Schrödinger Quantum Modes on the Taub–nut Background

Abstract

The Schrödinger equation is investigated in the Euclidean Taub-NUT geometry. The bound states are degenerate and an extra degeneracy is due to the conserved Runge-Lenz vector. The existence of the extra conserved quantities, quadratic in four-velocities implies the possibility of separating variables in two different coordinate systems. The eigenvalues and the eigenvectors are given in both cases in explicit, closed form.Pacs 04.62.+v *

A Note on Klein–gordon Equation in a Generalized Kaluza–klein Monopole Background

“…Although the existence of the global solution (26), (30) (with E fixed by (31)) still requires an existence proof, this agrees with the closed form solution known for C = D = 1 [11]. The expansion (30) has only one free coefficient, which is fixed by the normalization condition (14).…”

“…The radial equation (13) is similar to those of the non-relativistic quantum mechanics apart the term ρ ′′ /ρ . Unfortunately, there is only one solution of the equation (13) known in closed form, corresponding to a GPS monopole background [11] 1 . However, one can analyze the properties of the general solutions by using a combination of analytical and numerical methods, which is enough for most purposes.…”

“…The arguments here are similar to the case C = D = 1 [11]. Following the standard approach [17], we suppose that P (r) admits a power series expansion with p k real coefficients.…”

Hierarchy of Dirac, Pauli, and Klein–Gordon conserved operators in Taub-NUT background

“…Moreover, the Taub-NUT geometry possesses four Killing-Yano tensors, f (i) (iϭ1,2,3) and f Y , of valence 2, related to the hidden symmetries of the Taub-NUT geometry reflected by the existence of the nontrivial Stäckel-Killing tensors k (i) . 2,4,7,14 In this Kaluza-Klein geometry there is a pentad gauge fixing 15 where the massless Dirac field, , satisfies a simple gauge-covariant Dirac equation, D " ϭ0, where D " ϭi␥ 0 ‫ץ‬ t ϪD " s . 16,5,7,9 In the standard representation of the Dirac matrices ͓with diagonal ␥ 0 ͑Ref.…”

Geodesic Motions in Euclidean Taub-NUT Spinning Spaces

“…Moreover, the motion of well-separated monopole-monopole interactions is described approximately by the geodesics of this space [30][31][32][33]. The Euclidean Taub-NUT background contains also interesting specific features of the quantum theory in the case of the scalar fields [34] as well as for Dirac fields of spin-1 2 fermions [35][36][37]. There exist large algebras of conserved observables in both cases [38].…”