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Computational method for the quantum Hamilton-Jacobi equation: bound states in one dimension.
Chia-Chun Chou1, Robert E Wyatt
1Institute for Theoretical Chemistry and Department of Chemistry and Biochemistry, The University of Texas at Austin, Austin, TX 78712, USA.
The Journal of Chemical Physics
|November 15, 2006
Summary
A new computational method accurately solves the quantum Hamilton-Jacobi equation for quantum momentum and wave functions. This approach, validated with the harmonic oscillator and Morse potential, offers a reliable tool for quantum mechanics problems.
Area of Science:
- Quantum mechanics
- Computational physics
- Theoretical chemistry
Background:
- The quantum Hamilton-Jacobi equation is a fundamental tool in quantum mechanics.
- Accurate numerical solutions are crucial for understanding quantum systems.
- Existing methods may face challenges with singularities.
Purpose of the Study:
- To present an accurate computational method for the one-dimensional quantum Hamilton-Jacobi equation.
- To develop a technique for synthesizing bound state wave functions.
- To demonstrate the method's efficacy on solvable quantum mechanical models.
Main Methods:
- Numerical integration of the quantum Hamilton-Jacobi equation using the Mobius propagation scheme.
- Accurate handling of singularities within the propagation scheme.
- Synthesis of wave functions via phase integral and antithetic cancellation.
Main Results:
- Accurate computation of quantum momentum functions.
- Accurate synthesis of bound state wave functions.
- Excellent agreement with exact analytical results for harmonic oscillator and Morse potential.
Conclusions:
- The proposed computational method accurately solves the quantum Hamilton-Jacobi equation.
- The method effectively obtains both quantum momentum and wave functions.
- This approach shows promise for solving a wide range of quantum mechanical problems.
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