Abstract: A class of densely defined self-adjoint linear operator a^N is introduced in the functional space L2(M)of normal martingale square-integrable, where a is a positive number, and N is the number operator in L2(M), a^N is called the a-level power-number operator of N. Firstly, the analytical properties of a^N are discussed: a sufficient and necessary condition that a^N is bounded is given, and in this case, {a^N; 0≤1} are unit operator on L2(M). Secondly, a^N is compact operator if and only if 0<1; the construction and the spectrum of {a^N; a>0} are discussed: {aⁿ; a>0} is the spectrum of a^N and all of its eigenvector forms an orthonormal basis of L2(M), 1 is the unique spectrum of {a^N; a>0} and the vacuum Z_ is the unique common eigenvector of 1. And then the dependence of a^N on a is discussed. Finally, a uniform convergence sequence of a^N for a∈(0,1)and a strong convergence sequence of a^N is constructed when a>1 by means of the quantum Bernoulli noise indexed by Γ.