Matrix algorithm for solving Schrödinger equations with position-dependent mass or complex optical potentials

Area of Science:

• Quantum mechanics
• Computational physics
• Molecular physics

Background:

• Solving quantum mechanical Hamiltonians is crucial for understanding molecular and solid-state systems.
• Traditional methods face challenges with non-Hermitian, PT-symmetric, or position-dependent mass Hamiltonians.

Purpose of the Study:

• To develop a novel matrix representation for low-dimensional quantum mechanical Hamiltonians.
• To accurately reproduce bound-state energies and wave functions for various Hamiltonian types.
• To extend the applicability to systems with position-dependent mass.

Main Methods:

• Representing Hamiltonians using moderately sized finite matrices.
• Achieving machine-precision accuracy for energies and wave functions.
• Applying the method to non-Hermitian, PT-symmetric, and position-dependent mass Hamiltonians.

Main Results:

• The matrix method accurately reproduces the lowest O(10) bound-state energies and wave functions.
• The approach successfully handles Hamiltonians that are neither Hermitian nor PT symmetric, allowing spectral analysis.
• Demonstrated effectiveness for position-dependent mass models, including molecular inversion motion and solid-state effective-mass models.

Conclusions:

• The finite matrix representation offers a robust and accurate method for solving diverse quantum mechanical problems.
• This technique is particularly valuable for complex systems in molecular physics, quantum chemistry, and solid-state physics.
• The study validates the method's performance through comparisons with established analytical and numerical results.