mechanics of Born, Heisenberg, and Jordan 1925-1926 with the wave equation of Schrodinger in 1926. The collected and expanded works of John von Neumann showed that the two theories were mathematically equivalent, and it is this realization that is today understood as first quantization. Qualitative mathematical preliminaries To understand the term first quantization one must first understand what it means for something to be quantum in the first place. The classical theory of Newton is a second order nonlinear differential equation that gives the deterministic trajectory of a system of mass, . The acceleration, , in Newton's second law of motion, , is the second derivative of the system's position as a function of time. Therefore, it is natural to seek solutions of the Newton equation that are at least second order differentiable. Quantum theory differs dramatically in that it replaces physical observables such as the position of the system, the time at which that observation is made, the mass, and the velocity of the system at the instant of observation with the notion of operator observables. Operators as observables change the notion of what is measurable and brings to the table the unavoidable conclusion of the Max Born probability theory. In this framework of nondeterminism, the probability of finding the system in a particular observable state is given by a dynamic probability density that is defined as the absolute value squared of the solution to the Schrodinger equation. The fact that probability densities are integrable and normalizable to unity imply that the solutions to the Schrodinger equation must be square integrable. The vector space of infinite sequences, whose square summed up is a convergent series, is known as (pronounced "little ell two"). It is in one-to-one correspondence with the infinite dimensional vector space of square-integrable functions, , from the Euclidean space to the complex plane, . For this reason, and are often referred to indiscriminately as "the" Hilbert space. This is rather misleading because is also a Hilbert space when equipped and completed under the Euclidean inner product, albeit a finite dimensional space. Types of systems Both the Newton theory and the Schrodinger theory have a mass parameter in them and they can thus describe the evolution of a collection of masses or a single constituent system with a single total mass, as well as an idealized single particle with idealized single mass system. Below are examples of different