Spectral Theorem approach to the Characteristic Function of Quantum Observables.

Abstract

Using the spectral theorem we compute the Quantum Fourier Transform (or Vacuum Characteristic Function) $\\langle \\Phi, e^{itH}\\Phi\\rangle$ of an observable $H$ defined as a self-adjoint sum of the generators of a finite-dimensional Lie algebra, where $\\Phi$ is a unit vector in a Hilbert space $\\mathcal{H}$. We show how Stone s formula for computing the spectral resolution of a Hilbert space self-adjoint operator, can serve as an alternative to the traditional reliance on splitting (or disentanglement) formulas for the operator exponential.