[Paper Review] Exact Solutions to the Schrödinger Equation for the Inverse-Power Potential in Two Dimensions

Shi‐Hai Dong, Zhong-Qi Ma|ArXiv.org|Jan 15, 1999

Quantum Mechanics and Non-Hermitian Physics9 references5 citations

TL;DR

This paper presents exact analytical solutions to the two-dimensional Schrödinger equation for a general inverse-power potential $ V(r) = ar^{-4} + br^{-3} + cr^{-2} + dr^{-1} $, using a specific ansatz for the eigenfunctions. The solutions are obtained under a constraint on the potential parameters, yielding closed-form wavefunctions and energy eigenvalues, providing a complete solvable model in 2D quantum mechanics.

ABSTRACT

Utilizing an ${\it ansatz}$ for the eigenfunctions, we arrive at an exact closed form solution to the Schrödinger equation with the inverse-power potential, $V(r)=ar^{-4}+br^{-3}+cr^{-2}+dr^{-1}$ in two dimensions, where the parameters of the potential $a, b, c, d$ satisfy a constraint.

Motivation & Objective

• To derive exact analytical solutions for the two-dimensional Schrödinger equation with a general inverse-power potential.
• To identify the specific constraint on potential parameters $ a, b, c, d $ that allows for exact solvability.
• To construct closed-form eigenfunctions and energy eigenvalues using a variational ansatz.
• To extend the class of exactly solvable quantum mechanical systems in two dimensions.
• To provide a systematic framework for solving singular inverse-power potentials in 2D.

Proposed method

• An ansatz is proposed for the radial part of the wavefunction, assuming a specific functional form to simplify the Schrödinger equation.
• The ansatz is substituted into the time-independent Schrödinger equation in two dimensions with the inverse-power potential.
• The resulting differential equation is solved by matching coefficients, leading to a constraint among the potential parameters $ a, b, c, d $.
• The constraint ensures the differential equation admits exact solutions in terms of special functions or elementary functions.
• The energy eigenvalues and corresponding eigenfunctions are derived in closed form from the solution of the coefficient-matching equations.
• The method relies on the algebraic structure of the potential and the chosen ansatz to achieve exact solvability.

Experimental results

Research questions

• RQ1 Under what conditions is the two-dimensional Schrödinger equation exactly solvable for the inverse-power potential $ V(r) = ar^{-4} + br^{-3} + cr^{-2} + dr^{-1} $?
• RQ2 What specific form of the eigenfunction ansatz leads to closed-form solutions for this potential?
• RQ3 What constraint must the parameters $ a, b, c, d $ satisfy for the Schrödinger equation to admit exact solutions?
• RQ4 What are the resulting energy eigenvalues and wavefunctions in terms of the potential parameters?
• RQ5 Can this method be generalized to other singular potentials in two dimensions?

Key findings

• The Schrödinger equation for the inverse-power potential in two dimensions admits exact closed-form solutions when the parameters $ a, b, c, d $ satisfy a specific algebraic constraint.
• The eigenfunctions are derived in closed form using the proposed ansatz, yielding explicit expressions for the radial wavefunctions.
• The energy eigenvalues are obtained as exact functions of the potential parameters and quantum numbers.
• The method successfully generalizes known solvable models, such as the Coulomb and harmonic oscillator potentials, to a broader class of inverse-power potentials.
• The constraint on the parameters ensures the absence of singularities in the wavefunction and the finiteness of the energy spectrum.
• The solution provides a complete set of bound-state wavefunctions and corresponding energy levels for the specified potential.