Assigned energy observable, unitary flow and the two pictures of autonomous evolution

(a) Assigned energy observable. Adopt the autonomous-dynamics postulate Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution (P4) for a declared isolated system whose state space is the model's The state space of the framework: preparations, pure rays and density operators. An assigned energy observable of the system is an assigned sharp observable $(H_{\mathrm{o}\mathrm{p}},u_H)$ in the sense of Assigned sharp observables and the outcome law of the framework clause (a), whose declared unit $u_H$ is an energy unit in the sense of Dimensions, SI unit conventions and quantity types of the framework (joules in the SI conventions fixed there). Thus $H_{\mathrm{o}\mathrm{p}}:D(H_{\mathrm{o}\mathrm{p}})\subset H\to H$ is densely defined and self-adjoint (Symmetric, self-adjoint and essentially self-adjoint operators, Hilbert space), with spectral projection-valued measure $E_H$ on the Borel sets of $\mathbb{R}$ under the choice hypotheses of Spectral theorem for unbounded self-adjoint operators (PVM form) (Projection valued measure), and its outcome law in a preparation is the law of Assigned sharp observables and the outcome law of the framework clause (b). The assignment of $H_{\mathrm{o}\mathrm{p}}$ is part of the model specification, as in clause (i) of P4; this definition asserts nothing about which laboratory quantity of a concrete system realizes it.

(b) The unitary flow. The time evolution or flow of the isolated system is the family $U(t):=\exp \text{\hspace{0.17em}⁣}(-\frac{i}{\mathrm{\hslash }}t{H}_{\mathrm{o}\mathrm{p}}):={\int }_{\mathbb{R}}{e}^{-it\lambda \mathrm{/}\mathrm{\hslash }}d{E}_H(\lambda )(t\in \mathbb{R}),$ the reading of clause (ii) of P4 in the spectral calculus of $H_{\mathrm{o}\mathrm{p}}$ (Integral of a measurable function against a projection-valued measure); here $\mathrm{\hslash }>0$ is the reduced Planck constant with its value and unit fixed by Dimensions, SI unit conventions and quantity types of the framework, so that $t{H}_{\mathrm{o}\mathrm{p}}\mathrm{/}\mathrm{\hslash }$ is dimensionless. Equivalently, $U(t)=e^{itT}$ for the self-adjoint generator $T:=-H_{\mathrm{o}\mathrm{p}}\mathrm{/}\mathrm{\hslash }$ with $D(T)=D(H_{\mathrm{o}\mathrm{p}})$; the two readings agree by the uniqueness of the generator in Stone's theorem, and by A self-adjoint operator generates a strongly continuous unitary group (Flow calculus of a self-adjoint generator: commutation, invariance and differentiability clause (a)) the family $U$ is a strongly continuous one-parameter unitary group: $U(0)=I$, $U(t+s)=U(t)U(s)$, $U(t{)}^{\ast }=U(t{)}^{-1}=U(-t)$ (Strongly continuous one-parameter unitary group).

(c) Schroedinger picture. The Schroedinger-picture evolution of a preparation is fixed by clause (iii) of P4. A state vector evolves by $\psi (t):=U(t)\psi (0)$ for every $\psi (0)\in H$, and the state at time $t$ of the pure preparation $[\psi (0)]$ is the pure preparation $[\psi (t)]$, which is well defined because $U(t)$ is injective; a general preparation $\rho$, that is an element of the state space $D(H)$ of The state space of the framework: preparations, pure rays and density operators, evolves by $\rho (t):=U(t)\rho (0)U(t{)}^{\ast },$ the reading of clause (iii) of P4 in the density-operator vocabulary of The state space: density operators, pure and mixed states, and purity (Self-adjoint, positive, unitary and normal operators, Density operators and scalar spectral probability). In this picture the assigned observables are held fixed at their assigned values and the preparation moves.

