Cylindrical Potential Well Basics

• Cylindrical potential wells are regions with cylindrical symmetry where quantum confinement produces discrete energy states, often solved using Bessel functions.
• The analysis involves separation of variables in cylindrical coordinates, enabling studies of resonant cavity effects, Casimir–Polder interactions, and plasma sheath formation.
• Applications extend to nanowires, quantum dots, and magnetic systems, demonstrating their broad relevance in quantum mechanics, nanotechnology, and condensed matter physics.

A cylindrical potential well is a spatial confinement region characterized by cylindrical symmetry, where the potential takes a specified value (often zero) inside a defined cylindrical volume and another value (often infinite or a functional barrier) outside. This geometry features prominently in quantum mechanics, condensed matter physics, electromagnetism, nanotechnology, and plasma physics, serving as a prototypical system for the paper of quantum states, electromagnetic field quantization, Casimir and Casimir–Polder effects, domain wall dynamics, and plasma sheath formation. The analysis of cylindrical wells often relies on separable differential equations in cylindrical coordinates, leading to solutions in terms of Bessel functions, Fourier expansions, and, in generalizations, confluent hypergeometric (Kummer) functions or Dunkl Bessel functions. Cylindrical potential wells are central to the modeling of nanowires, quantum dots, atom–cavity interactions, electron gases, and physical settings with rotational or axial symmetry.

1. Mathematical Definition and Quantum Formalism

The classic formulation considers the time-independent Schrödinger equation in cylindrical coordinates $(\rho, \varphi, z)$ for a particle of mass $m$:

$$\left[ -\frac{\hbar^2}{2m} \left( \frac{1}{\rho} \frac{\partial}{\partial \rho}\left( \rho \frac{\partial}{\partial \rho} \right) + \frac{1}{\rho^2} \frac{\partial^2}{\partial \varphi^2} + \frac{\partial^2}{\partial z^2} \right) + V(\rho, \varphi, z) \right] \Psi = E \Psi$$

where $V(\rho, \varphi, z) = 0$ inside the cylinder $(\rho < a, 0 < z < H)$ and $V \to \infty$ elsewhere. The boundary conditions enforce $\Psi(\rho = a) = 0$ and $\Psi(z=0) = \Psi(z=H) = 0$. Separation of variables yields solutions:

• Angular: $$X(\varphi) = \frac{1}{\sqrt{2\pi}}\exp(i p \varphi)$$, $p=0,\pm1,\pm2,\ldots$
• Axial: $\psi_z(z) = \sqrt{2/H}\sin(n_z\pi z / H)$, $n_z=1,2,\ldots$
• Radial: $$R(\rho) = \frac{\sqrt{2}}{a J_{n+1}(j_{n,n_r})} J_n\left(\frac{j_{n,n_r}\rho}{a}\right)$$, with $j_{n,n_r}$ the $n_r$-th zero of $J_n$.

Corresponding energy eigenvalues are:

$$E_{n_r, n_\varphi, n_z} = \frac{\hbar^2}{2m}\left( \frac{j_{n,n_r}^2}{a^2} + \frac{n_z^2\pi^2}{H^2} \right)$$

This framework can be further generalized to include singular potentials (e.g., $U(r) = \alpha/r$), resulting in radial solutions involving Kummer functions (Villegas, 2012), or the inclusion of reflection operators via Dunkl derivatives (Mota et al., 6 Aug 2025).

2. Casimir–Polder Potentials and Resonant Cavity Enhancement

Cylindrical metallic cavities are powerful tools for manipulating long-range Casimir–Polder (CP) potentials arising from quantum electromagnetic field fluctuations. The CP interaction for a particle in state $|n\rangle$ in a thermal environment is expressed as (Ellingsen et al., 2010):

$$U(\mathbf{r}) = \sum_n p_n [ U_n^{nr}(\mathbf{r}) + U_n^{res}(\mathbf{r}) ]$$

where $U_n^{nr}$ (non-resonant) involves a Matsubara frequency sum with the Green tensor, and $U_n^{res}$ (resonant) contains thermal photon number contributions and transition rates.

Sharp resonant enhancement occurs when the cavity radius matches $R_{mj}^{(\prime)} = (c/\omega) j_{mj}^{(\prime)}$ ($j_{mj}^{(\prime)}$: zeros of $J_m$ or $J'_m$), resulting in near-divergent Green tensor and drastic amplification of both CP potential and atomic transition rates. Observed with low-lying Rydberg states, values exceeding 30 kHz are attainable, exceeding detection limits of current atom-optic experiments. Enhancement depends sensitively on material dissipation, as encoded in the Drude permittivity:

$$\varepsilon(\omega) = 1 - \frac{\omega_p^2}{\omega(\omega + i\gamma)}$$

with the resonance peak shifting and scaling as $$U \propto |\varepsilon(\omega)|^{1/4} \sqrt{\cos(\arg \varepsilon(\omega))}$$.

Thermal non-equilibrium further modifies CP potentials by including real photon absorption and emission, leading to spatially oscillatory and non-uniform atom-surface forces, with implications for atom trapping, guiding and precision tests of the thermal Casimir anomaly (Ellingsen et al., 2010, Bezerra et al., 2011).

