The Time Independent Schrodinger Equation Without Using Energy as a Starting Point

The time-independent Schrodinger equation is written terms of energy, namely using V(x) the potential and E, the energy eigenvalue. In fact, the equation can be written as KE average(x) + V(x) = E and so can be expressed in terms of a classical energy conservation equation at each point. KE average, however, in such a case is complicated and seems to contain various kinds of kinetic energy motion which only yields on average the classical kinetic energy. The purpose of this note, is to try to determine the form of the time-independent Schrodinger equation without any concept of energy. Only at the end of the discussion, will comparisons be made with classical conservation of energy. In such a discussion, it is assumed that particles undergo a zitterbewegung type motion which can be modeled by exp(ikx) hbar k being the momentum and that there are forces (related to a V(x)) in the one dimensional problem. The objective is to try to understand microscopically what is happening without immediate resort to a conservation equation.