Adopted isolated prepared unitary model

For a specified finite-dimensional complex Hilbert space $\mathcal{H}$, $H=H^{\ast }$ in J, and adopted $\mathrm{\hslash }>0$ in J s, take a prepared finite density matrix ${\rho }_0$ (positive semidefinite with trace one). For time $t$ measured in s, adopt isolated evolution $U(t)=e^{-itH\mathrm{/}\mathrm{\hslash }},{\rho }_t=U(t){\rho }_{0}U(t{)}^{\ast },$ and the Born expectation ${\mathbb{E}}_t[A]=\mathrm{Tr}({\rho }_{t}A)$ for each bounded self-adjoint matrix observable $A$. This adopts the finite subsystem's preparation and unitary time evolution as a model assumption. It inherits the full Axiom of Choice convention, stated in The Axiom of Choice, of Normal-state quantum measurement interpretation for the wider Hilbert-space spectral-measure framework, although this finite bounded-observable clause uses no unbounded spectral measurement. No actual apparatus, measured value, thermalization, or infinite-volume Hamiltonian is asserted.

The finite unitary and Gibbs conventions are stated in Finite unitary dynamics and stationary Gibbs states as formulation context; they do not prove that this postulate describes nature.