Common Convention Translations

This page is a quick crosswalk for comparing formulas written in different conventions. Use it when a textbook, paper, notebook, or software package appears to disagree with a page by a sign, factor of $\hbar$, Fourier normalization, tensor-product order, or basis choice.

The canonical convention definitions live in Conventions Overview. This page only records common translations.

Default Snapshot

The default conventions are:

• keep $\hbar$ explicit unless a page declares otherwise;
• use $e>0$ for the elementary positive charge and $q=-e$ for an electron;
• use $\hat{\mathbf p}\to\hat{\mathbf p}-q\mathbf A$ for minimal electromagnetic coupling;
• use the physics inner product, conjugate-linear in the bra and linear in the ket;
• use symmetric position-momentum Fourier factors with phases $e^{ipx/\hbar}$ and $e^{-ipx/\hbar}$;
• use $[A,B]=AB-BA$ and $\{A,B\}=AB+BA$;
• use left-to-right tensor-product subsystem ordering;
• use the standard $S_z$ spin-half basis and $S_i=\hbar\sigma_i/2$;
• use the Condon-Shortley phase convention for spherical harmonics and angular momentum tables;
• use density operators $\rho$ with $\rho^\dagger=\rho$, $\rho\ge0$, and $\operatorname{Tr}\rho=1$.

Translation Table

Issue	Default convention	Common alternate convention	Translation check
Reduced Planck constant	$\hbar$ explicit	$\hbar=1$	Restore $\hbar$ by dimensional analysis: commutators, phases, momenta, and spectra are the first places to check.
Speed of light	$c$ explicit outside declared relativistic units	$c=1$	Restore $c$ by dimensions before mixing nonrelativistic and relativistic formulas.
Electric charge	$e>0$, electron charge $q=-e$	Electron charge denoted by $e$ with sign included locally	Check whether $e$ means positive elementary charge or signed electron charge before using electromagnetic Hamiltonians.
Minimal coupling	$\hat{\mathbf p}\to\hat{\mathbf p}-q\mathbf A$	Opposite charge sign or different vector-potential sign	Translate the sign of $q\mathbf A$ together with the stated charge convention.
Fourier variable	momentum $p$ with $px/\hbar$	wave number $k$ with $kx$	Use $p=\hbar k$ and include the Jacobian in probability densities.
Fourier sign	$\psi(x)$ from $e^{ipx/\hbar}$ and $\phi(p)$ from $e^{-ipx/\hbar}$	Opposite forward transform sign	Translate the transform pair as a whole; do not change one sign while keeping the old momentum operator.
Inner-product slot	$\langle\phi\vert\psi\rangle$ linear in $\psi$	Linear in the first argument	Complex conjugate scalar-linearity statements and check adjoint definitions.
Matrix elements	$A_{mn}=\langle m\vert A\vert n\rangle$	Row-vector or opposite-index convention	Check which index labels the output component before importing matrices.
Commutator	$[A,B]=AB-BA$	Formula written using $[B,A]$ or opposite Lie-bracket sign	Reverse the sign when the order is reversed.
Heisenberg equation	$dA_H/dt=(i/\hbar)[H,A_H]$ for no explicit time dependence	$dA_H/dt=[A_H,H]/(i\hbar)$	These are equivalent because $[A_H,H]=-[H,A_H]$.
Tensor-product basis	$\lvert ij\rangle=\lvert i\rangle_A\otimes\lvert j\rangle_B$	Little-endian or software-specific bit order	Translate by a basis permutation, not by changing physics.
Local operators	$O_A\otimes I_B$ acts on the first factor	Identities suppressed or subsystem order reversed	Restore identity factors before comparing matrices.
Spin operators	$S_i=\hbar\sigma_i/2$	Dimensionless spin operators or $\hbar=1$	Check whether eigenvalues are $\pm1$, $\pm1/2$, or $\pm\hbar/2$.
Pauli basis	$\lvert+\rangle_z=(1,0)^T$, $\lvert-\rangle_z=(0,1)^T$	Basis labels or phases changed	Compare the displayed Pauli matrices; basis changes conjugate all matrices consistently.
Angular momentum	$J_\pm=J_x\pm iJ_y$ and Condon-Shortley phases	Different spherical-harmonic phase convention	Translate spherical-harmonic and Clebsch-Gordan signs together.
Density operator	Abstract $\rho$; matrix entries are basis dependent	Density matrix treated as primary object	Name the basis before interpreting populations, coherences, or matrix entries.
Measurement notation	Projectors $P_a$, POVM effects $E_a$	Projectors written $\Pi_a$ or effects written $M_a$	Identify whether the symbol denotes a projector, effect, Kraus operator, or outcome label.

