Coulomb problem in non-commutative quantum mechanics - Exact solution

Abstract

We investigate consequences of space non-commutativity in quantum mechanics of the hydrogen atom. We introduce rotationally invariant noncommutative space $\hat{\bf R}^3_0$ - an analog of the hydrogen atom ($H$-atom) configuration space ${\bf R}^3_0\,=\, {\bf R}^3\setminus \{0\}$. The space $\hat{\bf R}^3_0$ is generated by noncommutative coordinates realized as operators in an auxiliary (Fock) space ${\cal F}$. We introduce the Hilbert space $\hat{\cal{H}}$ of wave functions $\hat{\psi}$ formed by properly weighted Hilbert-Schmidt operators in ${\cal F}$. Finally, we define an analog of the $H$-atom Hamiltonian in $\hat{\bf R}^3_0$ and explicitly determine the bound state energies $E^\lambda_n$ and the corresponding eigenstates $\hat{\psi}^\lambda_{njm}$. The Coulomb scattering problem in $\hat{\bf R}^3_0$ is under study.