Tables

The Tables section collects compact data that is useful enough to keep at hand but too convention-dependent to quote without context. Its entries include matrices, transform pairs, commutator identities, angular-momentum coefficients, unit systems, and other reusable reference data.

A table answers what is the entry under a stated convention? It is not the canonical home for the derivation or physical interpretation. Each page therefore points to the concept, formula, or model page where the result is explained.

Find a Table by Task

Task	Start here	Check before use
Insert a numerical physical constant	Constants	Exact versus measured status, units, and quoted precision
Translate between unit systems	Units	Which constants have been set to one
Calculate with a spin-$1/2$ or qubit	Pauli Matrices	Basis order and rotation convention
Construct a spin-$j$ representation	Spin Matrices	Basis order, $\hbar$ convention, and ladder phases
Simplify an operator commutator	Commutators	Operator order and domain assumptions
Identify a standard operator	Common Operators	Representation and boundary conditions
Recognize a standard Hamiltonian	Common Hamiltonians	Degrees of freedom and omitted terms
Transform between position and momentum	Fourier Transforms	Exponential sign and normalization
Look up a low-order angular function	Spherical Harmonics	Phase and normalization convention
Couple two angular momenta	Clebsch–Gordan Coefficients	State phases and coefficient ordering
Recouple three or more angular momenta	Wigner Symbols	Symbol convention and admissibility rules
Check angular-momentum labels or identities	Angular Momentum Tables	Dimensionless versus $\hbar$-carrying operators
Identify a special-function family	Special Functions	Argument, weight, normalization, and domain
Decompose a small symmetry representation	Character Tables	Group, conjugacy-class order, and irrep labels
Calculate with common quantum gates	Quantum Gates	Qubit ordering and global-phase convention

Before Using Any Table

Most apparent disagreements between reference tables are convention changes rather than physical contradictions. Record the following information before importing an entry into a calculation:

1. Basis and ordering. A matrix is meaningless without the ordered basis used for its rows and columns. For a composite system, also record the tensor-factor order.
2. Units and dimensions. Check whether $\hbar$, $c$, $k_{\mathrm B}$, or another constant has been set to one, and restore dimensions before comparing numerical results.
3. Normalization. Fourier kernels, continuum eigenstates, spherical harmonics, and special functions admit several standard normalizations.
4. Phase. Eigenvectors and coupled angular-momentum states have phase freedom. Relative phases become fixed only after a convention is chosen consistently.
5. Orientation and signs. Verify the sign of the Fourier exponential, the handedness of coordinates, the Levi-Civita convention, and the order of noncommuting factors.
6. Domain and regularity. Operator identities involving unbounded operators require a common invariant domain; distributional transform identities require a declared interpretation.
7. Precision and provenance. Numerical data should state significant digits, source, and edition or release. Generated arrays should identify the generating convention and validation method.

When any of these fields is absent, treat the table as an orientation aid rather than calculation-ready data.

Fundamental Conventions

Constants and units

• Constants distinguishes exact SI defining constants from experimentally determined values and rounded working values.
• Units compares SI, natural, atomic, and thermal conventions without treating a unit choice as a physical approximation.

The Units and Constants guide explains dimensional reasoning. The Common Convention Translations page gives rapid translations among several common choices.

Operators and Algebra

• Pauli Matrices gives the matrices, products, eigenspaces, projectors, and vector identities used for spin-$1/2$ and two-state systems.
• Spin Matrices gives spin-$j$ matrix elements and selected low-dimensional representations.
• Commutators gathers algebraic rules and common quantum operator algebras, together with domain warnings.
• Common Operators identifies standard operators, their action, and representation-dependent caveats.
• Common Hamiltonians lists compact model Hamiltonians and the assumptions encoded in each expression.
• Angular Momentum Tables organizes quantum numbers, dimensions, ladder actions, and coupling rules.

Use the Operator and Hamiltonian Library when the operator itself is the object of interest. Those cards specify Hilbert space, domain, spectrum, symmetries, and physical role; the tables here optimize for comparison and retrieval.

Angular and Symmetry Data

• Spherical Harmonics records low-order functions and their symmetry properties.
• Clebsch–Gordan Coefficients records coefficients for selected low-angular-momentum couplings.
• Wigner Symbols summarizes notation and selected identities for $3j$, $6j$, and $9j$ symbols.
• Character Tables supplies small-group examples and the orthogonality data used to decompose representations.

These are especially sensitive to phase and label conventions. A coefficient copied from one source can be incompatible with states copied from another even when both sources are internally correct. Convert the entire calculation to one convention; do not repair isolated signs by inspection.

Transforms and Special Functions

• Fourier Transforms gives transform pairs and operational rules in both the position–momentum and standard mathematical conventions.
• Special Functions maps common function families to their differential equations, weights, and quantum applications.

Transform tables often mix ordinary functions with distributions. For example, the transform of a constant is a delta distribution, not an ordinary function. The table must state the function space or distributional interpretation when this distinction matters.

Quantum Information

• Quantum Gates gives common one- and two-qubit matrices and their actions.

Gate matrices depend on computational-basis order. For two qubits, the ordering

$$\lbrace \lvert 00\rangle,\lvert 01\rangle, \lvert 10\rangle,\lvert 11\rangle\rbrace$$

is common but not universal. Circuit diagrams can also reverse the visual order of tensor factors. Verify the action on basis kets before importing a matrix.

Table, Formula Card, or Canonical Page?

