Hydrogen atom in rotationally invariant noncommutative space

Abstract

We consider the noncommutative algebra which is rotationally invariant. The hydrogen atom is studied in a rotationally invariant noncommutative space. We find the corrections to the energy levels of the hydrogen atom up to the second order in the parameter of noncommutativity. The upper bound of the parameter of noncommutativity is estimated on the basis of the experimental results for 1s-2s transition frequency

Exact Noncommutative Two-Dimensional Hydrogen Atom

“…As a result, we obtain the solution of the Schrodinger equation for this system and calculate the energy levels. Using the spectrum obtained and experimental data, we estimated the noncommutativity parameter Θ, which has an order of magnitude of 10 −34 m 2 , and the noncommutative effects will be relevant to a length smaller than 10 −17 m. This result has the same order of magnitude obtained in [29], in which the authors studied the hydrogen atom in rotationally invariant noncommutative space. In this way, they founded the corrections to the energy levels of the hydrogen atom up to the second order in the parameter of noncommutativity.…”

Harmonic Oscillator Chain in Noncommutative Phase Space with Rotational Symmetry

“…Noncommutative algebra (1)-(3) with θ ij , η ij , γ ij being constants is not rotationally invariant [6,7]. Different generalizations of commutation relations (1)-(3) were considered to solve the problem of rotational symmetry breaking in noncommutative space [8,9,10,11]. Many papers are devoted to studies of position-dependent noncommutativity [12,13,14,15,16,17,18], spin noncommutativity [19,20].…”

Rotationally invariant noncommutative phase space of canonical type with recovered weak equivalence principle

“…In the canonical version of noncommutative space θ ij , η ij are elements of constant matrices. Different problems were examined in noncommutative space, among them free particles [1][2][3][4], classical systems with various potentials [5][6][7][8][9], the Landau problem [10][11][12][13][14][15], many-particle systems [1,[16][17][18][19][20], gravitational quantum wells [21,22], and many others.…”

“…In noncommutative space of canonical type there is a problem of rotational symmetry breaking. Therefore, different types of noncommutative algebras were considered to preserve the rotational symmetry (see, for instance, [9,[23][24][25]). Among them rotationally invariant noncommutative algebras with position-dependent noncommutativity (see, for instance [26][27][28][29][30][31][32], and reference therein), with involving spin degrees of freedom (see, for instance [33][34][35][36], and reference therein) have been widely studied.…”

“…Note that the last term in (49) appears because of noncommutativity of operators x 2 and [θ × p] 2 under the square root in X. The details of the calculation of the expansion of 1/X over the parameters of noncommutativity can be found in [9]. So, one can write the expression for ΔH,…”