Completeness

These eigenfunctions form a complete set, in the sense that any other function, f (x) can be expressed as a linear combination of such functions as shown:

Now, if you know the form of this function f (x), you can find coefficients c_n by using the orthonormality condition of the eigenfunctions as shown:

Thus,

As you can see in the previous equation, these coefficients depend on the form of eigenfunctions and some function f (x), which means that these coefficients depend on the physical conditions of the system under study. We will study several examples in the next chapter for finding such coefficients based on the physical conditions of the system.