Conventions Overview

Conventions pages are canonical. Unless a page declares an alternative, the defaults collected here determine how symbols, transforms, products, units, and matrix elements are read.

A different convention is not usually a different physical claim. It becomes dangerous when formulas from two conventions are combined without translation. Every local exception should therefore:

1. state the alternative before it is used;
2. give the translation to the default convention;
3. use the alternative consistently within its declared scope;
4. restore or restate the default when the scope ends.

Convention Hierarchy

Use the following order when interpreting a formula.

1. A local declaration wins. A page may set $\hbar=c=1$, change Fourier normalization, or adopt a source’s angular-momentum phase convention.
2. A chapter declaration applies within that chapter. Long derivations may declare a shared notation once and link back to it.
3. The detailed convention page supplies the sitewide default.
4. This overview supplies the fallback summary.

An undeclared departure from the defaults is an error, not an invitation for the reader to guess.

Quick Convention Ledger

Units and constants

• Keep $\hbar$ explicit on orientation, introductory, undergraduate, and core formalism pages.
• Advanced pages may use $\hbar=1$ only after declaring it.
• Keep $c$ explicit outside relativistic and field-theory bridge contexts unless natural units are declared.
• Restore constants by dimensions and by comparison with a convention-complete formula, not by memory alone.

The Units and Constants article expands these defaults. This ledger states the conventions used unless a page declares a local departure.

Inner products and bra-ket notation

Use the physics convention:

$$\langle\phi|\psi\rangle.$$

It is conjugate-linear in $\phi$ and linear in $\psi$. Thus

$$\begin{aligned} \langle a\phi|\psi\rangle &=a^*\langle\phi|\psi\rangle,\\ \langle\phi|b\psi\rangle &=b\langle\phi|\psi\rangle. \end{aligned}$$

Kets represent vectors, bras their adjoint linear functionals, and $\langle a|A|b\rangle$ is a matrix element. The vertical bar does not by itself imply a position eigenstate, conditional probability, or absolute value.

For the vector language, see State Vectors. Inner-Product Conventions owns the slot linearity, adjoint, and translation rules.

Operators and adjoints

• Operators act on kets from the left.
• Products compose right to left: $AB|\psi\rangle=A(B|\psi\rangle)$.
• The adjoint is defined by the inner product and, for unbounded operators, depends on domains.
• Use $A^\dagger$ for the adjoint and $A^*$ for complex conjugation of a scalar, function, or matrix entry when no ambiguity arises.
• Do not use “Hermitian” as a substitute for self-adjointness when domains matter.

For the domain-sensitive distinction between symmetric and self-adjoint operators, see Self-Adjoint Operators.

Commutators and anticommutators

Use

$$\begin{aligned} [A,B]&=AB-BA,\\ \{A,B\}&=AB+BA. \end{aligned}$$

The canonical position–momentum relation is

$$[X,P]=i\hbar I.$$

Reversing the order reverses the sign. For fermionic creation and annihilation operators, braces denote an anticommutator, not a set.

The displayed definitions are the operative sitewide convention; the focused commutator translator remains under review.

Fourier transforms and plane waves

The symmetric position–momentum convention below is the current sitewide default. A more detailed Fourier translator remains under review. Before combining formulas, identify:

• whether the transform variable is wave number $k$ or momentum $p=\hbar k$;
• which exponential sign appears in the forward transform;
• where factors of $2\pi$ and $\hbar$ are placed;
• whether the transform is unitary under the stated measure.

Position–momentum plane waves use

$$\langle x|p\rangle =\frac{1}{\sqrt{2\pi\hbar}} \exp\left(\frac{ipx}{\hbar}\right)$$

in one spatial dimension under the symmetric unitary convention adopted by the canonical page.

Wavefunction normalization

For a normalized pure state,

$$\langle\psi|\psi\rangle=1.$$

In position representation,

$$\int dx\,|\psi(x)|^2=1$$

when the state is square integrable and the measure is $dx$. In curvilinear coordinates, on discrete configuration spaces, or for generalized eigenstates, the measure or normalization changes.

The coordinate-space measures, boundary terms, and generalized-state distinctions are developed in Normalization Conventions.

