Operators and Commutators Formulas

These cards collect the operator identities most often needed in undergraduate quantum calculations: canonical brackets, commutator algebra, uncertainty bounds, exponentials of noncommuting operators, projection identities, and unitary transformations.

Each card is a lookup layer. It states assumptions, useful special cases, failure modes, and a link to the one canonical explanation or derivation.

Choose a card

Need	Card	Core statement
Identify a canonical pair or representation	Canonical Commutation Relations	$[X_i,P_j]=i\hbar\delta_{ij}I$
Expand or simplify a commutator	Commutator Identities	$[A,BC]=[A,B]C+B[A,C]$
Bound two outcome spreads	Uncertainty Relations	$\Delta_\rho A\,\Delta_\rho B\geq\frac12\left\lvert\langle[A,B]\rangle_\rho\right\rvert$
Combine or conjugate exponentials	Baker–Campbell–Hausdorff	$\log(e^Ae^B)=A+B+\frac12[A,B]+\cdots$
Select a subspace or spectral sector	Projection Identities	$P^2=P=P^\dagger$
Check or transform a reversible map	Unitary Identities	$U^{-1}=U^\dagger$

How the cards fit together

The cards form a common calculation chain.

1. Specify the operator class. Decide whether the object is self-adjoint, unitary, projective, positive, bounded, or merely a formal expression.
2. Fix the algebra. State canonical commutators and use product, inverse, power, and Jacobi identities without changing operator order.
3. Extract a physical consequence. A commutator expectation can bound variances; a projector expectation can give a yes–no probability.
4. Build finite transformations. Hermitian generators exponentiate to unitaries, and BCH controls products or conjugations of noncommuting exponentials.
5. Check domains and convergence. Finite matrix algebra is not automatically valid for unbounded differential operators.

For example, begin with

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

The commutator card gives

$$[X,P^n] = ni\hbar P^{n-1}$$

on a suitable common domain. The uncertainty card gives

$$\Delta X\,\Delta P\geq\frac{\hbar}{2}.$$

The BCH and unitary cards give the finite translation

$$e^{iaP/\hbar}Xe^{-iaP/\hbar} = X+aI.$$

These are related results, but their canonical derivations belong to different pages.

Algebraic core

The commutator convention is

$$[A,B]\equiv AB-BA.$$

It is bilinear, antisymmetric, and obeys the product rules

$$[A,BC] = [A,B]C+B[A,C],$$ $$[AB,C] = A[B,C]+[A,C]B.$$

The Jacobi identity is

$$[A,[B,C]] +[B,[C,A]] +[C,[A,B]] = 0.$$

These identities are exact in any associative algebra. For unbounded operators they are meaningful only on vectors for which all displayed products are defined, or on a specified common invariant core.

Exponentiation converts infinitesimal generators into finite transformations. When $[A,B]\ne0$,

$$e^{A+B}\ne e^Ae^B$$

in general. BCH supplies the correction:

$$\log(e^Ae^B) = A+B+\frac12[A,B]+\cdots.$$

For conjugation, use Hadamard’s lemma directly:

$$e^ABe^{-A} = B+[A,B] +\frac1{2!}[A,[A,B]] +\cdots.$$

Geometry and probability

Projectors and unitaries encode two different geometric operations.

An orthogonal projector selects a subspace:

$$P^2=P=P^\dagger.$$

It generally loses information because components in $\ker P$ are removed. For a state $\rho$,

$$p=\operatorname{Tr}(\rho P)$$

is the Born probability of the corresponding sharp alternative.

A unitary preserves the entire Hilbert-space geometry:

$$U^\dagger U=UU^\dagger=I.$$

It is reversible and preserves inner products:

$$\langle U\phi\vert U\psi\rangle = \langle\phi\vert\psi\rangle.$$

A nontrivial orthogonal projector is therefore not unitary. A projector keeps one sector; a unitary rotates or phases the whole space without discarding a sector.

The two classes interact through conjugation:

$$P'=UPU^\dagger.$$

The transformed $P'$ remains a projector and selects the transformed subspace $U(\operatorname{Ran}P)$.

Uncertainty workflow

For self-adjoint $A$ and $B$ in a normalized state $\rho$, first define

$$\delta_\rho A=A-\langle A\rangle_\rho I, \qquad \delta_\rho B=B-\langle B\rangle_\rho I.$$

The full two-observable bound is

$$\begin{aligned} (\Delta_\rho A)^2(\Delta_\rho B)^2 &\geq C_\rho(A,B)^2 \\ &\quad+ \frac14 \left\lvert \langle[A,B]\rangle_\rho \right\rvert^2. \end{aligned}$$

Use the shorter Robertson form when only a lower bound is needed:

$$\Delta_\rho A\,\Delta_\rho B \geq \frac12 \left\lvert \langle[A,B]\rangle_\rho \right\rvert.$$

Use the covariance-aware form when testing saturation or studying correlated Gaussian states. A vanishing commutator expectation in one state does not imply that the operators commute.

