We consider a macroscopic quantum system with unitarily evolving pure state $$\psi _t\in \mathcal {H}$$ ψ t ∈ H and take it for granted that different macro states correspond to mutually orthogonal, high-dimensional subspaces $$\mathcal {H}_\nu $$ H ν (macro spaces) of $$\mathcal {H}$$ H . Let $$P_\nu $$ P ν denote the projection to $$\mathcal {H}_\nu $$ H ν . We prove two facts about the evolution of the superposition weights $$\Vert P_\nu \psi _t\Vert ^2$$ ‖ P ν ψ t ‖ 2 : First, given any $$T>0$$ T > 0 , for most initial states $$\psi _0$$ ψ 0 from any particular macro space $$\mathcal {H}_\mu $$ H μ (possibly far from thermal equilibrium), the curve $$t\mapsto \Vert P_\nu \psi _t\Vert ^2$$ t ↦ ‖ P ν ψ t ‖ 2 is approximately the same (i.e., nearly independent of $$\psi _0$$ ψ 0 ) on the time interval [0, T]. And second, for most $$\psi _0$$ ψ 0 from $$\mathcal {H}_\mu $$ H μ and most $$t\in [0,\infty )$$ t ∈ [ 0 , ∞ ) , $$\Vert P_\nu \psi _t\Vert ^2$$ ‖ P ν ψ t ‖ 2 is close to a value $$M_{\mu \nu }$$ M μ ν that is independent of both t and $$\psi _0$$ ψ 0 . The first is an instance of the phenomenon of dynamical typicality observed by Bartsch, Gemmer, and Reimann, and the second modifies, extends, and in a way simplifies the concept, introduced by von Neumann, now known as normal typicality.

We consider a closed macroscopic quantum system in a pure state ψ t evolving unitarily and take for granted that different macro states correspond to mutually orthogonal subspaces H ν (macro spaces) of Hilbert space, each of which has large dimension. We extend previous work on the question what the evolution of ψ t looks like macroscopically, specifically on how much of ψ t lies in each H ν . Previous bounds concerned the absolute error for typical ψ 0 and/or t and are valid for arbitrary Hamiltonians H; now, we provide bounds on the relative error, which means much tighter bounds, with probability close to 1 by modeling H as a random matrix, more precisely as a random band matrix (i.e., where only entries near the main diagonal are significantly nonzero) in a basis aligned with the macro spaces. We exploit particularly that the eigenvectors of H are delocalized in this basis. Our main mathematical results confirm the two phenomena of generalized normal typicality (a type of long-time behavior) and dynamical typicality (a type of similarity within the ensemble of ψ 0 from an initial macro space). They are based on an extension we prove of a no-gaps delocalization result for random matrices by Rudelson and Vershynin [32].

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