Filter method without boundary-value condition for simultaneous calculation of eigenfunction and eigenvalue of a...

Area of Science:

• Quantum mechanics
• Computational physics
• Applied mathematics

Background:

• The stationary Schrödinger equation is fundamental in quantum mechanics.
• Solving the Schrödinger equation often requires imposing boundary conditions, which can complicate computations.
• Efficient and accurate numerical methods are crucial for quantum simulations.

Purpose of the Study:

• To present a novel method for the simultaneous computation of eigenfunctions and eigenvalues of the stationary Schrödinger equation.
• To develop a technique that does not require imposing boundary-value conditions.
• To demonstrate the method's efficacy and reliability for various boundary conditions.

Main Methods:

• A filter operator-based method is employed to compute eigenfunctions and eigenvalues.
• The filter operator selects eigenfunctions from a wave packet, analogous to a delta function.
• The method operates on a grid without requiring pre-defined boundary conditions.

Main Results:

• The filter method accurately computes eigenvalues and eigenfunctions for the stationary Schrödinger equation.
• Simulation results show strong agreement with analytical and other numerical solutions across different boundary conditions.
• The method is demonstrated to be robust, accurate, and reliable.

Conclusions:

• The developed filter method offers an effective way to solve the stationary Schrödinger equation without boundary conditions.
• The method's accuracy and reliability are validated against established techniques.
• Future work will explore the application of this filter method to higher-dimensional Schrödinger equation simulations.