The Heisenberg correspondence between classical Fourier components and quantal matrix elements is used to demonstrate remarkably accurate scaling laws, by which the matrix elements scale simply with the reduced energy ((E) over bar /B), where B is a characteristic energy parameter and (E) over bar is the average energy of the relevant states. Applications to the Morse oscillator, a cylindrically symmetric double well (champagne bottle) potential and to the spherical pendulum are described. The relevance of the scaling rules to modeling highly excited molecular vibrational states close to saddle points on the potential surface is discussed. Significant differences between Morse oscillator and vibron oscillator matrix elements are reported.