Differing Space and Time Interactions in Quantum Mechanics

 The wavefunction for a quantum free particle is exp(-iEt + ipx) In this case, E and p are linked through either E=pp/2m (nonrelativistic case) or EE =pp + momo (relativistic case c=1). We argue that there are different interactions involved with E and p. P (momentum) is energy flow and is involved in impulse type interactions with a potential usually V(x). For V(x,t) matters are more complicated, but will also be discussed. The impulse hits are associated with exp(ipx) and  lead to interfering plane waves and an overall crest-trough spatial scenario. There is a kinetic energy (nonrelativistic case) associated with each p, but a new overall energy En (n= eigenlevel) is created which yields the free form exp(-iEnt). This energy is linked to the cycling of exp(ipx)’s. It is also part of the new mass of the bound state system i.e. mo+En. This new  system could be considered as an overall  free particle and obtain a new overall momentum. (The initial bound state has 0 overall momentum.) Thus, time and space parts of the wavefunction behave in very different manners due to the nature of the interaction even though there is a link between 1/E and cycling of exp(ipx) terms. We call this case an impulse or space based type of interaction.

    It is also possible to have a time based interaction i.e. the absorption of a photon, phonon etc. In such a case, there is momentum conservation and so the whole system receives a new momentum, but this is usually ignored. The main effect is for the system to move from one bound state E(n1) to another E(n2). This has consequences on the spatial density and cycling of exp(ipx) terms in the wavefunction, but is not an impulse interaction in space. In the case of a system:  W(x,t)= Sum over n an(En) exp(-iEnt) Wn(x), one has time based interactions which knock the system from one bound state to another (as in heat bath interaction). Again there are links between each En and the spatial density, but there is an overall spatial density(x,t) and average energy(x,t) which depends on the interference of exp(-iEnt) terms. (This disappears if one integrates over all x.) This is similar to the diffraction/interference spatial pattern, but  in time.

  In the impulse scenario for a(p)=a(-p) one has cross terms in the spatial density of cos( (pi-pj)x) which create the crest/trough pattern seen. For the heat bath time type interaction, there exist cos( (Ei-Ej)t) type terms if an(En)=a*n(En) which are linked to the energy Ei-Ej of the intermediary photon/phonon etc and its time period. Thus in both cases, the quantum crest/trough pattern (in space or time) seems to be linked to the momentum or energy transfer. The energy transfer is known to be linked to periodic objects (photons/phonons) and it seems the momentum transfer pi-pj also has a similar periodicity, but in space. It is almost as if these “entities” are creating the unusual quantum crest/trough patterns.