Quality of Variational Trial States

Besides perturbation theory (which clearly requires the knowledge of the exact unperturbed solution), variational techniques represent the main tool for any investigation of the eigenvalue problem of some semibounded operator H in quantum theory. For a reasonable choice of the employed trial subspace of the domain of H, the lowest eigenvalues of H usually can be located with acceptable precision whereas the trial-subspace vectors corresponding to these eigenvalues approximate, in general, the exact eigenstates of H with much less accuracy. Accordingly, various measures for the accuracy of the approximate eigenstates derived by variational techniques are scrutinized. In particular, the matrix elements of the commutator of the operator H and (suitably chosen) different operators with respect to degenerate approximate eigenstates of H obtained by variational methods are proposed as new criteria for the accuracy of variational eigenstates. These considerations are applied to precisely that Hamiltonian for which the eigenvalue problem defines the well-known spinless Salpeter equation. This bound-state wave equation may be regarded as (the most straightforward) relativistic generalization of the usual nonrelativistic Schroedinger formalism, and is frequently used to describe, e.g., spin-averaged mass spectra of bound states of quarks.Comment: LaTeX, 7 pages, version to appear in Physical Review

Time ordering and counting statistics

The basic quantum mechanical relation between fluctuations of transported charge and current correlators is discussed. It is found that, as a rule, the correlators are to be time-ordered in an unusual way. Instances where the difference with the conventional ordering matters are illustrated by means of a simple scattering model. We apply the results to resolve a discrepancy concerning the third cumulant of charge transport across a quantum point contact.Comment: 19 pages, 1 figure; inconsequential mistake and typos correcte

Semi-Relativistic Hamiltonians of Apparently Nonrelativistic Form

We construct effective Hamiltonians which despite their apparently nonrelativistic form incorporate relativistic effects by involving parameters which depend on the relevant momentum. For some potentials the corresponding energy eigenvalues may be determined analytically. Applied to two-particle bound states, it turns out that in this way a nonrelativistic treatment may indeed be able to simulate relativistic effects. Within the framework of hadron spectroscopy, this lucky circumstance may be an explanation for the sometimes extremely good predictions of nonrelativistic potential models even in relativistic regions.Comment: 20 pages, LaTeX, no figure