Communication: Phase space wavelets for solving Coulomb problems.

Area of Science:

• Quantum mechanics
• Computational physics
• Mathematical physics

Background:

• The time-independent Schrödinger equation is fundamental in quantum mechanics.
• Previous phase space approaches utilized a periodic von Neumann basis with bi-orthogonal exchange (pvb).
• Efficient numerical methods are crucial for solving complex quantum systems.

Purpose of the Study:

• To extend the phase space approach for solving the Schrödinger equation.
• To introduce a wavelet scaling for phase space Gaussians, creating a novel wavelet pvb basis.
• To evaluate the efficiency of the new basis set for quantum mechanical problems.

Main Methods:

• Implementation of a wavelet scaling for phase space Gaussians.
• Utilizing a periodic von Neumann basis with bi-orthogonal exchange (pvb).
• Testing the new wavelet pvb basis on 1D Coulomb problems.

Main Results:

• The wavelet pvb basis offers a simple and effective alternative to existing wavelet methods.
• Basis set size was reduced by a factor of 13-60 compared to the Fourier grid method for 1D Coulomb problems.
• Significant basis set size savings are anticipated to increase with higher dimensionality.

Conclusions:

• The wavelet pvb basis is a computationally efficient method for solving the time-independent Schrödinger equation.
• This approach shows promise for tackling more complex and higher-dimensional quantum systems.
• The method provides a valuable tool for quantum mechanical simulations.