The primary difference between the
Newtonian Mechanics Model
of the universe and the
Quantum Mechanics Model
is that Newtonian Mechanics treats particles as points and Quantum Mechanics treats them as three dimensional "waves."
These are "waves" in a sea of probability. A particle is like a water molecule in the volume of the wave, i.e., its location is not fixed within the wave. As the wave moves, the molecule can be located at different places in the wave. In the Quantum Model, if the molecule is exactly located in the wave, neither the speed nor direction of the molecule's motion can be determined. This is the essence of the Uncertainty Principle of the universe and the One can locate a particle at an exact point in space but a wave is "spread out" allowing only a "most probable" location to be determined. This is contrary to what our five senses perceive but, as technology extends these senses, we observe phenomena that are only accurately described by the Quantum Mechanics Model. Quantum Mechanics is wierd and counter-intuitive but it works.
Because scientists must now deal with "waves," not "points," statistics is a necessary mathematical tool. The statistical probability that a particle will be located at a specific point in space replaces the exact three Newtonian spacial coordinates (x,y,z) for the particle. A Quantum Mechanical location statement for a particle is called a Wave Function.
The principle of Superposition requires that the mathematics of Quantum Mechanics reduce to those of Newtonian Mechanics under "normal" conditions. For example, when Quantum Mechanics is used to describe the motion of a ball on a pool table, the extra terms and factors in the Quantum Mechanical equations are trivial and, when these trivial terms and factors are ignored, the equations of motion look exactly like those of Newtonian Mechanics.
The Quantum Mechanics model is constructed using sets of Quantum Numbers and statistical Operators. The System under examination determines the Quantum Numbers which are specified. A System might be a single particle, a valence electron of an atom, an atomic nucleus, or a neutron star. These Quantum Numbers are "plugged into" the Operators resulting in a second set of Quantum Numbers. The "new" set of Quantum Numbers describe the System under "new" conditions, usually at a different time, past or future.