Phase-space approach to solving the time-independent Schrödinger equation

Area of Science:

• Quantum mechanics
• Computational physics
• Mathematical physics

Background:

• The time-independent Schrödinger equation is fundamental to quantum mechanics.
• Existing numerical methods like the Fourier grid method face challenges with convergence and scalability.
• The von Neumann (vN) lattice offers an alternative phase space representation.

Purpose of the Study:

• To develop a robust and efficient method for solving the time-independent Schrödinger equation.
• To address the convergence limitations of the standard von Neumann lattice method.
• To leverage classical phase space structures for accurate quantum computations.

Main Methods:

• Incorporation of periodic boundary conditions into the von Neumann (vN) lattice.
• Utilizing phase space Gaussians as basis functions.
• Application to a challenging two-dimensional potential problem.

Main Results:

• Successfully resolved the longstanding convergence problem of the vN method.
• Achieved high accuracy comparable to the Fourier grid basis.
• Demonstrated significant numerical savings, especially in higher dimensions.
• In the classical limit, achieved efficiency of one basis function per eigenstate.

Conclusions:

• The proposed method provides an accurate and efficient approach for solving the Schrödinger equation.
• It enables tailoring quantum calculations to classical phase space structures.
• The method shows great potential for computational savings in complex quantum systems.