[Paper Review] Coulomb problem in non-commutative quantum mechanics - Exact solution

Veronika Gáliková, P. Prešnajder|arXiv (Cornell University)|Dec 20, 2011

Noncommutative and Quantum Gravity Theories1 references3 citations

TL;DR

This paper presents an exact solution to the Coulomb problem in non-commutative quantum mechanics on a rotationally invariant non-commutative space $\\[\hat{\mathbf{R}}^{3}_{0}\ y$. By constructing a Hilbert space of wave functions as weighted Hilbert-Schmidt operators in an auxiliary Fock space and defining a non-commutative analog of the hydrogen atom Hamiltonian, the authors derive exact bound state energies $E^{\\\lambda}_{n}$ and eigenstates $\hat{\psi}^{\\\lambda}_{njm}$, showing that non-commutativity modifies the energy spectrum while preserving rotational symmetry through a deformation parameter $\lambda$. The solution generalizes the standard hydrogen atom spectrum via a modified confluent hypergeometric function with non-commutative corrections.

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 $\hatψ$ 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 $E^λ_n$ and the corresponding eigenstates $\hatψ^λ_{njm}$. The Coulomb scattering problem in $\hat{\bf R}^3_0$ is under study.

Motivation & Objective

• To extend the quantum mechanical description of the hydrogen atom to a non-commutative space that preserves rotational symmetry.
• To define a non-commutative configuration space $\hat{\mathbf{R}}^{3}_{0}$ as a deformation of $\mathbf{R}^{3}\setminus\{0\}$, with non-commuting coordinates satisfying $[\hat{x}_i, \hat{x}_j] = i\theta^{ij}$.
• To construct a Hilbert space $\hat{\mathcal{H}}$ of wave functions as weighted Hilbert-Schmidt operators in an auxiliary Fock space $\mathcal{F}$, ensuring unitarity and proper norm structure.
• To define a non-commutative analog of the Coulomb Hamiltonian acting on $\hat{\mathcal{H}}$, preserving rotational invariance through a deformation parameter $\lambda$.
• To exactly solve the non-commutative Schrödinger equation and determine the deformed bound state spectrum $E^{\lambda}_{n}$ and corresponding eigenstates $\hat{\psi}^{\lambda}_{njm}$.

Proposed method

• Introduce a rotationally invariant non-commutative space $\hat{\mathbf{R}}^{3}_{0}$ using non-commuting coordinates $\hat{x}_i$ in a Fock space $\mathcal{F}$, with $[\hat{x}_i, \hat{x}_j] = i\theta^{ij}$ and $\theta^{ij}$ chosen to preserve rotational symmetry.
• Define the Hilbert space $\hat{\mathcal{H}}$ as the space of weighted Hilbert-Schmidt operators in $\mathcal{F}$, with a norm defined via the trace of the operator product with its adjoint.
• Construct the non-commutative Coulomb Hamiltonian as a deformation of the standard hydrogen atom Hamiltonian, using operator realizations of $r^{-1}$ and kinetic energy in terms of creation and annihilation operators.
• Express the radial wave functions in terms of a generating function $\hat{R}$, which is expanded in powers of the number operator $\hat{N}$, and use commutator identities involving $\hat{a}_\alpha$, $\hat{a}^\dagger_\alpha$, and $\hat{N}$ to derive the radial equation.
• Solve the non-commutative radial Schrödinger equation by transforming it into a confluent hypergeometric-type differential equation in the variable $\varrho$, using a parameterization involving $\lambda$ and $\kappa$.
• Apply the Kummer relation to ensure single-valued solutions and identify bound states via the condition that the confluent hypergeometric function reduces to a polynomial, leading to quantized energy levels.

Experimental results

Research questions

• RQ1 How can the Coulomb problem in quantum mechanics be consistently formulated in a non-commutative space that preserves rotational symmetry?
• RQ2 What is the structure of the Hilbert space of wave functions in non-commutative quantum mechanics when the configuration space is $\hat{\mathbf{R}}^{3}_{0}$?
• RQ3 How is the non-commutative analog of the hydrogen atom Hamiltonian constructed, and what are its spectral properties?
• RQ4 What are the exact bound state energies and eigenstates in the non-commutative Coulomb problem, and how do they differ from the standard hydrogen atom spectrum?
• RQ5 How does the non-commutativity parameter $\lambda$ deform the energy levels and wave functions while preserving rotational invariance?

Key findings

• The non-commutative Coulomb problem admits an exact solution in the Hilbert space $\hat{\mathcal{H}}$ of weighted Hilbert-Schmidt operators in the auxiliary Fock space $\mathcal{F}$, ensuring unitarity and proper normalization.
• The bound state energies are given by $E^{\lambda}_{n} = -\frac{m e^4}{2\hbar^2 n^2} \cdot \left(1 + \lambda \kappa_n^2 \eta^2 \right)^{-1}$, where $\kappa_n = \sqrt{-2mE_n}/\hbar$, showing a deformation of the standard hydrogen spectrum due to non-commutativity.
• The eigenstates $\hat{\psi}^{\lambda}_{njm}$ are constructed as operator-valued functions in $\mathcal{F}$, with quantum numbers $n, j, m$, and depend on the non-commutativity parameter $\lambda$ through the radial function $R(\varrho)$.
• The radial wave function is expressed as $R(\varrho) = e^{\frac{1}{2}(D + \lambda\kappa^2)\varrho} F\left(j+1 + \frac{\alpha}{D}, 2j+2; -D\varrho\right)$, where $D = -2\kappa\sqrt{1 + \eta^2}$, and the confluent hypergeometric function ensures square-integrability for bound states.
• The energy spectrum is quantized when the first parameter of the confluent hypergeometric function becomes a non-positive integer, leading to the condition $n = j+1, j+2, \dots$, analogous to the commutative case but with modified dependence on $\lambda$.
• The solution preserves rotational symmetry through the use of a rotationally invariant non-commutative space and a Hamiltonian constructed from rotationally covariant operators, ensuring that the quantum numbers $j$ and $m$ remain good quantum numbers.