We can first assume that the operator for momentum is multiplication of p. Then we must have This is called the momentum representation . Making this substitution produces another operator for the same observable property. This creates a whole new way to find eigenfunctions and eigenvalues. Though the functions and operators are different, the results are the same - if they weren't our theory wouldn't be worthwhile.
Chemists usually use the position representation.
Solid state physics often employs the momentum representation. There are other representations possible. We will generally focus our attention from here on out with the position representation. As an example of how this process can form other operators, consider kinetic energy. All other operators are formed in a similar manner.