Operator Cards

These cards summarize standard quantum operators, their most common actions, assumptions, and convention warnings. They are lookup entries; full derivations and conceptual discussion live on the canonical pages linked from each card.

Cards

Card	Core Action	Main Caution
Position	$(\hat x\psi)(x)=x\psi(x)$	Domain and measure depend on configuration space.
Momentum	$\hat p=-i\hbar\,d/dx$	Self-adjointness depends on boundary conditions.
Angular Momentum	$[J_i,J_j]=i\hbar\epsilon_{ijk}J_k$	One component is diagonalized with $J^2$, not all three.
Spin	$S_i=\hbar\sigma_i/2$ for spin $1/2$	Spin acts on internal degrees of freedom.
Parity	$(\mathsf P\psi)(\mathbf r)=\psi(-\mathbf r)$	Parity must be a symmetry before eigenparity is conserved.
Time Reversal	$\mathsf T i\mathsf T^{-1}=-i$	Time reversal is antiunitary.
Translation	$U(\mathbf a)=e^{-i\mathbf a\cdot\mathbf P/\hbar}$	Active and passive conventions differ by signs.
Rotation	$U(R)=e^{-i\theta\hat{\mathbf n}\cdot\mathbf J/\hbar}$	State, vector, and coordinate rotations must be distinguished.
Creation-Annihilation	$[a,a^\dagger]=1$	Oscillator and many-body contexts use related but distinct notation.
Displacement	$D(\alpha)=e^{\alpha a^\dagger-\alpha^*a}$	Conventions differ for whether $D^\dagger aD=a+\alpha$ or the inverse action is quoted.
Squeeze	$S(\zeta)=e^{(\zeta^*a^2-\zeta a^{\dagger2})/2}$	Phase and sign conventions move between definitions.
Density Operator	$\rho\ge0$, $\operatorname{Tr}\rho=1$	Positivity and trace class are essential.

Reading Rule

For any operator card, identify the Hilbert space, domain, representation, boundary conditions, and whether the operator is an observable, a symmetry action, a generator, or a state object.

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, Academic Press, 1980.