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

Abstract

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 $\mathbf{X}$ and $\mathbf{Y}$, according to the Campbell-Baker-Hausdorff-Dynkin theorem, ${\mathrm{e}}^{\mathbf{X}+\mathbf{Y}}$ is not equivalent to ${\mathrm{e}}^{\mathbf{X}}{\mathrm{e}}^{\mathbf{Y}}$, but is instead given by the well-known infinite series formula. For a Lie algebra of a basis of three operators $\{\mathbf{X},\mathbf{Y},\mathbf{Z}\}$, such that $[\mathbf{X},\mathbf{Y}]=\kappa \mathbf{Z}$ for scalar $\kappa$ and cyclic permutations, here it is proven that ${\mathrm{e}}^{a\mathbf{X}+b\mathbf{Y}}$ is equivalent to ${\mathrm{e}}^{p\mathbf{Z}}{\mathrm{e}}^{q\mathbf{X}}{\mathrm{e}}^{-p\mathbf{Z}}$ for scalar $p$ and $q$. Extensions for ${\mathrm{e}}^{a\mathbf{X}+b\mathbf{Y}+c\mathbf{Z}}$ 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.

arXiv comment6 pages,0 figures

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