We will assume that the eigenfunctions form a complete set so that any function can be written as a linear combination of them. They can be said to form a basis set in terms of which … For any given physical problem, the Schrödinger equation solutions which separate (between time and space), are an extremely important set. It is standard subject in any textbook on mathematical physics that this is indeed a complete set… form a complete set of linearly independent functions. (This can be proven for many of the eigenfunctions we will use.) Energy eigenfunctions can also be used to represent a general solution. Because the eigenfunctions of the Sturm-Liouville problem form a complete set with respect to piecewise smooth functions over the finite two-dimensional domain, the preceding sums are the generalized double Fourier series expansions of the functions f(r, θ) and g(r, θ) in terms of the allowed eigenfunctions. However, this is where my question begins: Consider a set of energy eigenfunctions $\psi_n$ which satisfy by definition … Since the eigenfunctions … "Complete" means that any state in your space can be written as a superposition of energy eigenstates; that is, energy eigenstates span all the state space. Basis Set Postulate The set of functions Ψ j which are eigenfunctions of the eigenvalue equation. If we … Eigenfunctions, Eigenvalues and Vector Spaces.