Typical Macroscopic Long-Time Behavior for Random Hamiltonians

We consider a closed macroscopic quantum system in a pure state \(\psi _t\) evolving unitarily and take for granted that different macro states correspond to mutually orthogonal subspaces \({\mathcal {H}}_\nu \) (macro spaces) of Hilbert space, each of which has large dimension. We extend previous work on the question what the evolution of \(\psi _t\) looks like macroscopically, specifically on how much of \(\psi _t\) lies in each \({\mathcal {H}}_\nu \). Previous bounds concerned the absolute error for typical \(\psi _0\) 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 \(\psi _0\) 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 (Geom Funct Anal 26:1716–1776, 2016).