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Title: Bound states of hydrogen atom in a theory with minimal length uncertainty relations

Abstract

The following properties of bound states of hydrogen atom in a theory with noncommuting position operators are investigated: their number, multiplicities, accidental degeneracy, localization, and dependence on the values of deformation parameters.

Authors:
 [1]
  1. Center for Stochastic Processes in Science and Engineering, Virginia Tech, Blacksburg, Virginia 24061-0435 (United States) and Physics Department, Virginia Tech, Blacksburg, Virginia 24061-0435 (United States)
Publication Date:
OSTI Identifier:
20929724
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 48; Journal Issue: 5; Other Information: DOI: 10.1063/1.2423221; (c) 2007 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ATOMS; BOUND STATE; HYDROGEN; MULTIPLICITY; POSITION OPERATORS

Citation Formats

Slawny, J. Bound states of hydrogen atom in a theory with minimal length uncertainty relations. United States: N. p., 2007. Web. doi:10.1063/1.2423221.
Slawny, J. Bound states of hydrogen atom in a theory with minimal length uncertainty relations. United States. doi:10.1063/1.2423221.
Slawny, J. Tue . "Bound states of hydrogen atom in a theory with minimal length uncertainty relations". United States. doi:10.1063/1.2423221.
@article{osti_20929724,
title = {Bound states of hydrogen atom in a theory with minimal length uncertainty relations},
author = {Slawny, J.},
abstractNote = {The following properties of bound states of hydrogen atom in a theory with noncommuting position operators are investigated: their number, multiplicities, accidental degeneracy, localization, and dependence on the values of deformation parameters.},
doi = {10.1063/1.2423221},
journal = {Journal of Mathematical Physics},
number = 5,
volume = 48,
place = {United States},
year = {Tue May 15 00:00:00 EDT 2007},
month = {Tue May 15 00:00:00 EDT 2007}
}
  • This paper identifies a new class of shape invariant models. These models are based on extensions of conventional quantum mechanics that satisfy a string-motivated minimal length uncertainty relation. An important feature of our construction is the pairing of operators that are not adjoints of each other. The results in this paper thus show the broader applicability of shape invariance to exactly solvable systems.
  • We review our work on the minimal length uncertainty relation as suggested by perturbative string theory. We discuss simple phenomenological implications of the minimal length uncertainty relation and then argue that the combination of the principles of quantum theory and general relativity allow for a dynamical energy-momentum space. We discuss the implication of this for the problem of vacuum energy and the foundations of nonperturbative string theory.
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  • The energy spectrum of the Coulomb potential with minimal length commutation relations [X{sub i},P{sub j}]=i({Dirac_h}/2{pi}){l_brace}{delta}{sub ij}(1+{beta}P{sup 2})+{beta}{sup '}P{sub i}P{sub j}{r_brace} is determined both numerically and perturbatively for arbitrary values of {beta}{sup '}/{beta} and angular momenta l. The constraint on the minimal length scale from precision hydrogen spectroscopy data is of the order of a few GeV{sup -1}, weaker than previously claimed.
  • We studied energy spectrum for the hydrogen atom with deformed Heisenberg algebra leading to the minimal length. We developed the correct perturbation theory free of divergences. It gives a possibility to calculate analytically in the three-dimensional case the corrections to s levels of the hydrogen atom caused by the minimal length. Comparing our results with the experimental data from precision hydrogen spectroscopy an upper bound for the minimal length is obtained.
  • We investigated the hydrogen atom problem with deformed Heisenberg algebra leading to the existence of a minimal length. Using modified perturbation theory developed in our previous work [Stetsko and Tkachuk, Phys. Rev. A 74, 012101 (2006)] we calculated the corrections to the arbitrary s levels for the hydrogen atom. We obtained a simple relation for the estimation of the minimal length. We also compared the estimation of minimal length obtained here with the results obtained in previous investigations.