These reference types solve different problems:

Resource	Primary question	Typical content
Table	What is the entry under this convention?	Matrices, coefficients, transform pairs, or numerical values
Formula card	What equation should I use, and under what assumptions?	Formula, symbols, dimensions, limits, and failure modes
Operator card	What mathematical object is this?	Space, domain, spectrum, symmetries, and realizations
Model card	What physical problem has been defined?	Degrees of freedom, dynamics, parameters, observables, and regime
Canonical concept page	Why is the result true and what does it mean?	Motivation, derivation, interpretation, examples, and exercises

The one-canonical-home rule prevents a compact table from becoming a second, potentially divergent derivation. When more explanation is needed, follow the canonical link rather than inferring assumptions from a row of symbols.

Convention-Sensitive Comparisons

Fourier normalization

All three pairs below are standard:

$$\begin{aligned} \widetilde f(k) &= \int_{-\infty}^{\infty} e^{-ikx}f(x)\,dx, & f(x) &= \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{ikx}\widetilde f(k)\,dk, \\[4pt] \widehat f(k) &= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{-ikx}f(x)\,dx, & f(x) &= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{ikx}\widehat f(k)\,dk, \\[4pt] \phi(p) &= \frac{1}{\sqrt{2\pi\hbar}} \int_{-\infty}^{\infty} e^{-ipx/\hbar}\psi(x)\,dx, & \psi(x) &= \frac{1}{\sqrt{2\pi\hbar}} \int_{-\infty}^{\infty} e^{ipx/\hbar}\phi(p)\,dp. \end{aligned}$$

The factors are not decorative. They determine delta-function normalization, Parseval identities, convolution factors, and the dimensions of transformed functions.

Angular-momentum phases

The Condon–Shortley convention fixes a widely used relative phase for spherical harmonics and coupled states, but other consistent choices exist. Observable probabilities do not depend on an overall phase; interference amplitudes and tabulated coupling coefficients do depend on relative phases. Use spherical harmonics, Clebsch–Gordan coefficients, and Wigner symbols from a mutually compatible convention set.

Pauli matrices and rotations

With the ordered basis $\lbrace\lvert 0\rangle,\lvert 1\rangle\rbrace$ and $\sigma_z\lvert 0\rangle=+\lvert 0\rangle$, the standard matrix for $\sigma_y$ contains $-i$ in its upper-right entry. Reversing the basis order or changing the active-rotation sign changes several displayed matrices while leaving the represented physics equivalent.

Numerical constants

An exact SI value and a recommended measured value have different epistemic status. The Planck constant $h$ is exact in the present SI, whereas quantities such as particle masses are experimentally determined and periodically adjusted. Preserve more digits during intermediate work than the final uncertainty warrants, and cite the data release used.

Provenance and Verification

A mature table records how its entries were established:

• Exact editorial table: identities are checked algebraically against the declared definitions.
• Generated symbolic table: the generating script, software version, input convention, and independent checks are linked.
• Generated numerical table: the algorithm, precision, truncation, tolerances, and benchmark cases are recorded.
• Externally maintained data: the authoritative source, edition or release date, and retrieval date are stated.

Generation does not by itself establish correctness. A useful validation strategy combines at least two independent checks, such as a defining recurrence and an orthogonality relation, or direct matrix multiplication and a representation identity.

For convention-sensitive exact tables, verification should include:

1. one or more defining relations;
2. normalization or orthogonality;
3. a symmetry or composition law;
4. a low-dimensional case that can be checked directly;
5. agreement with an authoritative reference after translating conventions.

Common Mistakes

• Copying a matrix without copying its ordered basis.
• Treating a phase-convention difference as an observable disagreement.
• Mixing dimensionless spin matrices with operators that include $\hbar$.
• Using a Fourier pair from one convention and a convolution rule from another.
• Applying an operator identity outside a common domain.
• Reading a rounded decimal as an exact defining value.
• Combining Clebsch–Gordan coefficients and spherical harmonics from incompatible phase conventions.
• Assuming a computer-generated table is verified because it contains many digits.
• Using a table as evidence for an approximation that its scope does not state.

Exercises

Exercise 1: Two correct Fourier tables

One source defines its forward transform with no prefactor and its inverse with $1/(2\pi)$. Another assigns $1/\sqrt{2\pi}$ to both directions. Explain why a Gaussian, a delta function, and a convolution acquire different numerical factors in the two tables even though both conventions represent the same transform.

Solution

The two definitions distribute the same total normalization factor differently between the forward and inverse transforms. If $\widetilde f$ denotes the first convention and $\widehat f$ the symmetric convention, then

$$\widehat f(k)=\frac{1}{\sqrt{2\pi}}\,\widetilde f(k).$$

Every transform pair must inherit this rescaling. Delta distributions also rescale because their coefficient is fixed by the inverse transform, and the convolution theorem gains the factor required by the chosen normalization. Physical or mathematical predictions agree once every transform, inverse, delta normalization, and convolution rule is taken from one convention.

Exercise 2: A matrix without a basis

A table lists

$$M= \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}$$

but does not state its basis. Can you conclude that it exchanges the physical states called $\lvert 0\rangle$ and $\lvert 1\rangle$?

Solution

No. A matrix represents an operator only after an ordered basis and the map between basis vectors and physical states have been specified. In an ordered basis $\lbrace\lvert 0\rangle,\lvert 1\rangle\rbrace$, this matrix exchanges those vectors. In another basis the same array may represent a different named operator, and the same physical operator generally has a different matrix. The missing basis prevents a physical interpretation.

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, Dover, 1972.
• NIST Digital Library of Mathematical Functions, DLMF, maintained online.
• G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, 7th ed., Academic Press, 2013.
• 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.
• B. N. Taylor and A. Thompson, eds., The International System of Units (SI), NIST Special Publication 330, 2019.
• International Bureau of Weights and Measures, The International System of Units (SI Brochure), 9th ed., version 3.01, 2024.