A new method for solving one-dimensional steady-state equations in quantum mechanics based on duality
LI Shilin
ZHUANG Yuting
Abstract：There exists a dual relationship between the one-dimensional steady-state equations in quantum mechanics,which can be extended to any system where two potentials are located.The problem of solving the one-dimensional steady-state Schrödinger equation using the extended dual relationship can be transformed into a problem of finding the dual transformation between the equation to be solved and the equation with a known solution.This provides a new method for solving the one-dimensional steady-state Schrödinger equation.Starting from a one-dimensional harmonic oscillator system with known solutions,a solution to the one-dimensional steady-state Schrödinger equation under an unsolved periodic potential can be obtained using the dual relationship.And it can provide the conditions that the coefficient satisfies when the energy eigenvalue belongs to the bandgap or conduction band.
Keywords：quantum mechanics dualityone-dimensional steady-state equationexact solution
Publication Date：2025-12-15
Online Publishing Date：2026-04-01(First online date of this platform, not the publication date of the document)
Pages：6( 289-294 )