Cornell Interaction in the Two-body Pauli-Schrödinger-type Equation Framework: The Symplectic Quantum Mechanics Formalism

We investigate the quantum behavior of a quark-antiquark bound system under the influence of a magnetic field within the symplectic formulation of quantum mechanics. Employing a perturbative approach, we obtain the ground and first excited states of the system described by the Cornell potential, which incorporates both confining and non-confining interactions. After performing a Bohlin mapping in phase space, we solve the time-independent symplectic Pauli-Schrödinger-type equation and determine the corresponding Wigner function. Special attention is given to the observation of the confinement of the quark-antiquark, that is revealed in the phase space structure. Due to the presence of spin in the Hamiltonian, the results reveal that the magnetic field enhances the non-classicality of the Wigner function, signaling stronger quantum interference and a departure from classical behavior. The experimental mass spectra is used to estimate the intensity of the external field, leading to a value that is in order of the transient magnetic field measured in non-central heavy-ion collisions at RHIC and LHC.

💡 Research Summary

The paper presents a detailed study of a quark‑antiquark bound system (quarkonium) subjected to an external magnetic field, using the symplectic (phase‑space) formulation of quantum mechanics. Starting from the well‑known Cornell potential V(r)=−α/r+βr, the authors construct a non‑relativistic Pauli‑Schrödinger‑type Hamiltonian that includes minimal coupling to a uniform magnetic field B directed along the z‑axis and a Zeeman term −eħσ_z B accounting for the quark spin. By assuming that the spin interaction dominates over the orbital magnetic coupling, the Hamiltonian reduces to a two‑dimensional harmonic‑oscillator form plus the Cornell potential.

To treat the Coulomb‑like −α/r term, a Bohlin (Levi‑Civita) canonical transformation is applied, mapping the Cartesian coordinates (x,y) onto quadratic variables (q₁,q₂) while simultaneously redefining the momenta (p₁,p₂). This transformation preserves the classical dynamics and enables a straightforward quantization in phase space. After the mapping, the Hamiltonian becomes a sum of kinetic terms (p₁²+p₂²)/2m, a quadratic magnetic confinement term proportional to B²(q₁²+q₂²), and the linear and quadratic pieces of the Cornell potential expressed in the new variables.

In the symplectic framework, the wavefunction ψ(q,p) lives in a Hilbert space of phase‑space functions and evolves according to the symplectic Schrödinger equation H⋆ψ=Eψ, where the Moyal star product encodes quantum non‑commutativity. The Hamiltonian is split into an unperturbed part H₀ (the isotropic 2‑D harmonic oscillator) and a perturbation H₁ containing the magnetic‑field‑dependent cubic term and the β‑dependent quadratic term. Creation and annihilation operators a, a†, b, b† are introduced via the star product, obeying the usual commutation relations, which allows the exact solution of H₀. The zero‑order eigenfunctions are products of one‑dimensional phase‑space harmonic‑oscillator states ϕ_n(q,p) and the corresponding eigenvalues are α⁽⁰⁾_{n₁,n₂}=¼ω(n₁+n₂+1). The oscillator frequency ω is linked to the Cornell coupling α, the magnetic field B, the reduced mass m, and the energy E through ω²=−(±B/m²+8E/m).

First‑order perturbation theory is then employed to evaluate the corrections due to H₁. The magnetic‑field term contributes a shift proportional to B²/(16mω³)⟨(q₁²+q₂²)³⟩, while the confinement term contributes −β/(2ω²)⟨(q₁²+q₂²)²⟩. The spin projection σ_z=±1 introduces a sign change in the linear Zeeman contribution, leading to a B‑dependent splitting of the energy levels. The corrected energy spectrum thus contains both linear and quadratic dependencies on B, as well as the usual Cornell parameters.

The phase‑space wavefunction ψ is used to construct the Wigner function f_W(q,p)=ψ⋆ψ†. Because ψ consists of Gaussian factors multiplied by Laguerre polynomials, f_W exhibits the characteristic oscillatory structures of quantum interference. The presence of the magnetic field enhances the negative regions of the Wigner function, increasing its total “non‑classical volume.” This effect is interpreted as the magnetic field, together with the spin coupling, amplifying quantum interference and pushing the system further away from a classical description.

Finally, the authors compare their theoretical spectrum with experimental masses of heavy quarkonia (e.g., J/ψ, Υ families). By fixing α and β to reproduce the observed level spacing and using the first‑order corrected energies, they invert the formulas to estimate the magnitude of the external magnetic field. The resulting field strength, of order 10¹⁴–10¹⁵ Tesla, matches the transient magnetic fields estimated for non‑central heavy‑ion collisions at RHIC and the LHC.

In summary, the work demonstrates how the symplectic quantum‑mechanical approach, combined with a canonical Bohlin mapping, provides a transparent and analytically tractable framework for studying spin‑dependent magnetic effects in quarkonium. It quantifies the enhancement of non‑classical features in the Wigner function, links the theoretical model to experimental spectroscopy, and offers a novel bridge between high‑energy nuclear physics and phase‑space quantum information concepts.