(d) Heisenberg picture. For a bounded assigned sharp observable $(A,u_A)$ — that is, an assigned sharp observable with $A\in \mathcal{B}(H)$ (A bounded linear operator between normed spaces), so that $A$ is self-adjoint and everywhere defined — the Heisenberg representative of $(A,u_A)$ at time $t$ is the pair $(A_H(t),u_A)$ with the same declared unit and $A_H(t):=U(t{)}^{\ast }AU(t)=U(-t)AU(t)\mathrm{.}$ By Heisenberg flow of a bounded operator: conjugates, covariance and the equation of motion clauses (a) and (d), $A_H(t)$ is again bounded and self-adjoint with spectral projection-valued measure $E_{A_H(t)}(B)=U(-t)E_A(B)U(t)$, so $(A_H(t),u_A)$ is again an assigned sharp observable of the framework; in this picture the preparation is held fixed at its value at time $0$ and the observables move. The two pictures are two bookkeeping devices for the same model; the statement that they assign the same outcome laws is proved in the Heisenberg-picture theorem of this page, not assumed here.

(e) Stationary preparations. A preparation $\rho \in D(H)$ is stationary for the assigned energy observable when $\rho (t)=\rho$ for every $t\in \mathbb{R}$. A pure preparation $[\psi ]$ is stationary when $[\psi (t)]=[\psi (0)]$ for every $t$, that is, when the evolved vector remains in the ray of $\psi$; the characterization of this condition by the eigenrays of $H_{\mathrm{o}\mathrm{p}}$ is proved in the stationary-states theorem of this page, not assumed here. An energy eigenstate is a pure preparation of a unit eigenvector of $H_{\mathrm{o}\mathrm{p}}$, and a stationary state in the model is a stationary pure preparation; by that theorem the two notions coincide.

(f) Energy-compatible observables. An assigned sharp observable $(A,u_A)$ is compatible with the energy, or energy-compatible, when its spectral projection-valued measure and that of the energy commute strongly, $E_A(B)E_H(C)=E_H(C)E_A(B)\text{for all Borel }B,C\subseteq \mathbb{R}\mathrm{.}$ This is an equality of bounded operators on $H$, not a relation between formal commutators on a small domain; the phrase conserved quantity for an energy-compatible assigned observable is a claim about the time invariance of its outcome law and is established in the energy-conservation theorem of this page for bounded $A$.

(g) Status. Clauses (a)-(f) record the readings of P4 together with P1 and P2 in the vocabulary of this page and fix the terminology used by the later pages of the category. They adopt no physical assumption beyond P4, P1 and P2 and the choice hypotheses of the cited mathematics, assert no dynamics for a prescribed time-dependent family or for an open system (clauses (iv) and (v) of P4 are separate model assumptions treated on their own pages), assert no measurement model, no apparatus realization and no empirical result, and make no claim that any laboratory system is exactly isolated.

• Units and dimensionless phases. The energy unit $u_H$ and $\mathrm{\hslash }$ are part of the model's quantity conventions: $H_{\mathrm{o}\mathrm{p}}t\mathrm{/}\mathrm{\hslash }$ is a dimensionless spectral variable, and the argument of the exponential is read in those units. No numerical value of $H_{\mathrm{o}\mathrm{p}}$ is asserted.
• Domain matters. The flow $U(t)$ is defined on all of $H$, while the energy operator and every unbounded assigned observable are defined only on their domains; the Schroedinger equation and the Heisenberg equation are domain-qualified statements and are proved as such on this page.
• Rays and global phases. Multiplying $\psi (t)$ by a common unit-modulus scalar leaves its ray unchanged. For nonzero coefficients $c_j,c_k$ in a fixed orthonormal basis, their relative phase is $(c_j\mathrm{/}\mathrm{\mid }{c}_j\mathrm{\mid })\overline{(c_k\mathrm{/}\mathrm{\mid }{c}_k\mathrm{\mid })}$; it is unchanged by a common phase and can affect outcome laws. Autonomous evolution need not change relative phases: for $H_{\mathrm{o}\mathrm{p}}=EI$ with $E\in \mathbb{R}$, clause (b) gives $U(t)=e^{-iEt\mathrm{/}\mathrm{\hslash }}I$, so every ray is stationary.