An advance in the arithmetic of the Lie groups as an alternative to the forms of the Campbell-Baker-Hausdorff-Dynkin theorem

The exponential of an operator or matrix is widely used in quantum theory, but it sometimes can be a challenge to evaluate. For non-commutative operators X and Y , according to the Campbell-Baker-Hausdorff-Dynkin theorem, e X + Y is not equivalent to e X e Y , but is instead given by the well-known infinite series formula. For a Lie algebra of a basis of three operators { X , Y , Z }, such that [ X , Y ] = κ Z for scalar κ and cyclic permutations, here it is proven that e a X + b Y is equivalent to e p Z e q X e − p Z for scalar p and q . Extensions for e a X + b Y + c Z are also provided. This method is useful for the dynamics of atomic and molecular nuclear and electronic spins in constant and oscillatory transverse magnetic and electric fields.