Common Operators

This table identifies common operators and where their canonical explanations live.

Operator Table

Operator	Symbol	Common Action or Definition	Canonical Link
Position	$\hat x$	$(\hat x\psi)(x)=x\psi(x)$	Position Operator
Momentum	$\hat p$	$-i\hbar\,d/dx$ in position representation	Momentum Operator
Hamiltonian	$H$	generator of time evolution	Operator Library
Angular momentum	$\mathbf J$ or $\mathbf L$	generators of rotations	Angular Momentum
Spin	$\mathbf S$	intrinsic angular momentum	Spin
Projector	$P$	$P^2=P=P^\dagger$	Projectors
Density operator	$\rho$	positive trace-one operator	Density Operator
Number operator	$N$	$a^\dagger a$	Creation and Annihilation
Parity	$\Pi$	$(\Pi\psi)(x)=\psi(-x)$	Parity
Time evolution	$U(t,t_0)$	maps states from $t_0$ to $t$	Time-Evolution Operator

Domain Warning

Operators such as $\hat x$, $\hat p$, and $H$ are often unbounded. A formula can be algebraically correct but incomplete if it ignores domains and boundary conditions.

Common Mistakes

• Confusing an operator with its matrix in one basis.
• Treating $\hat p=-i\hbar d/dx$ as domain-free.
• Using local operators on tensor-product spaces without identity factors.
• Calling every Hermitian matrix a physical Hamiltonian without specifying the model.

Related Tables

• Commutators
• Common Hamiltonians
• Pauli Matrices
• Spin Matrices

References

• 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.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, revised and enlarged ed., Academic Press, 1980.