On optimum Hamiltonians for state transformations

For a prescribed pair of quantum states |psi_I> and |psi_F> we establish an elementary derivation of the optimum Hamiltonian, under constraints on its eigenvalues, that generates the unitary transformation |psi_I> --> |psi_F> in the shortest duration. The derivation is geometric in character and does not rely on variational calculus.Comment: 5 page

An alternative to the conventional micro-canonical ensemble

Usual approach to the foundations of quantum statistical physics is based on conventional micro-canonical ensemble as a starting point for deriving Boltzmann-Gibbs (BG) equilibrium. It leaves, however, a number of conceptual and practical questions unanswered. Here we discuss these questions, thereby motivating the study of a natural alternative known as Quantum Micro-Canonical (QMC) ensemble. We present a detailed numerical study of the properties of the QMC ensemble for finite quantum systems revealing a good agreement with the existing analytical results for large quantum systems. We also propose the way to introduce analytical corrections accounting for finite-size effects. With the above corrections, the agreement between the analytical and the numerical results becomes very accurate. The QMC ensemble leads to an unconventional kind of equilibrium, which may be realizable after strong perturbations in small isolated quantum systems having large number of levels. We demonstrate that the variance of energy fluctuations can be used to discriminate the QMC equilibrium from the BG equilibrium. We further suggest that the reason, why BG equilibrium commonly occurs in nature rather than the QMC-type equilibrium, has something to do with the notion of quantum collapse.Comment: 25 pages, 6 figure

A generalized quantum microcanonical ensemble

We discuss a generalized quantum microcanonical ensemble. It describes isolated systems that are not necessarily in an eigenstate of the Hamilton operator. Statistical averages are obtained by a combination of a time average and a maximum entropy argument to resolve the lack of knowledge about initial conditions. As a result, statistical averages of linear observables coincide with values obtained in the canonical ensemble. Non-canonical averages can be obtained by taking into account conserved quantities which are non-linear functions of the microstate.Comment: improved version, new titl