Filter Method for Solving Schrödinger Equation

The Filter Method is a numerical approach used to solve the stationary Schrödinger equation for quantum systems. It allows for the simultaneous computation of eigenfunctions and eigenvalues on a grid without the need for boundary-value conditions.

Description

The paper "Filter method without boundary-value condition for simultaneous calculation of eigenfunction and eigenvalue of a stationary Schrödinger equation on a grid" presents the Filter Method. The method efficiently computes the eigenfunctions and eigenvalues of a given quantum system using a tridiagonal matrix and iterative filtering.

This implementation provided in the code examples is based on the Kronig-Penney potential, which is a model used to study the behavior of electrons in a periodic structure. It also includes an example for a single harmonic potential.

Code Example - Kronig-Penney Potential

The first code example demonstrates how to calculate and plot the Kronig-Penney potential. It uses the kronig_penney_potential function to define the potential and then plots the potential versus x.

Code Example - Energy Spectrum

The second code example shows how to calculate the energy spectrum of the Kronig-Penney potential. It uses the eigenState function to compute the eigenstates and eigenenergies for a range of energy values. The energy spectrum is then plotted, and the corresponding wave functions and probability distributions are visualized.

Code Example - Single Harmonic Potential

The third code example illustrates how to calculate and plot the wave function, probability distribution, and energy level for a single harmonic potential. It uses the eigenState function with a single harmonic potential defined in the potensial_harmonik function.

Usage in Google Colab

To run the code examples in Google Colab, you can use the provided Colab notebook: Filter Method Colab Notebook. The notebook allows you to execute the code interactively and visualize the results.

Remember to set up the necessary environment and libraries in the Colab notebook before running the code.

References

• Filter Method journal (Filter method without boundary-value condition for simultaneous calculation of eigenfunction and eigenvalue of a stationary Schrödinger equation on a grid)