Non-Classical Representations of Potential, Kinetic and Total Energies

This chapter presents the first concrete consequences of the foregoing one. The possibility of configurations, x, for which the classical expression for the potential energy, V(x), is larger than the total energy, E, implies, that this expression, V(x), does no longer represent the contribution of the configuration x to the work storage of the system. For this, a ‘limiting function’, F(x), is introduced such, that the non-classical contribution of the configuration x to the work storage, Vncl(x) = F(x)V(x), is smaller than the total energy. The same is done for the momentum configurations and the kinetic energy. Moreover, since there are no trajectories, the non-classical representation of the energies become integral expressions, in agreement with Schrödinger’s vision. Then, the general properties of the limiting functions are deduced. Limiting amplitudes (dimensionless wave functions) are introduced, in order to find an integral relationship between the motions in space and in momentum space, as envisaged by Schrödinger, again.

Keywords: Characteristic length, Fourier transform, Limiting function, Nonclassical kinetic energy, Non-classical potential energy, Non-classical total energy, Normalization, Ordered sets, Weight amplitude, Weight function.