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Eigenvalue problem of the Schrödinger equation via the finite-difference time-domain method
1Department of Electrical and Electronic Engineering, University of Bristol, BS8 1TR, United Kingdom.
Summary
We developed an efficient quantum method to solve the Schrödinger equation, enabling faster calculations for large quantum systems like GaN quantum dots.
Area of Science:
- Quantum mechanics
- Computational physics
- Materials science
Background:
- Solving the time-independent Schrödinger equation is crucial for understanding quantum systems.
- Traditional methods can be computationally intensive for large-scale systems.
Purpose of the Study:
- To present an efficient numerical scheme for calculating the eigenvalue problem of the time-independent Schrödinger equation.
- To enable the study of large-scale quantum systems with reduced computational cost.
Main Methods:
- The eigenvalue problem is solved using an initial-value approach of the time-dependent Schrödinger equation.
- The finite-difference time-domain method is employed for wave function time evolution.
- Fast Fourier transformation is used to extract eigenenergies from the time-dependent wave function.
Main Results:
- The computational effort scales favorably with the number of grid points, making it suitable for large systems.
- Accurate confined energies and wave functions for a 3D GaN quantum dot were obtained in 7 hours.
- The method was successfully applied to a system with one million grid points.
Conclusions:
- The proposed method offers a highly efficient and scalable approach for quantum mechanical eigenvalue problems.
- This technique can be implemented on parallel computing systems for even more complex quantum system studies.
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