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 ${\hat{\mathbf{R}}}_0^3$ - an analog of the hydrogen atom ($H$-atom) configuration space ${\mathbf{R}}_0^3={\mathbf{R}}^3\setminus \{0\}$. The space ${\hat{\mathbf{R}}}_0^3$ is generated by noncommutative coordinates realized as operators in an auxiliary (Fock) space $\mathcal{F}$. We introduce the Hilbert space $\hat{\mathcal{H}}$ of wave functions $\hat{\psi }$ formed by properly weighted Hilbert-Schmidt operators in $\mathcal{F}$. Finally, we define an analog of the $H$-atom Hamiltonian in ${\hat{\mathbf{R}}}_0^3$ and explicitly determine the bound state energies $E_n^{\lambda }$ and the corresponding eigenstates ${\hat{\psi }}_{njm}^{\lambda }$. The Coulomb scattering problem in ${\hat{\mathbf{R}}}_0^3$ is under study.

18 pages

Cited by in corpus (1)

• Spectral triplets, statistical mechanics and emergent geometry in non-commutative quantum mechanics