Operator and Hamiltonian Library

An operator card is useful only when the formula is attached to its mathematical and physical setting. For an unbounded operator, the domain is part of the definition:

$$A:D(A)\subseteq\mathcal H\longrightarrow\mathcal H.$$

For a Hamiltonian, the Hilbert space, boundary conditions, external controls, approximation regime, and energy convention are equally essential. This library is a compact identification layer; it links to canonical derivations instead of duplicating them.

Relativistic Hamiltonians

The relativistic bridge pair illustrates the card format:

1. Dirac Hamiltonian records the free four-component operator and its minimally coupled prescribed-background form.
2. Pauli Hamiltonian records the leading two-component low-energy operator with the minimal $g=2$ magnetic term.

The Hamiltonian Cards index compares their spaces, regimes, charge conventions, and canonical owners. Their derivations are on the Covariant Dirac Equation, Minimal Coupling, Dirac to Pauli, and Pauli Equation pages.

Card contract

A mature operator or Hamiltonian card states:

1. Mathematical type: operator, form, matrix, generator, superoperator, or effective symbol.
2. Space and domain: Hilbert space, tensor factor, dense domain, boundary conditions, and codomain.
3. Defining action: abstract operation before a basis-dependent matrix or differential expression.
4. Parameters and conventions: dimensions, signs, unit system, ordering, gauge, basis, and energy zero.
5. Assumptions: prescribed backgrounds, weak-field or low-energy limits, neglected interactions, and regularity.
6. Reliability checks: self-adjointness, dimensions, symmetries, limiting cases, and gauge covariance.
7. Canonical owner: one visible page for derivation and interpretation.

A compact expression that omits data affecting its spectrum or evolution is not a complete card.

Dirac and Pauli are different objects

The free Dirac Hamiltonian acts on $L^2(\mathbb R^3,\mathbb C^4)$ with a first-order differential domain and has positive- and negative-energy sectors. In a prescribed electromagnetic background,

$$H_D=c\boldsymbol\alpha\cdot\boldsymbol\pi+\beta mc^2+q\Phi.$$

The Pauli Hamiltonian acts on two-component nonrelativistic spinors and is a controlled positive-energy approximation:

$$H_P= \frac{\boldsymbol\pi^2}{2m} +q\Phi -\frac{q\hbar}{2m}\boldsymbol\sigma\cdot\mathbf B$$

for minimal $g=2$. Treating the second as merely a smaller matrix representation of the first erases the $2mc^2$ sector separation and the approximation that eliminated the lower components.

Reliability checks

Before using either card:

• substitute the signed charge once; for an electron $q=-e$;
• distinguish canonical $\mathbf p$ from kinetic $\boldsymbol\pi$;
• preserve operator ordering when potentials vary;
• confirm the domain and boundary condition needed for self-adjoint evolution;
• keep prescribed classical fields distinct from dynamical QED;
• do not append Foldy–Wouthuysen or anomalous-moment corrections without a stated order and matching convention.

Further operator cards

The sidebar includes additional operator and Hamiltonian cards. Check each card’s mathematical setting, canonical owner, and editorial status; an accessible page is not by itself evidence that a formula has been reviewed.

References

• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• F. Schwabl, Advanced Quantum Mechanics, 3rd ed., Springer, 2005, doi:10.1007/3-540-28528-8.