Mini Translations

Natural Units to Explicit Constants

If a source sets $\hbar=1$, then

$$[\hat x,\hat p]=i$$

translates to

$$[\hat x,\hat p]=i\hbar.$$

Similarly,

$$U(t)=e^{-iHt}$$

translates to

$$U(t)=e^{-iHt/\hbar}.$$

The quickest check is dimensional: the argument of an exponential must be dimensionless, and a commutator of position and momentum has units of action.

Momentum Fourier Amplitudes from Wave-Number Amplitudes

Suppose a source uses

$$\tilde\psi(k) = \frac{1}{\sqrt{2\pi}} \int e^{-ikx}\psi(x)\,dx.$$

With $p=\hbar k$, the default momentum-space amplitude is

$$\phi(p) = \frac{1}{\sqrt{\hbar}} \tilde\psi(p/\hbar),$$

provided the sign convention matches. This factor is required because probabilities must agree:

$$\lvert\phi(p)\rvert^2\,dp = \lvert\tilde\psi(k)\rvert^2\,dk.$$

Missing this Jacobian is one of the most common normalization errors in position-momentum translations.

Physics and Mathematics Inner Products

With the physics convention,

$$\langle a\phi\vert\psi\rangle = a^*\langle\phi\vert\psi\rangle, \qquad \langle\phi\vert a\psi\rangle = a\langle\phi\vert\psi\rangle.$$

Some mathematics texts reverse which slot is linear. When comparing a theorem stated in that convention, do not copy sesquilinear identities mechanically. Check whether the adjoint, Riesz map, or matrix-element convention has been defined with the opposite slot.

Tensor-Product Bit Order

The default two-qubit coefficient order is

$$\lvert00\rangle,\quad \lvert01\rangle,\quad \lvert10\rangle,\quad \lvert11\rangle.$$

If a software convention orders the middle two basis states as $\lvert10\rangle,\lvert01\rangle$, then the same physical state has its middle coefficients swapped. This is a permutation of coordinate labels, not a different state.

Heisenberg Equation Signs

The default no-explicit-time-dependence form is

$$\frac{dA_H}{dt} = \frac{i}{\hbar}[H,A_H].$$

The form

$$\frac{dA_H}{dt} = \frac{1}{i\hbar}[A_H,H]$$

is the same statement because $1/i=-i$ and $[A_H,H]=-[H,A_H]$.

Translation Workflow

1. Identify the source convention explicitly. Do not infer it from one formula if the source may be mixing notation.
2. Translate one simple diagnostic formula first, such as $[\hat x,\hat p]$, $U(t)$, a Fourier pair, or a Pauli matrix.
3. Check dimensions, normalization, and basis order.
4. Compare a convention-independent result, such as a probability, spectrum, trace, or expectation value.
5. Link to the canonical convention page if the translated formula will appear in a page.

Common Mistakes

• Restoring $\hbar$ in one formula but not in the related phase or commutator.
• Changing a Fourier sign without changing the transform pair consistently.
• Treating $\phi(p)$ and $\tilde\psi(k)$ as the same function rather than related amplitudes.
• Comparing matrix elements without checking which index labels the output basis vector.
• Reading software bit strings as if they used the pedagogical tensor-product order.
• Treating $\sigma_i$ and $S_i$ as interchangeable.
• Copying spherical-harmonic or Clebsch-Gordan signs from a source with a different phase convention.
• Using a POVM effect as if it were automatically the state-update operator.

Canonical Pages

• Units and Constants
• Bra-Ket Notation
• Inner Product Conventions
• Fourier Transform Conventions
• Wavefunction Normalization
• Operator Conventions
• Commutator Conventions
• Tensor Product Ordering
• Spin and Pauli Matrix Conventions
• Angular Momentum Conventions
• Density Matrix Conventions
• Probability and Measurement Conventions

References

• P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed., Oxford University Press, 1958.
• 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.
• 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.
• D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific, 1988.

Exercises

1. A source writes $U(t)=e^{-iHt}$ after setting $\hbar=1$. What is the explicit-$\hbar$ form?

Solution

The explicit form is

$$U(t)=e^{-iHt/\hbar}.$$

The exponent must be dimensionless, and $Ht$ has units of action.

2. A wave-number amplitude $\tilde\psi(k)$ is normalized so that $\int\lvert\tilde\psi(k)\rvert^2\,dk=1$. What momentum amplitude $\phi(p)$ preserves normalization when $p=\hbar k$?

Solution

Use

$$\phi(p)=\frac{1}{\sqrt{\hbar}}\tilde\psi(p/\hbar).$$

Then $\lvert\phi(p)\rvert^2dp=\lvert\tilde\psi(k)\rvert^2dk$.

3. In the default two-qubit order, a state has coefficient column $(a,b,c,d)^T$. A source uses the order $\lvert00\rangle,\lvert10\rangle,\lvert01\rangle,\lvert11\rangle$. What is the source-order coefficient column for the same state?

Solution

The middle two basis labels are swapped. The source-order column is

$$(a,c,b,d)^T.$$