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Coulomb problem in non-commutative quantum mechanics

### Abstract

The aim of this paper is to find out how it would be possible for space non-commutativity (NC) to alter the quantum mechanics (QM) solution of the Coulomb problem. The NC parameter λ is to be regarded as a measure of the non-commutativity – setting λ = 0 which means a return to the standard quantum mechanics. As the very first step a rotationally invariant NC space , an analog of the Coulomb problem configuration space (R 3 with the origin excluded) is introduced. is generated by NC coordinates realized as operators acting in an auxiliary (Fock) space . The properly weighted Hilbert-Schmidt operators in form , a NC analog of the Hilbert space of the wave functions. We will refer to them as “wave functions” also in the NC case. The definition of a NC analog of the hamiltonian as a hermitian operator in is one of the key parts of this paper. The resulting problem is exactly solvable. The full solution is provided, including formulas for the bound states for E < 0 and low-energy scattering for E > 0 (both containing NC corrections analytic in λ) and also formulas for high-energy scattering and unexpected bound states at ultra-high energy (both containing NC corrections singular in λ). All the NC contributions to the known QM solutions either vanish or disappear in the limit λ → 0.

© 2013 AIP Publishing LLC

Received 22 March 2013
Accepted 11 April 2013
Published online 06 May 2013

Article outline:

I. INTRODUCTION
II. THE NON-COMMUTATIVE SPACE
A. The non-commutative configuration space
B. Hilbert space of NC wave functions
C. Orbital momentum in
D. The radial part and normalization in
E. The NC analog of Laplace operator in
F. The potential term in
III. THE COULOMB PROBLEM IN NC QM
A. Bound states
1. Bound states for *E* < 0, η = *i* |η|
2. Bound states for *E* > 2/λ^{2}, η = |η| > 1
B. Scattering *E* ∈ (0, 2/λ^{2}), η ∈ (0, 1)
C. Bound states revisited-poles of the S-matrix
1. Poles of the *S*-matrix for attractive potential
2. Poles of the *S*-matrix for repulsive potential
IV. COMMENTS AND CONCLUSIONS

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2013-05-06

2016-05-02

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