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Exact solution of three dimensional schrödinger equation with power function superposition potential.
Meihuan Fu1, Yongwen Liu1, Jianxin Shi1
1Public Foundational Courses Department, Nanjing Vocational University of Industry Technology, Nanjing, China.
Plos One
|November 28, 2023
Summary
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.
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.
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