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arXiv:hep-th/0608107 · Physica Scripta

Hyperfine splitting in noncommutative spaces

S A Alavi

Published Jun 17, 2008• 11 citations
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Abstract

We study the hyperfine splitting in the framework of the noncommutative quantum mechanics (NCQM) developed in the literature. The results show deviations from the usual quantum mechanics. We show that the energy difference between two excited F = I + 1/2 and the ground F = I - 1/2 states in a noncommutative space (NCS) is bigger than the one in the commutative case, so the radiation wavelength in NCSs must be shorter than the radiation wavelength in commutative spaces. We also find an upper bound for the noncommutativity parameter.

DOI
10.1088/0031-8949/78/01/015005

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We study the hyperfine splitting in the framework of the noncommutative quantum mechanics (NCQM) developed in the literature. The results show deviations from the usual quantum mechanics. We show that the energy difference between two excited F = I + 1/2 and the ground F = I - 1/2 states in a noncommutative space (NCS) is bigger than the one in the commutative case, so the radiation wavelength in NCSs must be shorter than the radiation wavelength in commutative spaces. We also find an upper bound for the noncommutativity parameter.
In this letter, firstly, the Schrddoto\\ddot{o}ddotodinger equation on noncommutative phase space is given by using a generalized Bopp's shift. Then the anomaly term of commutator of arbitrary physical observable operators on noncommutative phase space is obtained. Finally, the basic uncertainty relations for space-space and space-momentum as well as momentum-momentum operators in noncommutative quantum mechanics (NCQM), and uncertainty relation for arbitrary physical observable operators in NCQM are discussed.
This paper provides an examination of how are prediction of standard quantum mechanic (QM) affected by introducing a noncommutative (NC) structure into the configuration space of the considered system (electron in the Coulomb potential in the present case). The parameter controlling the extent of modification is denoted as {\lambda}. The coordinates in the NC space are realized via creation and annihilation operators acting in an auxiliary Fock space, this one being chosen in such a way that the rotational invariance of the system remains intact also in NCQM. Analog of Schr\"odinger equation for hydrogen atom is found and analytically solved, both for bound states and scattering. The exact formulas for NC corrections are given. None of the NC predictions contradicts experimentally verified QM results, since in the correspondence limit {\lambda} -> 0 both QM and NCQM coincide. Highly surprising feature of the NC version is the existence of bound states for repulsive potential at ultra-high energies. However, these disappear from the Hilbert space in the mentioned limit. The whole problem is solved also using a method analogous to that of Pauli. Besides rotational invariance, the dynamical symmetry related to the conservation of NC analog of Laplace-Runge-Lenz vector is being used and the results obtained this way are in the full agreement with those given by "Schr\"odinger-like" approach. The presented NC deformation of QM preserves all those mysterious properties of the Coulomb system that made it a distinguished key-stone of the modern physics.
We introduce new representations to formulate quantum mechanics on noncommutative coordinate space, which explicitly display entanglement properties between degrees of freedom of different coordinate components and hence could be called entangled state representations. Furthermore, we derive unitary transformations between the new representations and the ordinary one used in noncommutative quantum mechanics (NCQM) and obtain eigenfunctions of some basic operators in these representations. To show the potential applications of the entangled state representations, a two-dimensional harmonic oscillator on the noncommutative plane with both coordinate-coordinate and momentum-momentum couplings is exactly solved.