[Paper Review] Algebraic Solution of the Supersymmetric Hydrogen Atom

Andreas Wipf, A. Kirchberg+1 more|ArXiv.org|Nov 23, 2005

Quantum Mechanics and Non-Hermitian Physics2 references3 citations

TL;DR

This paper presents an algebraic solution to the ${\cal N}=2$ supersymmetric hydrogen atom in $d$ dimensions by constructing a supersymmetrized Laplace-Runge-Lenz vector, which extends the $SO(d)$ rotational symmetry to a hidden $SO(d+1)$ dynamical symmetry. The bound state spectrum, degeneracies, and wave functions are derived algebraically using group-theoretic methods, with energies determined by the second-order Casimir operator of $SO(d+1)$ and dependence on the fine structure constant and fermion number.

ABSTRACT

The N=2 supersymmetric extension of the Schrödinger-Hamiltonian with 1/r-potential in d dimension is constructed. The system admits a supersymmetrized Laplace-Runge-Lenz vector which extends the rotational SO(d) symmetry to a hidden SO(d+1) symmetry. It is used to determine the discrete eigenvalues with their degeneracies and the corresponding bound state wave functions.

Motivation & Objective

• To extend the Pauli-Fock-Bargmann algebraic approach to the supersymmetric hydrogen atom with ${\cal N}=2$ supersymmetry.
• To construct a supersymmetrized version of the Laplace-Runge-Lenz vector in $d$ dimensions that closes the $SO(d+1)$ dynamical symmetry algebra.
• To determine the discrete energy spectrum, degeneracies, and bound state wave functions using group-theoretic methods in the context of supersymmetric quantum mechanics.
• To generalize the non-relativistic Coulomb problem to $d$ dimensions while preserving supersymmetry and hidden symmetry.
• To identify the irreducible representations of $SO(d+1)$ that describe the bound states and to clarify the role of fermion number and Casimir invariants.

Proposed method

• Construct the ${\cal N}=2$ supersymmetric Hamiltonian in $d$ dimensions by extending the Schrödinger Hamiltonian with a $1/r$ potential using supercharges acting between bosonic and fermionic subspaces.
• Define a supersymmetrized Laplace-Runge-Lenz vector operator $\boldmathe{C}$ that commutes with the Hamiltonian and transforms as a vector under $SO(d)$.
• Introduce the Hermitian operator $\boldmathe{K} = \frac{1}{2} \frac{\boldmathe{C}}{\sqrt{-H}}$ on the negative energy subspace to close the $SO(d+1)$ algebra with angular momentum generators $L_{ab}$.
• Combine $L_{ab}$ and $K_a$ into $SO(d+1)$ generators $L_{AB}$ via a block matrix construction, ensuring closure of the $so(d+1)$ Lie algebra.
• Express the Hamiltonian as $H = -\frac{\eta^2}{(d-1)^2 + 4\mathcal{C}_{(2)}}$, where $\mathcal{C}_{(2)}$ is the second-order Casimir operator of $SO(d+1)$.
• Identify the allowed irreducible representations of $SO(d+1)$ as completely symmetric representations, with constraints from $n-1$ Casimir invariants in $d=2n$ or $d=2n-1$ dimensions.

Experimental results

Research questions

• RQ1 How can the $SO(d+1)$ dynamical symmetry of the non-supersymmetric hydrogen atom be extended to the ${\cal N}=2$ supersymmetric case in $d$ dimensions?
• RQ2 What is the algebraic structure of the supersymmetrized Laplace-Runge-Lenz vector, and how does it close the $SO(d+1)$ symmetry algebra?
• RQ3 How do the bound state energies and degeneracies of the supersymmetric hydrogen atom depend on the fine structure constant, space dimension $d$, fermion number, and Casimir invariants?
• RQ4 Which irreducible representations of $SO(d+1)$ are realized in the supersymmetric hydrogen atom, and how are they selected by the symmetry constraints?
• RQ5 Can the algebraic method used for the non-supersymmetric case be generalized to supersymmetric systems with extended supersymmetry and matrix Hamiltonians?

Key findings

• The supersymmetrized Laplace-Runge-Lenz vector generates a hidden $SO(d+1)$ symmetry algebra when combined with angular momentum generators, extending the $SO(d)$ rotational symmetry.
• The Hamiltonian is algebraically expressed as $H = -\frac{\eta^2}{(d-1)^2 + 4\mathcal{C}_{(2)}}$, where $\mathcal{C}_{(2)}$ is the second-order Casimir operator of $SO(d+1)$, enabling algebraic determination of the spectrum.
• Bound states transform under completely symmetric irreducible representations of $SO(d+1)$, with degeneracies determined by group-theoretic counting.
• The energy spectrum depends on the fine structure constant $\eta$, space dimension $d$, fermion number $\textbf{N}$, and the Casimir $\mathcal{C}_{(2)}$, reflecting a new 'accidental' degeneracy across particle-number sectors in higher dimensions.
• The supercharge and super-Hamiltonian are constructed as matrix operators, with the supercharge equivalent to a dimensionally reduced Dirac operator in $2d$ dimensions.
• The system realizes a wide class of supersymmetric models, including the supersymmetric oscillator and lattice Wess-Zumino models, with the hydrogen atom as a specific subsector.