Analytical evaluation of the numerical coefficients in the Zassenhaus product formula and its applications to quantum and statistical mechanics

TL;DR

This paper analyzes the Zassenhaus product formula to determine key coefficients, enabling applications in solving differential equations and quantum mechanics through a closed-form approach.

Contribution

It provides a method to explicitly compute coefficients in the Zassenhaus formula, facilitating practical applications in physics and mathematics.

Findings

Derived explicit formulas for coefficients in the Zassenhaus expansion

Demonstrated applications in quantum mechanics and differential equations

Provided a closed-form solution approach for operator exponentials

Abstract

This paper studies the exponential of the sum of two non-commuting operators as an infinite product of exponential operators involving repeated commutators of increasing order. It will be shown how to determine two coefficients in front of the nested commutators in the Zassenhaus formula. The knowledge of one coefficient is enough to generate a closed formula that has several applications in solving problems ranging from linear differential equations, quantum mechanics to non-linear differential equations.