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

A note on the distributions in quantum mechanical systems