Dynamics and Hamiltonians

Closed autonomous dynamics is represented by a one-parameter unitary group, but the word “continuous” must name the right topology and the differential equation must name its domain.

Reading sequence

Begin with State Vectors for Hilbert-space norms and unitary operators. Stone’s theorem is then the meeting point of two short branches:

1. On the operator branch, read Self-Adjoint Operators and then The Unbounded Spectral Theorem to obtain the domain-sensitive generator and its bounded functional calculus.

2. On the dynamics branch, read Strongly Continuous Unitary Groups, which separates vector-norm continuity from operator-norm continuity, proves the translation example, and defines the strong derivative domain.

3. With both branches complete, Stone’s Theorem proves the exact two-way result

   $$U(t)=e^{-itH/\hbar} \quad\longleftrightarrow\quad H\text{ self-adjoint}.$$

The group exists and is continuous on every Hilbert-space vector. The strong Schrödinger equation

$$i\hbar\frac{d}{dt}U(t)\psi=HU(t)\psi$$

holds for $\psi\in D(H)$, generally a proper dense subspace.

Boundaries to retain

• Strong continuity permits unbounded generators; norm continuity is equivalent to a bounded generator.
• A time-dependent Hamiltonian generally produces $U(t,s)$, not a group depending only on $t-s$.
• A projective ray action needs a coherent unitary lift before Stone applies.
• A symmetric differential expression is not enough; the Hamiltonian must be self-adjoint on a declared domain.

The same theorem supplies the position and momentum generators used in the Weyl CCR.