Quantum Mechanics Retains Precise Position Information in Steady State Scenarios?

Quantum mechanics is usually associated with loss of spatial information. For example, in the case of a superposition Sum over p a(p)exp(ipx), one has the Heisenberg uncertainty relation (standard deviation x) * (standard deviation p) >= hbar/2 and even in the case of a precise p, the probability exp(ipx) suggests an impulse hit can occur within dx =hbar/p. We suggest here, however, that there exists an example in which quantum mechanics retains precise spatial information even in the case in which classical mechanics does not. In particular, we consider the situation of a dx=hbar/p = 1 (i.e. in units of 2*3.14) and a one dimensional set of creation points for a particle: x=0, x=⅓, x=2/3. In such a case, the phase shift of each exp(ipx + phase shift) will identify the precise position (or pretty close if one considers the notion of a wavepacket) of the creation point in a steady stream scenario. In a classical situation, a steady stream scenario will completely wash out any information of the creation points of the particles as there is no distinguishing them in time. If one wishes to consider all different kinds of x creation points, then different p values are required. Thus, discerning space implies a loss of precise information of p.