An acceptable wave function Ψ associated with a moving particle must be:

Concept:

In one dimension, wave functions are often denoted by the symbol ψ(x,t). 

They are functions of the coordinate x and the time t. But ψ(x,t) is not a real, but a complex function, the Schroedinger equation does not have real, but complex solutions.

The wave function of a particle, at a particular time, contains all the information that anybody at that time can have about the particle. 

But the wave function itself has no physical interpretation. It is not measurable. However, the square of the absolute value of the wave function has a physical interpretation. 

In one dimension, we interpret |ψ(x,t)|² as a probability density, a probability per unit length of finding the particle at a time t at position x.

The probability of finding the particle at time t in an interval ∆x about the position x is proportional to |ψ(x,t)|²∆x.

This interpretation is possible because the square of the magnitude of a complex number is real. 

For the probability interpretation to make sense, the wave function must satisfy certain conditions. We should be able to find the particle somewhere, we should only find it at one place at a particular instant, and the total probability of finding it anywhere should be one. This leads to the requirements listed below.

• The wave function must be single valued and continuous. The probability of finding the particle at time t in an interval ∆x must be some number between 0 and 1.
• We must be able to normalize the wave function.  We must be able to choose an arbitrary multiplicative constant in such a way, so that if we sum up all possible values ∑|ψ(x_i,t)|²∆x_i  we must obtain 1.