Hello all. I'm in an introductory QM course as a physics major. As I understand it, one has an eigenfunction of an operator if one applies that operator to the function and gets a scaler multiple of it. Using the analogy of a rotation matrix, the eigenvectors represent the axis of rotation. I understand that if an operator doesn't change the vectors angle, it must be pointed along the axis of rotation. So what about the "direction" of eigenfunctions in QM? If the Hamiltonian operator just scales the function, and the eigenvalue is the energy, what sort of significance can be placed on the "direction" of the eigenfunction, as related to the operator? Why is the eigenvalue the energy? I guess I don't understand why eigenvalues/functions are used. What properties do they have that serve the math-needs of the problems at hand? Sorry for unclear questions, but I'm having problems verbalizing what I mean. Thanks in advance for the help.