So the equations of QM give eigenfunctions and eigenvalues. The eigenfunctions form a complete set with which any state is a combination of such. When measuring, the superposition of states collapse to one of the eigenfunctions. And the probability that some state with be measured in a particular eigenfunction is formed like an inner product of two states, etc. All this to ask the question: the eigenfunction is a function that maps one manifold to another. And each eigenfunction is a different manifold from the others. There is an inner product between these manifolds to form the probability of going from one to the other. I wonder if the inner product tells us that all the separate eigenfunctions rest within a larger manifold. This would be a manifold of manifolds. Is this a valid way of looking at things? If so, then is there some more general equation that specifies this manifold of eigenfunctions, perhaps some symmetry principle? Thanks.