If you're into eigenfunctions, pick up any textbook on quantum mechanics. The hamiltonian is a linear operator whose eigenfunctions are the stationary states of the system ("stationary" because an isolated system in a stationary state will never leave that state) and whose eigenvalues are the observable values of the energy of the system. In general, there is a correspondence between observable quantities and Hermitian operators on wavefunctions: "measurement" is the application of a Hermitian operator to the wavefunction, and the values you may observe are the eigenvalues of the operator. So, for instance, energy is quantized in some systems because their hamiltonian has discrete eigenvalues.