When we are dealing with waves on a large scale, say sound waves or waves on water, we tend to get sine waves that spread out over distance and dissipate as they get further from the source.
The waves associated with, say a photon or a particle, does not tend to dissipate in this way and remain confined to a region in space. To do this requires an infinite number of frequencies in a narrow band.
Wave function: A function giving a complex number (a+ i b) for each point in space and time.
The equation represents the condition that the wave function must satisfy to represent the behavior of an object under the influence of one or more forces.
The square of the wave function (a2 + b2) is the relative probability of finding the object at that point in space-time.
The phase of the wave function is the angle arctan(a/b)
Schrödinger function - behavior of non relativistic systems.
|(,t)|2 = probability of finding particle at position at time t.
Dirac equation - behavior of spin 1/2 particles