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Exact solution of three dimensional schrödinger equation with power function superposition potential.

Meihuan Fu1, Yongwen Liu1, Jianxin Shi1

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Researchers found an analytical solution for the Schrödinger equation using power function superposition potentials. This method provides exact energy levels for quantum systems, aiding quantum theory development.

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Area of Science:

  • Quantum Mechanics
  • Theoretical Physics
  • Mathematical Physics

Background:

  • The Schrödinger equation is central to quantum mechanics.
  • Analytical solutions for potentials like harmonic oscillator and Coulomb are crucial.
  • Power function superposition potentials represent a significant class of potentials.

Purpose of the Study:

  • To find an analytical solution for the Schrödinger equation with power function superposition potentials.
  • To develop a general method applicable to various quantum systems.
  • To obtain exact formulas for bound-state energy levels.

Main Methods:

  • Decomposition of the second-order radial Schrödinger equation into a first-order Ricatti equation.
  • Construction of two specific forms of power function superposition potentials.
  • Derivation of exact analytical solutions for these potentials.

Main Results:

  • An exact analytical solution was obtained for the Schrödinger equation with two specific power function superposition potentials.
  • A general formula for bound-state energy levels was derived.
  • Calculated energy levels for diatomic molecules matched existing methods.

Conclusions:

  • The developed method offers an effective approach to solving the Schrödinger equation for power function superposition potentials.
  • The findings contribute to the advancement of quantum theory and potential applications.
  • The exact energy level formulas derived are valuable for analyzing quantum systems.