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New exact solutions of the 3D Schrödinger equation

This paper constructs a class of exact solutions to the 3D Schrödinger equation for a two-parameter "quadratic funnel" potential within the $\Psi$ -model, deriving explicit energy spectra and eigenfunctions that exhibit the Aharonov-Bohm effect and Dirac string magnetic fields, while analyzing the resulting vortex probability currents and system properties using the Wigner-Vlasov formalism.

Original author: E. E. Perepelkin, B. I. Sadovnikov, N. G. Inozemtseva, A. S. Medvedev

Published 2026-11-04

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CC BY 4.0

Original author: E. E. Perepelkin, B. I. Sadovnikov, N. G. Inozemtseva, A. S. Medvedev

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the author. For technical accuracy, refer to the original paper. Read full disclaimer

Quantum mechanics is the set of rules that governs the behavior of the smallest things in the universe, from atoms to the particles inside them. For nearly a century, physicists have relied on a fundamental equation, known as the Schrödinger equation, to predict how these tiny objects move and interact. While this equation is the bedrock of modern technology, solving it exactly is incredibly difficult. Most of the time, scientists must rely on approximations or computer simulations to get an answer. However, finding an exact solution is like finding a perfect map of a complex terrain; it allows researchers to see the underlying structure of the system without any guesswork. This is particularly important when trying to understand how quantum systems behave in the presence of magnetic fields or when they form complex patterns of probability, which are the chances of finding a particle in a specific place.

In a recent study, researchers have constructed a new class of exact solutions for this equation in three-dimensional space. They focused on a specific type of invisible landscape, or potential, that acts on a quantum particle. They call this landscape a "quadratic funnel." Imagine a funnel that is not just a simple cone, but one with a shape that changes based on two specific parameters, creating a unique environment for the particle to move through. The researchers did not just find one solution; they discovered a whole family of solutions that describe how a particle behaves in this funnel. These solutions reveal that the particle does not just sit still or move in a simple circle. Instead, it flows in a complex, swirling pattern that moves from one pole of a sphere to the other, creating a vortex-like current.

A key feature of this work is how the researchers handled the concept of probability. In the quantum world, we cannot say exactly where a particle is, only where it is likely to be. This likelihood is described by a wave function. The researchers found that for their specific funnel-shaped potential, the probability of finding the particle is always positive, meaning it never dips into negative values, which is a rare and useful property for certain types of calculations. They also showed that this system naturally creates a magnetic effect, similar to the famous Aharonov-Bohm effect, where a particle is influenced by a magnetic field even if it never directly touches the source of that field. In their model, this magnetic influence comes from a singular line of force, which they describe as a "Dirac string," running through the center of the system.

The team explored what happens when they combine different versions of these solutions. By adding together waves that swirl in opposite directions, they could create new states where the swirling motion cancels out, leaving the particle in a state with no net rotation. Conversely, they could combine waves to create a system with a strong, intrinsic magnetic moment, essentially turning the quantum system into a tiny magnet. They also investigated what happens when the flow of probability moves between the poles of the sphere. By superimposing waves moving in opposite directions along this axis, they created a standing wave pattern, where the probability density forms a stable, stationary shape with specific nodal points, or places where the probability is zero.

One of the most striking aspects of their findings is the connection to ancient mathematics. The researchers discovered that the specific shapes and energy levels of these quantum states are governed by Pythagorean triples. These are sets of three whole numbers that fit the famous rule of the right-angled triangle, such as 3, 4, and 5. In their model, the parameters that define the shape of the funnel and the energy of the particle must relate to each other in this specific mathematical way. This means that the quantum system only exists in stable forms when these numbers align perfectly, much like how a musical instrument only produces a clear note when its strings are tuned to specific ratios.

The study also clarifies the forces at play within this system. The researchers showed that the particle is subject to two opposing forces. One is an external force pushing the particle inward, trying to compress it. The other is a "quantum pressure" force that pushes outward, resisting the compression. In the stable states they found, these two forces balance each other perfectly, allowing the particle to exist in a steady state. When the particle is moving in a swirl, this balance is maintained by a specific type of acceleration that increases as the particle gets closer to the center, a behavior that is quite different from how objects move in our everyday world.

By using a mathematical framework that connects classical fluid dynamics with quantum mechanics, the authors were able to visualize these abstract concepts as flows of probability. They described the movement of the particle as a fluid that can be potential, meaning it flows smoothly, or vortex-like, meaning it spins. They showed how these flows can be manipulated to create systems with or without magnetic properties. The work provides a clear, exact picture of how a quantum system behaves in a complex, funnel-shaped environment, offering a new tool for understanding the fundamental nature of matter and the forces that shape it. This precise understanding could help in the future design of quantum technologies, where controlling the exact behavior of particles is essential.

