Taylor's theorem for shift operators

Abstract

Abstract An expansion equivalent to Taylor's series is derived for functions of two-operators of the form F(A + B) where B isa shift operator with regard to A; the result is obtained as a special case of a theorem due to Kermack and McCrea. The development in powers of Binvolves finite differences of the function F(A) instead of derivatives as in the case of commuting variables. Exponential functions can be factorized into two terms, each depending on one operator only. The result allows a Gaussian potential energy to be expressed as a product of exponentials involving only creation and destruction operators. Explicit results are derived for the matrix elements of such a Gaussian function in the representations which diagonalize the Hamiltoniana of the harmonic oscillator in one and two dimensions; they can be expressed in terms of Jacobi polynomials (finite hypergeometric series).