels are not the same objects as resolvent Green functions.

Canonical Systems

Canonical Systems provides spectra and operator identities for standard exactly solvable models.

• Particle-in-a-Box Spectrum
• Finite Square-Well Equations
• Harmonic Oscillator Spectrum
• Harmonic Oscillator Ladder Operators
• Coherent-State Expansion
• Hydrogen Spectrum
• Hydrogen Radial Wavefunctions
• Rigid-Rotor Spectrum
• Landau Levels

A spectrum card is valid only with its Hamiltonian, configuration space, boundary conditions, quantum-number ranges, and energy-zero convention.

Spin and Angular Momentum

Spin and Angular Momentum collects commutators, matrices, ladder actions, coupling rules, and tensor-operator structure.

• Angular Momentum Algebra
• Spin-Half Matrices
• Ladder-Operator Action
• Addition of Angular Momentum
• Wigner–Eckart Theorem

Check whether a source uses dimensionful angular-momentum operators or dimensionless generators, and whether Pauli matrices or physical spin operators are being displayed.

Approximation and Scattering

Approximation and Scattering gathers perturbative corrections, bounds, semiclassical quantization, transition rates, and scattering observables.

• First-Order Perturbation Theory
• Second-Order Perturbation Theory
• Fermi Golden Rule
• Variational Bound
• WKB Quantization
• Scattering Cross Section
• Born Approximation
• Optical Theorem

These formulas are especially assumption-sensitive. Record normalization, dimensionality, asymptotic-state convention, approximation order, and the parameter controlling the approximation.

Density Matrices and Open Systems

Density Matrices and Open Systems covers mixed-state expectations, subsystem reduction, channels, Markovian generators, and entropy. Use the Density Operators, Partial Trace, and Quantum Operations routes above for their canonical explanations.

• Density-Matrix Expectation
• Partial Trace
• Kraus Map
• Lindblad Equation
• Von Neumann Entropy

Trace preservation, complete positivity, subsystem ordering, basis independence, and logarithm base are common hidden conventions.

Many-Body and Quantum Statistical Mechanics

Many-Body and Quantum Statistical Mechanics collects equilibrium occupations, second-quantized operators, and correlation functions.

• Fermi–Dirac Distribution
• Bose–Einstein Distribution
• Second-Quantized One-Body Operator
• Correlation Functions

Check ensemble, chemical-potential convention, operator ordering, real versus imaginary time, and Fourier normalization before combining formulas.

Quantum Information

Quantum Information provides single-qubit geometry, gates, state-comparison measures, entropies, and stabilizer identities.

• Bloch Vector
• Quantum Gates
• Fidelity
• Trace Distance
• Entropy Identities
• Stabilizer Identities

Fidelity has squared and unsquared conventions. Qubit ordering, global phase, logarithm base, and normalized versus unnormalized distance conventions must also be checked.

Quantum Matter

Quantum Matter collects periodic-system, band-structure, response, and topological quantities.

• Bloch Theorem
• Density of States
• Effective Mass
• Berry Curvature
• Chern Number

Brillouin-zone measure, band indices, cell normalization, charge-sign convention, and Berry-connection gauge convention are essential context.

Relativistic QM

Relativistic QM collects wave equations and matrix identities at the bridge from single-particle quantum mechanics to quantum field theory.

• Klein–Gordon Equation
• Dirac Equation
• Pauli Equation
• Gamma-Matrix Identities

Metric signature, gamma-matrix convention, electromagnetic-coupling sign, natural units, and spinor normalization can all change the displayed equations.

Choosing the Right Reference Layer

Use the layers for different questions:

Question	Reference layer
What does this term mean?	Glossary
What does this glyph denote here?	Symbol Index
Which compact equation applies?	Formula card in this compendium
What are the fixed entries of a matrix or transform?	Tables
Why is the result true?	The card’s canonical derivation link or Derivation Index
Where is the result used?	Worked example, model card, or canonical topic page

This separation preserves one canonical home for explanations and derivations while allowing the same result to be found by task, symbol, model, or method.

Reliability Rules

• Never detach a displayed equation from its assumptions or convention.
• Treat domains and boundary conditions as part of an operator formula.
• Distinguish exact identities from perturbative, asymptotic, and phenomenological formulas.
• State whether continuous-outcome expressions are probabilities or probability densities.
• Preserve degeneracy labels and projectors when an outcome is not one-dimensional.
• Check units before importing a natural-unit expression into numerical work.
• Link to the canonical derivation instead of reproducing it in several reference cards.
• Prefer an explicit limitation over a compact formula that silently overclaims.

References

• L. E. Ballentine, Quantum Mechanics: A Modern Development, World Scientific, 1998.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, 2 vols., Wiley, 1977.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics, vols. I and II, Academic Press, 1975–1980.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.