Tensor-product ordering

Tensor factors are ordered as declared. For two subsystems,

$$|\psi\rangle_A\otimes|\phi\rangle_B$$

places $A$ before $B$. Matrix representations, computational-basis bit strings, partial traces, and circuit wire order must use the same ordering. Swapping factors is an operation, not a typographical rearrangement.

The displayed $A$-then-$B$ order is the fallback convention. A calculation that chooses another wire or factor order must declare it before use.

Spin and angular momentum

• Use $\mathbf S=\hbar\boldsymbol\sigma/2$ for spin-$1/2$.
• Use the standard Pauli matrices and right-handed Cartesian orientation.
• Use the Condon–Shortley phase convention for spherical harmonics and angular-momentum tables unless an alternative is declared.
• State basis ordering before writing spin matrices or coupled-basis coefficients.
• State the Clebsch–Gordan, $3j$, or reduced-matrix-element convention when comparing tables.

The Spin and Pauli-Matrix Conventions and Angular-Momentum Conventions articles expand these defaults and explain how to compare alternative conventions.

Density operators and measurement

• Density operators are positive trace-class operators with unit trace.
• Expectation values use $\operatorname{Tr}(\rho A)$ when defined.
• Projective measurements, POVMs, effects, and instruments are not interchangeable terms.
• Outcome probabilities and post-measurement state updates are separate parts of a measurement model.
• Label subsystem traces explicitly when ambiguity is possible.

For the canonical state operator, see Density Operators. The bullets above govern notation; the corresponding measurement theory stays with its canonical formalism pages.

How to Compare Two Sources

Before deciding that two formulas disagree, make a convention ledger.

1. Identify the physical quantity. Compare energies with energies, amplitudes with amplitudes, and probability densities with densities under the same measure.
2. Record units. Note whether $\hbar$, $c$, $k_B$, or other constants have been set to one.
3. Record normalization. Check states, delta functions, Fourier transforms, and continuum measures.
4. Record signs. Check exponential phases, metric signature, charge conventions, commutator order, and time-evolution sign.
5. Record basis order. Check tensor factors, spin bases, coordinate order, and coupled versus uncoupled states.
6. Record phase choices. Overall state phases are unobservable, but relative phases and table conventions affect intermediate components.
7. Translate one complete expression. Do not replace isolated symbols while leaving their measures or normalization factors unchanged.
8. Compare an invariant output. Probabilities, spectra, cross sections, traces, and expectation values should agree after translation.

This workflow is usually faster than rederiving both results.

Dimensions as a Translation Tool

Dimensions detect many missing constants. In SI-like units,

$$[\hbar]=\text{energy}\times\text{time}, \qquad [c]=\frac{\text{length}}{\text{time}}.$$

For example, the phase in a time-evolution operator must be dimensionless:

$$U(t)=\exp\left(-\frac{iHt}{\hbar}\right).$$

If a source writes $e^{-iHt}$, it has likely declared $\hbar=1$. Restoring $\hbar$ in that exponential does not automatically restore every hidden constant elsewhere; the dimensions of fields, couplings, and integration measures may also depend on the unit convention.

Dimension checking cannot determine dimensionless factors such as $2\pi$, signs, or phase conventions. Those require the full ledger.

Basis and Representation Labels

Use labels when the representation is not obvious:

$$\psi(x)=\langle x|\psi\rangle, \qquad \widetilde\psi(p)=\langle p|\psi\rangle.$$

The state $|\psi\rangle$ has not changed when its components are expressed in a new basis. Tildes, subscripts, or arguments should distinguish representations when the same symbol would otherwise be ambiguous.

For composite systems, attach subsystem labels:

$$A\otimes I_B, \qquad \operatorname{Tr}_B\rho_{AB}.$$

Omitting an identity factor is acceptable only when context fixes which subsystem the operator acts on.

Local Declarations

A declaration should be near the first formula it controls. Good declarations include:

• “In this section, set $\hbar=c=1$.”
• “Use metric signature $(+,-,-,-)$.”
• “The forward Fourier transform carries $e^{-ikx}$.”
• “Order the computational basis as $|00\rangle,|01\rangle,|10\rangle,|11\rangle$.”
• “Clebsch–Gordan coefficients follow the Condon–Shortley convention.”

A declaration such as “standard conventions are used” is inadequate when multiple standards coexist.

