Coulomb problem in non-commutative quantum mechanics

Abstract

We investigate consequences of space non-commutativity in quantum mechanics of the hydrogen atom. We introduce rotationally invariant noncommutative spaceR 3 0 -an analog of the hydrogen atom (H-atom) configuration space R 3 0 = R 3 \ {0}. The spaceR 3 0 is generated by noncommutative coordinates realized as operators in an auxiliary (Fock) space F. We introduce the Hilbert spaceĤ of wave functionsψ formed by properly weighted Hilbert-Schmidt operators in F. Finally, we define an analog of the H-atom Hamiltonian inR 3 0 and explicitly determine the bound state energies E λ n and the corresponding eigenstatesψ λ njm . The Coulomb scattering problem inR 3 0 is under study.

Magnetic monopoles and symmetries in noncommutative space

“…where the +1 was added to ensure that r 2 − x 2 i = O(λ 2 ) (so there is no term linear in λ). This construction of 3D rotational invariant space R 3 λ was developed in [4] and explored by the present authors and their collaborator in [5,6,7,8,9,10,11].…”

Noncommutative field theories on $ \mathbb{R}_{\lambda}^3 $: towards UV/IR mixing freedom

“…This is a natural one for R 3 λ : it is constructed in terms of derivations of the algebra, which are all inner, supplemented by a well defined multiplicative operator. A different proposal is suggested in [85,86] which is not based on derivations of the algebra. This issue is presently under investigation, together with the analysis of the commutative limit.…”

“…This is an interesting point; our Laplacian is a natural one for R 3 λ : it is constructed in terms of derivations of the algebra supplemented by multiplicative operators. We signal the reference [85,86] where a different Laplacian is proposed for R 3 λ in the context of noncommutative quantum mechanics, to study the hydrogen atom. It would be interesting to apply such proposal to QFT.…”

Commutative–non-commutative dualities

“…Now the momenta are still commuting, but the coordinates are noncommuting, as one would have expected. The Hamiltonian is still given by (9) and the same factorization as in (10) applies. Now, however, the creation and annihilation operators are…”

“…for the space-time commutation relations is normally adopted, where θ µν are constants. Generalizations to fuzzy-and Snyder-space have been explored in [9][10][11][12] and [13] respectively. The latter offers the possibility of avoiding the breaking of rotational and Lorentz symmetries.…”