3. Generalizations: Singular Potentials and Symbolic Solution Methods

For cylindrical wells with non-zero or singular potentials (e.g., $U(r) = \alpha/r$), the radial Schrödinger equation is not solvable in terms of elementary functions. Instead, the solution is constructed using special functions (Villegas, 2012). For the case above, the solution is:

$$y(r) \propto e^{-\lambda r} r^\nu \text{KummerM}(a, b, \lambda r)$$

Energy levels are determined by the roots of the Kummer function, subject to vanishing boundary conditions at $r=R$ and regularity at $r=0$. Computer algebra systems are employed for symbolic integration and numerical root finding, facilitating studies of complex potential shapes and boundary conditions beyond analytic capabilities.

In systems with reflection symmetry, the Dunkl derivative formalism introduces parity constraints, leading to wavefunctions classified by reflection operator eigenvalues and with exact solutions again in terms of Bessel functions whose order is determined by both quantum numbers and Dunkl parameters (Mota et al., 6 Aug 2025).

4. Cylindrical Wells in Condensed Matter and Magnetic Nanowires

Cylindrical potential wells underlie the physics of quantum wires, nanotubes, and electron gases on curved surfaces. They govern electronic states, momentum distributions, and transport properties (Baltenkov et al., 2016, Filgueiras et al., 2015). In quantum nanowires and nanotubes, the aspect ratio $H/R_0$ determines the dimensionality: $H\gg R_0$ yields quasi-1D conduction with momentum mainly in the transverse plane, $R_0\gg H$ favors quasi-2D behavior. Measured via photoionization or momentum-resolved electron spectroscopy, the predicted anisotropy in momentum distribution serves as a diagnostic signature of quantum confinement (Baltenkov et al., 2016).

In cylindrical nanowires, geometric constrictions create harmonic pinning wells for magnetic domain walls, with spring constants determined by notch geometry and detailed depinning fields shaped by surface roughness (Dolocan, 2015). Surface-induced asymmetry results in shifted minima and asymmetric depinning fields. Interwire stray fields can induce secondary potential wells and trapped bound domain wall states, with thermal activation effects included via Arrhenius–Neel kinetics.

5. Collective Phenomena and Particle Arrays in Cylindrical Wells

Hard sphere arrays confined by cylindrical harmonic wells (e.g., via rotation in liquid-filled tubes) exhibit rich equilibrium structures, including zigzag transitions and bifurcation sequences analogous to those seen in trapped ion chains (Winkelmann et al., 2019). With centripetal force modeled as $F_n = r_n$ and compressive contact forces, force balance and geometric connectivity yield iterative relations for equilibrium tilt angles and energies. Bifurcation diagrams and linearised approximations reveal multiple stable and unstable buckled states, including symmetrical and asymmetrical displacement profiles, with connections to soliton-like excitations and Peierls–Nabarro potentials.

Such systems are valuable models for studying phase transitions, collective excitations, and ordered assembly in confined geometries, bridging the gap between particulate classical mechanics and quantum many-body physics.

6. Plasma Sheath Potentials and Curvature Effects

In plasma physics, cylindrical (and spherical) potential wells arise in the calculation of sheath potentials near curved electrodes, with direct implications for thermionic cooling and space-charge effects on hypersonic vehicle surfaces (Ghosh et al., 11 Feb 2025). The governing Poisson equation in normalized radial coordinate $\xi$ is:

$$\frac{d^2\Phi}{d\xi^2} + \frac{1}{\xi} \frac{d\Phi}{d\xi} = \text{source terms}$$

with the source terms capturing contributions from wall-emitted electrons, plasma electrons (Boltzmann distributed), and ions (Mach number dependent). Integration and shooting methods yield sheath formation conditions (minimum Mach number, potential derivative at the surface, net current per density), demonstrating the strong influence of curvature and geometry on field structure and emission efficiency.

Generalization of Takamura's method from Cartesian to cylindrical coordinates enables higher-fidelity modeling of curved plasma sheath dynamics relevant to advanced material and device engineering.

7. Spheroidal Generalization and Limiting Cases

Spheroidal quantum wells provide the unifying context for cylindrical wells (Usov, 2023). By solving the Helmholtz equation in spheroidal coordinates and employing power-series expansions for angular and radial spheroidal functions, one recovers the cylindrical well as the limit $a/b \gg 1$ (elongated spheroid aspect ratio). The radial equation reduces to the Bessel equation with quantization by zeros $J_m(q R) = 0$. This approach demonstrates the mathematical robustness of the eigenvalue structure and provides a framework for modeling intermediate geometries between spherical quantum dots and cylindrical wires.

The connection between boundary conditions, separable solutions, and quantization in these systems underlines the broader applicability of cylindrical wells to nanoscience, materials design, and quantum engineering.

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Cylindrical potential wells thus serve as fundamental and versatile archetypes for spatial confinement, quantum state quantization, field phenomena, and collective dynamics across a wide range of physical regimes, with exact analytic solutions and experimentally observable consequences deeply shaped by geometry, material properties, and boundary conditions.