The wave or state functions describes the motion and properties of a particle, such as an electron or photon. The magnitude squared of the wave-function is the probability density of finding a particle in a volume.
The Uncertainty Principle defines the limits of accuracy as a relationship between position and momentum.
The expected value of an observation, such as the presence of a particle in a volume is calculated via an operator.
The Schroedinger equation describes the motion of particles according to quantum mechanics.
The Schroedinger equation can be separated into the time dependent and and spatially dependent forms. The time-independent, spatially dependent form solution is as follows.