Rigorous Models

A quantum model consists of more than a Hamiltonian formula. Its Hilbert space specifies the states, its operator domain makes the Hamiltonian self-adjoint, and its spectral measure determines energies and evolution. This chapter works through familiar systems with those data explicit. The ordinary wavefunctions, angular-momentum constructions, and scattering calculations remain at their physical subject pages; the emphasis here is on the mathematical statements that make those calculations reliable.

Enter this chapter

Use Operators, Domains, and Spectra for the general operator machinery and Dynamics and Hamiltonians for construction and evolution theorems. It is also possible to begin with a familiar model and follow its required-background links when a domain or convergence claim needs justification.

Three questions organize the examples. What changes when the state space is infinite-dimensional? Which domain realizes the displayed differential expression? Which part of the spectrum is described by the eigenfunctions being used? Answering these questions often resolves an apparent paradox without changing any of the elementary physics.

Finite-dimensional models establish the baseline

Finite-Dimensional Systems identifies what matrix quantum mechanics guarantees automatically: everywhere-defined operators, equivalent norms, and quantitative continuity of spectra and unitary evolution. It also shows why finite matrices cannot satisfy the canonical position–momentum commutator exactly.

A Qubit as a Two-Dimensional Subspace distinguishes an abstract qubit from an encoded two-level sector of a larger system. The decisive condition is invariance, and a leakage estimate quantifies its failure. A compressed Hamiltonian can miss coupling to the discarded levels even when its matrix entries appear harmless.

Spin-j Systems uses the spin matrices to generate the complete finite observable algebra. It then examines why small commutators in a large-spin scaling do not by themselves guarantee a sharply localized classical state.

Differential models make domains visible

Particle on a Line fixes the configuration-space measure and explains how coordinate changes, scaling, and regularity enter the model. Momentum on the Real Line develops explicit resolvents and their one-sided kernels.

Momentum on an Interval follows quasiperiodic boundary phases through translations, gauge transformations, and the squared operator. Particle in a Box and Boundary Conditions compares Dirichlet, Neumann, and Robin Hamiltonians. Together they explain why a hard-wall kinetic energy cannot simply be identified with the square of any chosen self-adjoint interval momentum.

The Free-Particle Hamiltonian has continuous spectrum and a resolvent that improves regularity. The Harmonic Oscillator has compact resolvent and a discrete eigenbasis, but its domain still requires weighted square summability of expansion coefficients. A complete basis does not make every vector an admissible input to an unbounded operator.

Singular potentials test the meaning of a Hamiltonian formula

Delta-Function Potential as Self-Adjoint Extension treats the interaction as a matching condition. A rank-one resolvent formula supplies its bound-state pole and the limit in which a repulsive barrier separates the line into two half-lines. The form-construction theorem itself belongs to Quadratic Form Methods.

The Coulomb Hamiltonian illustrates a different mechanism: the singularity preserves the full-space $H^2$ operator domain, and a completed square gives a sharp energy lower bound. Hydrogen Atom as a Schrödinger Operator carries this realization into radial coordinates. It distinguishes the physical origin condition from mere radial normalizability and the bound-state subspace from the continuum.

These examples should not be grouped merely as “singular potentials.” Their mathematical constructions differ: a point trace is naturally a form perturbation, whereas the three-dimensional Coulomb potential is relatively operator bounded with bound zero.

Periodicity replaces isolated levels by bands

Periodic Schrödinger Operators constructs a direct integral over quasimomentum for a bounded real one-dimensional periodic potential. Each cell operator has a discrete spectrum; the whole-line operator has absolutely continuous bands. The monodromy discriminant determines their locations and shows why some gaps can close.

This is also a useful test of the word “eigenstate.” A normalized vector in one cell fibre is not a normalized whole-line Bloch wave. The direct-integral measure, rather than a countable sum of those fibre states, controls normalizable packets.

A model comparison to carry forward

Compare a normalized constant function in a finite box with an exponentially decaying cusp on the line. The first has zero Neumann energy but infinite Dirichlet mean energy. The second has finite free kinetic form energy but lies outside the free Hamiltonian operator domain; an appropriate point interaction changes the matching requirement. In both cases the Hilbert vector remains meaningful. The question is which observable, with which realization, admits the claimed expectation or differential action.

That habit transfers beyond exactly solvable models: specify the operator, distinguish its form domain, identify the spectral sector, and state the topology of any limiting procedure.

References

• Hall, Brian C., Quantum Theory for Mathematicians, Graduate Texts in Mathematics 267, Springer, 2013, doi:10.1007/978-1-4614-7116-5 — finite systems, canonical observables, and concrete Schrödinger models.
• Teschl, Gerald, Mathematical Methods in Quantum Mechanics, 2nd ed., Graduate Studies in Mathematics 157, American Mathematical Society, 2014, author’s text — self-adjoint realizations and their spectral analysis.