Technical Summary: New Exact Solutions of the 3D Schrödinger Equation

Problem Statement
The paper addresses the scarcity of exact analytical solutions to the three-dimensional (3D) Schrödinger equation, particularly for systems requiring a positive, gauge-invariant Weyl-Stratonovich quasi-probability density function. While the Wigner function is a standard tool for phase-space descriptions, it often yields negative values, violating classical probability intuition. Previous work identified a unique quantum system with a "quadratic funnel" potential that yields a positive quasi-probability density for a specific ground state wave function ( $f\Psi$ ) within the $\Psi$ -model. The objective of this work is to construct a complete class of exact solutions (including the full energy spectrum and eigenfunctions) for a two-parameter extension of this potential, denoted as $U(r, p_1, p_2)$ , within the framework of the $\Psi$ -model of micro and macro systems, building upon the foundation of that unique positive-density state.

Methodology
The authors employ the Wigner-Vlasov formalism, which establishes a connection between classical and quantum systems through a self-consistent chain of Vlasov equations. The methodology proceeds as follows:

Formalism Construction: The authors utilize the first two equations of the Vlasov chain to derive conservation laws for probability, momentum, and energy. By applying the Helmholtz theorem to decompose the probability flux velocity field into potential and vortex components, they derive a generalized Schrödinger equation (Eq. 1.11) that includes scalar and vector potentials.
Gauge Invariance: The study explicitly incorporates gauge invariance for scalar and vector potentials. This allows the transformation between a system with a probability flux velocity field and one with an electromagnetic field (specifically a "Dirac string" magnetic field).
Potential Definition: The core of the analysis focuses on a two-parameter "quadratic funnel" potential (Eq. 2.5) and its bounded variant (Eq. 3.5). These potentials are derived from specific phase functions and probability flux configurations.
Solution Construction:
- Singular Solutions: The authors solve the time-independent Schrödinger equation for the general two-parameter potential, yielding solutions involving generalized Laguerre polynomials and specific angular dependencies.
- Bounded Solutions: A special case is analyzed where the dissipation sources vanish, leading to solutions involving associated Legendre polynomials. These solutions are constrained by Pythagorean triples $(l, \epsilon, \lambda)$ , where $l^2 + \epsilon^2 = \lambda^2$ .
Superposition Analysis: The paper investigates superpositions of eigenfunctions with opposite vortex and potential fluxes to create standing wave analogues and systems with zero net probability current.

Key Contributions and Results

Exact Eigenfunctions and Energy Spectrum: The paper derives explicit expressions for the wave functions and energy eigenvalues for the two-parameter potential.
- Theorem 1 provides singular solutions (Eq. 2.4) characterized by parameters $\kappa$ and $\lambda$ , with energy levels dependent on these parameters and the system's characteristic size $\sigma$ .
- Theorem 3 presents bounded solutions (Eq. 3.4) where the quantum numbers are linked by Pythagorean triples. These solutions generalize the harmonic oscillator, reducing to it when specific parameters vanish.
Gauge-Invariant Solutions: Theorem 2 and Theorem 4 extend these solutions to the electromagnetic Schrödinger equation. The authors demonstrate that the wave functions remain valid when a vector potential (representing a "Dirac string" magnetic field) is introduced, maintaining the gauge-invariant properties of the Weyl-Stratonovich function.
Probability Flux and Vortex Structures: The study details the structure of the probability flux field.
- The solutions exhibit vortex structures flowing from "white holes" to "black holes" (poles of the sphere) or circulating azimuthally.
- Superpositions of counter-propagating fluxes result in standing wave patterns for the probability density, with nodal points determined by the quantum numbers.
Intrinsic Magnetic Moments: The paper calculates the intrinsic magnetic moment ( $\mu_S$ $μ_{S}$ ) for various configurations:
- Pure vortex states possess a magnetic moment proportional to the Bohr magneton and the parameter $\lambda$ (Eq. 2.16).
- Superpositions of opposite vortices can cancel the magnetic moment, resulting in a system with zero net current and zero magnetic moment (Eq. 2.17, 2.21).
Quantum Potential and Forces: The analysis includes the quantum potential ( $Q$ ) and demonstrates how it balances external forces. In superposition states with zero net current, the system is shown to be in equilibrium where the external force is exactly balanced by the quantum pressure force.

Significance and Claims
The paper claims to provide a rigorous mathematical framework for constructing exact 3D solutions to the Schrödinger equation that are consistent with the $\Psi$ -model. The primary significance lies in:

Completeness: It moves beyond the single ground-state solution found in previous work to provide a complete set of eigenfunctions and the full energy spectrum for the extended potential.
Phase Space Context: The solutions are constructed within the $\Psi$ -model, which was previously established to yield a positive, gauge-invariant Weyl-Stratonovich quasi-probability density for a specific non-Gaussian ground state, addressing the issue of negative probabilities in standard Wigner functions for that specific system.
Hydrodynamic Interpretation: The results offer a hydrodynamic interpretation of quantum phenomena, linking probability currents to vortex fields, intrinsic magnetic moments, and quantum pressure forces within a classical-like conservation law framework.
Generalization: The bounded solutions generalize the harmonic oscillator, introducing a new class of "modified spherical harmonics" constrained by Pythagorean triples.

The authors conclude that this approach aligns with established methods in continuum mechanics and provides a robust tool for testing numerical algorithms and understanding the fundamental principles of quantum systems with complex potentials.

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