Quantum Mechanical Problems with q-Deformations and over the p-Adic Number Fields

H. Ahmedov, I. H. Duru

Enrico G. Beltrametti, Gianni Cassinelli

Wilson A. Zúñiga-Galindo

We present a new non-Archimedean realization of the Fock representation of the q-oscillator algebras where the creation and annihilation operators act on complex-valued functions, which are defined on a non-Archimedean local field of arbitrary characteristic, for instance, the field of p-adic numbers. This new realization implies that many quantum models constructed using q-oscillator algebras are non-Archimedean models, in particular, p-adic quantum models. In this framework, we select a q-deformation of the Heisenberg uncertainty relation and construct the corresponding q-deformed Schrödinger equations. In this way we construct a p-adic quantum mechanics which is a p-deformed quantum mechanics. We also solve the time-independent Schrödinger equations for the free particle, and a particle in a non-Archimedean box. In the last case, we show the existence of a discrete sequence of energy levels. We determine the eigenvalues of Schrödinger operator for a general radial potential. By choosing the potential in a suitable form we recover the energy levels of the q-hydrogen atom.

C. Alacoque, Philippe Ruelle, E. Thiran +2 more

Andrei Yu. Khrennikov

An extension of the formalism of quantum mechanics to the case where the canonical variables and functions are valued in a field of p-adic numbers is considered. A new p-adic integral calculus is used for the realization of the Gauss representation in p-adic quantum mechanics.