  Based on 1) the spectral resolution of the energy operator; 2) the linearity
of correspondence between physical observables and quantum Hermitian operators;
3) the definition of conjugate coordinate-momentum variables in classical
mechanics; and 4) the fact that the physical point in phase space remains
unchanged under (canonical) transformations between one pair of conjugate
variables to another, we are able to show that <t_s|E_s>, the proper-time
rest-energy transformation matrices, are given as a*exp[-iE_s t_s/\hbar], from
which we obtain the proper-time rest -energy evolution equation
i\hbar{\partial/\partial t_s} |Psi>= \hat{E_s}|Psi>. For special relativistic
situations this equation can be reduced to the usual i\hbar{\partial/\partial
t}|Psi>=\hat{E}|Psi> dynamical equations, where t is the "reference time" and E
is the total energy. Extension of these equations to accelerating frames is
then provided.
