Quantization of a Free Particle in N-Dimensional Space

A basic classical mechanical system in n space dimensions, \(n \in {\mathbb {N}}^{*}\) , is given by a point-particle of mass \(m > 0\) , solely interacting with an external potential V. The case of a vanishing potential, corresponds to a “free” particle. It is tempting to qualify the corresponding system as “simple,” in the sense that not much can be learned from this system. According to classical mechanics, since there is no external force, such particles move uniformly in straight lines, i.e., perform geodesic motion in Euclidean space. So particles are somehow aware of the geometry of the surrounding space. This is not really simple. Similar is true for quantum mechanics. The Hamiltonian of the corresponding quantum system is a multiple of the Laplace operator that is also tied to Euclidean geometry, signaling the “awareness” of the quantum system of Euclidean geometry.