Commutator Conventions

The default commutator convention is

$$[A,B]=AB-BA.$$

The default anticommutator convention is

$$\{A,B\}=AB+BA.$$

These sign conventions are used throughout operator algebra, uncertainty relations, spin, symmetry generators, canonical quantization, and time evolution. For conceptual development, see Commutators. For algebraic identities, see Commutators and Anticommutators.

Canonical Relation

When position and momentum are represented as operators, the default convention is

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

In the position representation, this is consistent with

$$\hat p=-i\hbar\frac{d}{dx}.$$

Pages may omit hats when no confusion is possible, but pages comparing classical variables with operators should use hats or explicit prose.

Product Rules

The commutator acts as a derivation in either slot:

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

and

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

The Jacobi identity is

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

Time Evolution Sign

With Schrödinger evolution

$$i\hbar\frac{d}{dt}\lvert\psi(t)\rangle=H\lvert\psi(t)\rangle,$$

the Heisenberg equation for an operator with no explicit time dependence is written

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

Equivalent forms such as $dA_H/dt=-(i/\hbar)[A_H,H]$ are the same convention rearranged.

Lie Algebra Conventions

Angular momentum and spin use

$$[J_i,J_j] =i\hbar\sum_k\epsilon_{ijk}J_k.$$

For dimensionless Pauli matrices,

$$[\sigma_i,\sigma_j] =2i\sum_k\epsilon_{ijk}\sigma_k.$$

The Levi-Civita symbol convention is $\epsilon_{xyz}=+1$, equivalently $\epsilon_{123}=+1$ when $(1,2,3)$ correspond to $(x,y,z)$.

Common Mistakes

• Reversing the commutator sign when copying formulas from a source using $[B,A]$.
• Forgetting that $\{A,B\}$ means anticommutator here, not a Poisson bracket.
• Treating $[\hat x,\hat p]=i\hbar$ as a statement about ordinary numbers.
• Dropping domain assumptions for unbounded operators.
• Mixing the Heisenberg equation sign with a different Schrödinger-equation phase convention.

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.

Exercises

1. Prove $[A,B]=-[B,A]$.

Solution

Using the definition,

$$[B,A]=BA-AB=-(AB-BA)=-[A,B].$$

2. If $[H,A]=0$ and $A$ has no explicit time dependence, what does the Heisenberg equation say?

Solution

The Heisenberg equation gives

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

If the commutator vanishes, $dA_H/dt=0$. The observable is conserved in Heisenberg time evolution.