The common system for a wave function is Ѱ. Although Ѱ is a complex number, |Ѱ|² is a real number and corresponds to the probability density of finding a particle in a given place at a given time. The SI units for the wave function depend on the system. For one particle in three dimensions the units are m-3/2. The units are required so that an integral of |Ѱ|2 over a region of three-dimensional space is a unitless probability.
The wave function is central to quantum mechanics. It is the fundamental postulate of quantum mechanics. The wave function is a source of the mysterious consequences and philosophical difficulties in the interpretations of quantum mechanics. These topics continue to be in debate today.