Finite-matrix checks

For finite matrices, the following checks are quick and decisive.

Object	Primary checks	Secondary diagnostics
Self-adjoint observable $A$	$A^\dagger=A$	Real eigenvalues; orthogonal eigenspaces
Orthogonal projector $P$	$P^2=P$ and $P^\dagger=P$	Eigenvalues $0,1$; $\operatorname{Tr}P=\operatorname{rank}P$
Unitary $U$	$U^\dagger U=UU^\dagger=I$	Singular values one; $\lvert\det U\rvert=1$
Commutator $C=[A,B]$	$C=AB-BA$	$\operatorname{Tr}C=0$
Density operator $\rho$	$\rho^\dagger=\rho$, $\rho\geq0$, $\operatorname{Tr}\rho=1$	Eigenvalues nonnegative and sum to one

Secondary diagnostics do not replace the defining tests. In particular,

$$\lvert\det U\rvert=1$$

does not prove unitarity, and

$$\operatorname{Tr}P\in\mathbb Z$$

does not prove idempotence.

For floating-point matrices, use residual norms such as

$$\epsilon_P = \lVert P^2-P\rVert,$$ $$\epsilon_U = \lVert U^\dagger U-I\rVert,$$

and scale tolerances to the matrix norm, dimension, conditioning, and machine precision.

Infinite-dimensional rule

Three cautions recur throughout this category.

Domains

For unbounded $A$ and $B$, the formal expressions $AB$, $BA$, and $[A,B]$ may have different domains. An algebraic identity is usable only on a common domain where every term exists.

Topology and convergence

An infinite sum of projectors may converge strongly without converging in operator norm. A BCH series may be a formal local series rather than a global operator identity. State the mode of convergence.

Isometry versus unitary

In infinite dimension,

$$U^\dagger U=I$$

does not alone imply $UU^\dagger=I$. The first identity defines an isometry; unitarity also requires surjectivity, equivalently both adjoint identities.

These are structural qualifications, not technical decorations. Ignoring them can turn a correct finite-matrix manipulation into a false operator claim.

Convention checks

Before using a formula, confirm:

• the commutator convention is $[A,B]=AB-BA$;
• products act on states from right to left;
• canonical variables include the factor of $\hbar$ used here;
• generators are dimensionless inside exponentials;
• active and passive unitary conventions are identified;
• projectors are orthogonal unless oblique projection is explicitly intended;
• the state used in an uncertainty bound is normalized;
• approximation orders in BCH are counted with one common small parameter.

Dimensional analysis is especially useful for exponentials:

$$e^{-itH/\hbar}$$

has a dimensionless exponent, while $e^{-itH}$ silently assumes $\hbar=1$.

Common routing mistakes

• Looking up a canonical commutator when the real task is a domain or boundary condition problem.
• Using the uncertainty card to discuss apparatus error or disturbance.
• Using BCH for a time-ordered continuous evolution without checking the time-ordering problem.
• Multiplying two projectors and calling the result an intersection without checking commutation.
• Treating a positive POVM effect as a projector.
• Calling a norm-preserving antilinear symmetry unitary rather than antiunitary.
• Using an active operator transformation where a passive basis change was intended.
• Assuming finite-dimensional row, determinant, or trace tests extend unchanged to infinite-dimensional operators.
• Reproducing a long derivation from a card instead of following its canonical link.

Canonical explanations

Topic	Canonical home
Canonical brackets	Canonical Commutation Relations
Commutator meaning	Commutators
General uncertainty derivation	General Uncertainty Relations
Position–momentum specialization	Position–Momentum Uncertainty
BCH derivation and applications	Baker–Campbell–Hausdorff Formula
Projection linear algebra	Projectors
Quantum meaning of projectors	Projectors in Core Formalism
Unitary linear algebra	Unitary Operators
Closed-system evolution	Unitary Time Evolution

Continue by subject

• States and Probability gives normalization, Born probabilities, expectations, variances, currents, and continuity.
• Dynamics gives Schrödinger, Heisenberg, propagator, and picture-change formulas.
• Spin and Angular Momentum gives angular-momentum commutators, ladder actions, spin matrices, and coupling identities.
• Density Matrices and Open Systems gives mixed-state expectations, partial traces, channels, purity, and master equations.
• Quantum Information gives fidelity, trace distance, qubit geometry, and gate formulas.
• Commutator Table provides a condensed lookup of frequently used operator brackets.

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.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.
• R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd ed., Cambridge University Press, 2013.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics, Volume I: Functional Analysis, revised and enlarged ed., Academic Press, 1980.