This becomes especially clear by a comparison to classical mechanics. Imagine a single particle that can be described with k numbers (take for example a free particle described by its position and velocity vector, resulting in k = 6 ). In classical mechanics, n such particles can simply be described by k ⋅ ⋅ n numbers. The dimension of the classical many-body system scales linearly with the number of particles n . In quantum mechanics, however, the many-body-system is in general in a superposition of combinations of single particle states - all the k n different combinations have to be accounted for. The dimension of the quantum many body system therefore scales exponentially with n , much faster than in classical mechanics.