Quantum Bound States, Resolution, Information and Interference

 Free quantum particles may be represented by a wavefunction of exp(ipx) where “px” suggests a resolution scale L with p=2(3.14)hbar/L. In this note, we suggest that such a resolution may already be present in classical mechanics. exp(ipx) is a periodic form which incorporates this resolution and through a modulus of 1 ensures equal probability to be at any point x. In order to use exp(ipx), it seems that there must be a physical mechanics associated with the resolution.

   Given that exp(ipx)=cos(px) + isin(px), we argue that two different probability cycling or flows exist, one related to energy (pp/2m) and time and the other to flux- momentum and space i.e. sin(px). The link between the two is: cos(px)cos(px)+sin(px)sin(px) =1 i.e. the two are out of phase. This appears to be similar to the two dimensions x and ict. Each cycle or flow has its own frequency with positive and negative mutually exclusive subdivisions within resolution pockets, 1/E for time and L with p=2(3.14)hbar/L for space. The frequencies are associated by positive and negative probabilities (linked perhaps with forward and backward motions etc) and lead to interference if one considers different momentum values. 

    We attmept to consider these issues in this note. In addition, we examine ln(W) representing information, with W(x), being the wavefunction . The change in information is -i dx <p ave at x>. Thus, it seems that momentum flow carries information in a quantum bound state.