Hilbert-Space Quantum Mechanics

The Hilbert-space formalism separates three structures: a complete linear space of representatives, a projective space of pure states, and an algebra of observables on which states assign expectations. This chapter connects those structures and explains the hypotheses needed when finite matrices become infinite-dimensional operators. Core Formalism retains the first encounter with state vectors, global phase, and interference.

Enter the chapter

Complete the probability on-ramp before entering the chapter: State Vectors supplies normalized Hilbert-space representatives; Projectors and Probability Amplitudes prepare the Born Rule; and Transition Probabilities supplies the operational meaning of squared overlaps used below.

Use Main Mathematical Objects to distinguish states, effects, observables, instruments, channels, and representations in one qubit model. Continue with Hilbert Space as State Space to see why completeness matters for approximations and why a bare Hilbert space does not specify a quantum model. Then Physical States as Rays establishes three equivalent descriptions:

$$\text{complex ray} \quad\longleftrightarrow\quad \text{normalized vectors modulo }U(1) \quad\longleftrightarrow\quad \text{rank-one orthogonal projector}.$$

The intrinsic transition probability is

$$\operatorname{Tr}(P_\psi P_\phi) = \frac{|\langle\psi|\phi\rangle|^2} {\|\psi\|^2\|\phi\|^2},$$

which is the invariant preserved in Wigner’s theorem. The ray page derives this equivalence and transition probability; the later pages develop their geometric and operator consequences.

Geometry, operators, and expectations

1. Projective Hilbert Space constructs the norm quotient topology, local Hilbert-space charts, and horizontal tangent vectors. It states the domain boundary for geometric motion generated by an unbounded Hamiltonian.
2. Superposition Revisited distinguishes finite vector sums, convergent orthonormal series, and convergence in an operator graph norm. Its diagonal energy example shows why a convergent superposition need not belong to the Hamiltonian domain.
3. Pure States as Rank-One Projectors proves operator purity tests and convex extremality, then derives exact norm comparisons between two pure preparations.
4. Expectation Functionals turns vector expectations into positive normalized functionals. It distinguishes finite first moments, quadratic-form domains, and the stronger domain required by the literal operator action.

For symmetry applications, the ray description leads directly to Transition-Probability Preserving Maps and then Wigner’s Theorem.

Assumptions that matter beyond matrices

Separability Assumptions explains countable bases and why continuous position outcomes do not imply a nonseparable Hilbert space. It also distinguishes topological separability from separability of a composite quantum state.

Direct Sums and Superselection shows that an orthogonal decomposition becomes a sector restriction only after the observable algebra is specified. The same full-space pure vector can induce a mixed functional on a block-diagonal algebra.

Infinite-Dimensional Caveats collects explicit failure cases: weak limits that lose normalization, small state errors with large energy errors, spectral values without eigenvectors, and formal operator products whose domains do not match. Use it to test a calculation before importing a finite-matrix inference into a limit.

The main checkpoint is concrete: given a sequence of preparations and an observable, identify the state topology, the observable domain or moment condition, and the algebra of accessible measurements. Each choice changes which conclusion the calculation can support.

References

• B. C. Hall, Quantum Theory for Mathematicians, Graduate Texts in Mathematics 267, Springer, 2013 — the relation between physical postulates and mathematical state structure.
• G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, 2nd ed., Graduate Studies in Mathematics 157, American Mathematical Society, 2014 — Hilbert-space and operator foundations.