Quantum Bound States, Steady Stream and the Dirac Delta Function

 Classical mechanics usually employs two variables x,t to describe motion in the form of x(t). Derivatives of this function yield velocity, acceleration etc. For a bound system, velocity (momentum) changes from one x to the next, but one does not necessarily need time to describe the scenario as one could use the pair of variables  p(x),x instead of x(t),t. An issue which arises is that p(x) has double values as it is positive when the bound particle is moving to the right and negative when it is moving to the left. In such a case, time distinguishes between the two motions which occur at the same spatial point. From the point of view of energy, however, such a degeneracy at x is irrelevant because kinetic energy is pp/2m. Thus, it seems one may remove time from a bound state problem as long as one uses an energy (i.e. energy conservation) picture. 

    As a result, we propose using a p(xi), xi model to describe a bound state with p(xi) (momentum at xi) being different at each xi. In order to calculate average of the absolute value of p at xo we suggest using a Dirac delta function delta(p-p(xo)) i.e.    <p at xo>  related to {Integral  dp   p  delta(p-p(xo))} . If one expands the delta function in terms of the basis exp(ipx), it seems that quantum mechanics for the bound state arises as a classical steady stream picture in terms of energy conservation, with exp(ipx) representing a steady stream picture of a particle with a constant momentum. We discuss informational probability properties of exp(ipx) which make it the basis of choice.