If a quoted or historically important formula uses another convention, preserve the source’s meaning, state the source convention, and translate before combining it with local formulas.

Translation Checks

Check 1: Inner-product linearity

Another source uses the mathematics convention, linear in the first argument and conjugate-linear in the second. It writes $(\phi,\psi)_{\mathrm{math}}$. How does this relate to the physics bra-ket convention?

Solution

With both conventions representing the same complex Hilbert-space inner product,

$$\langle\phi|\psi\rangle_{\mathrm{phys}} =(\psi,\phi)_{\mathrm{math}}.$$

Equivalently,

$$(\phi,\psi)_{\mathrm{math}} =\langle\psi|\phi\rangle_{\mathrm{phys}} =\langle\phi|\psi\rangle_{\mathrm{phys}}^*.$$

The arguments reverse because the slot chosen to be linear is reversed. Norms and absolute transition amplitudes agree:

$$(\psi,\psi)_{\mathrm{math}} =\langle\psi|\psi\rangle_{\mathrm{phys}}.$$

Check 2: Wave number versus momentum

A source uses

$$f(x) =\frac{1}{\sqrt{2\pi}} \int dk\,\widehat f(k)e^{ikx}.$$

Rewrite the integral using $p=\hbar k$ and define a momentum-normalized amplitude $\widetilde f(p)$ so that

$$\int dp\,|\widetilde f(p)|^2 =\int dk\,|\widehat f(k)|^2.$$

Solution

Because $dp=\hbar\,dk$, norm preservation requires

$$\widetilde f(p) =\frac{1}{\sqrt{\hbar}}\, \widehat f\left(\frac{p}{\hbar}\right).$$

Then

$$\begin{aligned} f(x) &=\frac{1}{\sqrt{2\pi\hbar}} \int dp\\ &\quad\times\widetilde f(p) \exp\left(\frac{ipx}{\hbar}\right),\\ \int dp\,|\widetilde f(p)|^2 &=\int dk\,|\widehat f(k)|^2. \end{aligned}$$

Changing variables alters both the measure and the amplitude normalization. Replacing $k$ by $p/\hbar$ in the exponential alone would be incomplete.

Check 3: Restoring Planck’s constant

In units with $\hbar=1$, a source writes

$$i\frac{d}{dt}|\psi(t)\rangle=H|\psi(t)\rangle, \qquad [X,P]=iI.$$

Restore $\hbar$.

Solution

The convention-complete equations are

$$\begin{aligned} i\hbar\frac{d}{dt}|\psi(t)\rangle &=H|\psi(t)\rangle,\\ [X,P]&=i\hbar I. \end{aligned}$$

The first factor follows because $Ht/\hbar$ is dimensionless. The second follows because $XP$ has dimensions of action. These checks fix the power of $\hbar$, though not every dimensionless sign or normalization convention.

Common Convention Errors

• Combining a $k$-space measure with a momentum-normalized wavefunction.
• Restoring $\hbar$ in an exponent but not in commutators or integration measures.
• Treating a global phase change as a disagreement while overlooking a changed relative phase.
• Comparing matrix entries before aligning basis order.
• Reversing $[A,B]$ without reversing its sign.
• Using a mathematics inner product as though its first slot were conjugate-linear.
• Omitting the radial or angular Jacobian in wavefunction normalization.
• Comparing Clebsch–Gordan tables from different phase conventions coefficient by coefficient.
• Calling a gauge choice a change in the physical electromagnetic field.
• Inferring a different prediction from two formulas whose amplitudes differ only by state normalization.

Convention Coverage

Inner-Product Conventions is the focused sitewide owner for this convention. Canonical pages supply the physics behind several other entries: State Vectors for ket and bra language, Normalization Conventions for coordinate-space measures, Self-Adjoint Operators for operator domains, and Density Operators for mixed-state notation.

Focused translators cover units, bra-ket notation, Fourier transforms, wavefunction normalization, operators, commutators, tensor ordering, spin, angular momentum, density matrices, and measurement. This page supplies the shared defaults; a local departure must still be declared before use.

References

• G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, 7th ed., Academic Press, 2013.
• L. E. Ballentine, Quantum Mechanics: A Modern Development, World Scientific, 1998.
• 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.
• D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific, 1988.