Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution

Adopt as physical assumptions of the framework the following statement of isolated autonomous dynamics. Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice), which are the choice hypotheses of the cited spectral and group suppliers.

(i) Energy operator. An isolated autonomous system of the sector has a declared energy operator $H_{\mathrm{o}\mathrm{p}}:D(H_{\mathrm{o}\mathrm{p}})\subset H\to H$ that is self-adjoint (Adjoint of a densely defined operator, Symmetric, self-adjoint and essentially self-adjoint operators) and a declared energy unit in which its spectral coordinate is read. The energy operator is part of the model specification of the system; for a massive particle it is fixed by the kinematics postulate P5, and singular or unbounded model Hamiltonians require their own self-adjointness statements.

(ii) Evolution. The time evolution of the model is the strongly continuous one-parameter unitary group $U(t):=\exp \text{\hspace{0.17em}⁣}(-\frac{i}{\mathrm{\hslash }}t{H}_{\mathrm{o}\mathrm{p}})(t\in \mathbb{R}),U(t+s)=U(t)U(s),U(t{)}^{\ast }=U(t{)}^{-1},$ defined by the spectral calculus of the self-adjoint generator (Spectral theorem for unbounded self-adjoint operators (PVM form), Integral of a measurable function against a projection-valued measure, A self-adjoint operator generates a strongly continuous unitary group, Strongly continuous one-parameter unitary group). The energy unit enters the argument of the exponential so that $H_{\mathrm{o}\mathrm{p}}t\mathrm{/}\mathrm{\hslash }$ acts in the dimensionless spectral variable.

(iii) State evolution. A vector state evolves continuously, $\psi (t):=U(t)\psi (0)$ for every $\psi (0)\in H$, and the Schroedinger equation $i\mathrm{\hslash }\frac{d}{dt}\psi (t)=H_{\mathrm{o}\mathrm{p}}\psi (t)$ holds in the norm-differentiable sense exactly for initial vectors $\psi (0)\in D(H_{\mathrm{o}\mathrm{p}})$, where the derivative is the norm limit of the difference quotients; for other vectors the equation is not asserted, although the evolution remains continuous. A density-operator preparation evolves as $\rho (t)=U(t)\rho (0)U(t{)}^{\ast }$, which is a density operator for every $t$ and satisfies $\mathrm{tr}\rho (t)=1$ (Density operators and scalar spectral probability).

(iv) Time-dependent families. A prescribed time-dependent family $t\mapsto H(t)$ of self-adjoint operators is a different model assumption: a norm-continuous finite-dimensional Hermitian family is used only through the unitary two-time propagator that its family defines, and the exponential of the integral of $H(t)$ is not identified with that propagator unless the family commutes with itself at different times. The general unbounded time-dependent case requires a separate existence and domain theorem of its own and is not adopted by this postulate; the driven-dynamics page states its own assumptions for the models it treats.

(v) Open systems. A non-unitary evolution of the reduced description of a system coupled to an environment is an extension obtained from an explicitly chosen environment model and a declared reduction, not a replacement for clause (ii); no environment model is adopted by this postulate.

(vi) Status. Clauses (i)-(v) are adopted assumptions of the isolated-system idealization. They assert nothing about the origin of the energy operator of a concrete system, about the accuracy of the idealization for a laboratory preparation, or about which laboratory interactions realize an isolated evolution; those are tested identifications in the sense of P7.

• Domain matters. The Schroedinger equation is a statement about vectors in $D(H_{\mathrm{o}\mathrm{p}})$; for a vector outside the domain only the unitary group acts, and the expression $H_{\mathrm{o}\mathrm{p}}\psi$ is undefined. Statements about the derivative of expected values require the common-domain hypotheses stated with them.
• Autonomous only. An isolated system evolves autonomously; driven evolution is an additional model assumption of the pages that treat it.
• Units. The energy unit is declared with the observable; energies and times enter the phase $H_{\mathrm{o}\mathrm{p}}t\mathrm{/}\mathrm{\hslash }$ so that the combination is dimensionless in the declared units.