A note on the distributions in quantum mechanical systems

In this paper, we study the distributions and the affine distributions of the quantum mechanical system. Also, we discuss the controllability of the quantum mechanical system with the related question concerning the minimum time needed to steer a quantum system from a unitary evolution $U(0)=I$ of the unitary propagator to a desired unitary propagator $U_f$. Furthermore, the paper introduces a description of a $\mathfrak{k} \oplus \mathfrak{p}$ sub-Finsler manifold with its geodesics, which equivalents to the problem of driving the quantum mechanical system from an arbitrary initial state $U(0)=I$ to the target state $U_f$, some illustrative examples are included. We prove that the Lie group $G$ on a Finsler symmetric manifold $G/K$ can be decomposed into $KAK$.