Accuracy of Approximate Eigenstates
نویسندگان
چکیده
Besides perturbation theory, which 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 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 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 the variational methods are proposed as new criteria for the accuracy of variational eigenstates. These considerations are applied to that Hamiltonian the eigenvalue problem of which defines the spinless Salpeter equation. This bound-state wave equation may be regarded as the most straightforward relativistic generalization of the usual nonrelativistic Schrödinger formalism, and is frequently used to describe, e.g., spin-averaged mass spectra of bound states of quarks. PACS : 03.65.Ge, 03.65.Pm, 11.10.St, 12.39.Ki ∗ E-mail : [email protected] † E-mail : [email protected]
منابع مشابه
Quality of Variational Trial States
Various measures for the accuracy of approximate eigenstates of semibounded self-adjoint operators H in quantum theory, derived, e.g., by some variational technique, are scrutinized. In particular, the matrix elements of the commutator of the operatorH and (suitably chosen) different operators with respect to degenerate approximate eigenstates of H obtained by the variational methods are propos...
متن کاملDevelopment of an approximation scheme for quasi-exactly solvable double-well potentials
We make use of a recently developed method to, not only obtain the exactly known eigenstates and eigenvalues of a number of quasi-exactly solvable Hamiltonians, but also construct a convergent approximation scheme for locating those levels, not amenable to analytical treatments. The fact that, the above method yields an expansion of the wave functions in terms of corresponding energies, enables...
متن کاملThe quantum discrete self-trapping equation in the Hartree approximation
We show how the Hartree approximation (HA) can be used to study the quantum discrete self-trapping (QDST) equation, which in turn provides a model for the quantum description of several interesting nonlinear effects such as energy localization, soliton interactions, and chaos. The accuracy of the Hartree approximation is evaluated by comparing results with exact quantum mechanical calculations ...
متن کاملIsospin: An Approximate Symmetry on the Quark Level
Isospin is an approximate symmetry which treats the up and down quarks as different eigenstates of the same particle. The mathematical structure for describing the isospin of a system is identical to that of angular momentum. We explore the implications of isospin. Specifically, we use isospin to predict the ratios of cross-sections in pion-nucleon scattering with incredible accuracy. We also d...
متن کاملHigh Accuracy Relative Motion of Spacecraft Using Linearized Time-Varying J2-Perturbed Terms
This paper presents a set of linearized equations was derived for the motion, relative to an elliptical reference orbit, of an object influenced by J2 perturbation terms. Approximate solution for simulations was used to compare these equations and the linearized keplerian equations to the exact equations. The inclusion of the linearized perturbations in the derived equations increased the high ...
متن کامل