Numerical study of non-relativistic quantum systems and small oscillations induced in a helically twisted geometry

TL;DR

This paper develops a fully numerical framework to study bound states of a non‑relativistic scalar in a three‑dimensional helically twisted (torsional) geometry by solving a radial Sturm–Liouville problem derived from the curved‑space Schrödinger equation with minimal coupling to a magnetic field and Aharonov–Bohm flux. By transforming to a one‑dimensional radial equation with an effective potential that encodes geometric and gauge effects, the authors perform finite‑difference discretization on a truncated domain to obtain spectra and eigenfunctions for four model potentials (no external potential, Cornell, Kratzer, and Morse small‑oscillation). They systematically examine the dependence of low‑lying levels on the torsion parameter $\omega$, azimuthal quantum number $m$, and external potential parameters, highlighting how torsion can produce confinement even in the absence of external interactions and how gauge/torsion couplings break $m\to -m$ symmetry when the magnetic sector is active. The results establish a robust numerical baseline for spectral engineering in torsionful geometries and offer a flexible computational template for exploring quantum bound states in defect‑like curved spaces. The approach has potential applications in mesoscopic physics, condensed matter analogs, and quantum systems in curved backgrounds where geometry and gauge fields jointly shape confinement and spectra.

Abstract

We investigate bound states of a non-relativistic scalar particle in a three-dimensional helically twisted (torsional) geometry, considering both the free case and the presence of external radial interactions. The dynamics is described by the Schrödinger equation on a curved spatial background and, when included, by minimal coupling to a magnetic vector potential incorporating an Aharonov--Bohm flux. After separation of variables, the problem reduces to a one-dimensional radial eigenvalue equation governed by an effective potential that combines torsion-induced Coulomb-like and centrifugal-like structures with magnetic/flux-dependent terms and optional model interactions. Because closed-form analytic solutions are not reliable over the parameter ranges required for systematic scans, we compute spectra and eigenfunctions numerically by formulating the radial equation as a self-adjoint Sturm--Liouville problem and solving it with a finite-difference discretization on a truncated radial domain, with explicit convergence control. We analyze four representative scenarios: (i) no external potential, (ii) Cornell-type confinement, (iii) Kratzer-type interaction, and (iv) the small-oscillation regime around the minimum of a Morse potential. We present systematic trends of the low-lying levels as functions of the torsion parameter, magnetic field, and azimuthal sector, and we show that geometric couplings alone can produce effective confinement even in the absence of an external interaction.

Figures (21)

• Figure 1: Effective potential $V_{\mathrm{eff}}^{(\mathrm{I})}(r)$ [Eq. \ref{['eq:Veff-I']}] for Model I ($V(r)=0$) in the helically twisted geometry in the presence of a uniform magnetic field and an Aharonov--Bohm flux. The azimuthal vector potential is kept throughout, $A_{\varphi}(r)=-(B_0/2)\,r^{2}+\Phi_{B}/(2\pi)$. The curves correspond to $m=-1,0,1$ (as labeled in the plot). At large $r$, the magnetic contribution contained in $A_\varphi$ produces an effective parabolic wall, ensuring confinement in all sectors. At short distances, the dominant $1/r^{2}$ structure (including the $-1/(4r^2)$ term and the $A_\varphi$--dependent pieces) competes with the torsion Coulomb--like term $\propto -2m\omega k/r$, leading to an $m$--dependent shift of the well depth and its equilibrium position. Parameters: $\hbar=\mu=e=k=1$, $\omega=1$, $B_0=0.5$, and $\Phi_B=0.5$ (dimensionless units).
• Figure 2: Numerical (finite-difference) energy spectrum $E$ as a function of the torsion parameter $\omega$ in the helically twisted geometry for Model I ($V(r)=0$). The energies are obtained by solving the radial Sturm--Liouville eigenvalue problem \ref{['eq:SL']} with Dirichlet boundary conditions \ref{['eq:BC']}, using the full geometric/gauge contribution encoded in $V_1(r)$ [Eq. \ref{['eq:V1']}] through the effective potential $V_{\mathrm{eff}}(r)$ [Eq. \ref{['eq:Veff-general']}]. The three panels correspond to fixed azimuthal quantum numbers: (a) $m=-1$, (b) $m=0$, and (c) $m=1$. In each panel we display the lowest three radial levels, $n_r=0,1,2$ (color-coded), computed from the tridiagonal finite-difference discretization described in Sec. \ref{['LANDAU']}. Parameters: $\hbar=\mu=e=k=1$, $B_0=0.5$, and $\Phi_B=0.5$ (dimensionless units).
• Figure 3: Discrete numerical spectrum $E$ as a function of the azimuthal quantum number $m$ for Model I ($V(r)=0$) in the helically twisted geometry. For each integer $m\in[-5,5]$ we display the lowest six radial levels $n_r=0,1,2,3,4,5$ (color-coded markers). The eigenvalues are obtained by solving the radial Sturm--Liouville problem \ref{['eq:SL']} with Dirichlet boundary conditions \ref{['eq:BC']}, using a tridiagonal finite-difference discretization and direct diagonalization. Parameters: $\hbar=\mu=e=k=1$, $\omega=1$, $B_0=0.5$, and $\Phi_B=0.5$. The radial grid is uniform in $r\in[r_{\min},r_{\max}]$.
• Figure 4: Numerical probability densities $\rho(r)=|f(r)|^2$ (reduced radial problem) for Model I ($V(r)=0$) in the helically twisted geometry, obtained from the finite-difference solution of the Sturm--Liouville problem \ref{['eq:SL']} with Dirichlet boundary conditions \ref{['eq:BC']}. We fix the azimuthal sector to $m=-1$ and display three representative torsion values: $\omega=0.5$ (black), $\omega=1.0$ (blue), and $\omega=2.0$ (red). Panels (a)--(c) correspond to the lowest radial modes $n_r=0$, $n_r=1$, and $n_r=2$, respectively. The eigenfunctions are normalized according to Eq. \ref{['eq:norm_f']}. Parameters: $\hbar=\mu=e=k=1$, $B_0=0.5$, and $\Phi_B=0.5$.
• Figure 5: Effective potential $V_{\mathrm{eff}}^{(\mathrm{II})}(r)$ [Eq. \ref{['eq:Veff-II']}] for Model II (Cornell interaction, $V(r)=a/r+b\,r$) in the helically twisted geometry in the presence of a uniform magnetic field and an Aharonov--Bohm flux. The azimuthal vector potential is kept throughout, $A_{\varphi}(r)=-(B_0/2)\,r^{2}+\Phi_{B}/(2\pi)$. For $a>0$, the Coulomb term $a/r$ is repulsive and increases the inner wall at short distances, competing with the torsion--induced Coulomb--like term $\propto -2m\omega k/r$, which is attractive for $m>0$ and repulsive for $m<0$. At large radii, the linear confinement $b\,r$ adds to the magnetic parabolic wall (encoded in $A_\varphi$), leading to a steeper overall rise of $V_{\mathrm{eff}}^{(\mathrm{II})}(r)$. Parameters: $a=1.0$, $b=0.02$, $\hbar=\mu=e=k=1$, $\omega=1$, $B_0=0.5$, and $\Phi_B=0.5$ (dimensionless units).