Scaling and the Handling of Time in Plane Waves in a Potential

In (1) (2), an expression for fp, the momentum wavefunction, for a particle in a box with no potential (except at the walls) is given. Here W(x)= Sum over p fp exp(ipx), where W(x) is the spatial wavefunction. The description given suggests that even for a “free” particle in a box, there are all possible momentum states with E=(hbar k)2/2m being an average energy. The spatial wavefunction is proportional to sin(kx-b(n)) where b(n) is a constant depending on n, the energy level and the center of the box is at x=0. The spatial result is interesting because it seems one is considering exp(ipx + i p*p/2m t) scattering from the walls to create a resonance, which is of the same form ( exp(ikx) and exp(-ikx)) as the constituent plane waves except now one has k which is the root mean square momentum. The energy also scales from p*p/2m for each plane wave to an overall average E. In this note, we try to investigate this behaviour for a particle in a box and for other potentials and consider issues in handling exp(ip*p/2m t). For a plane wave, time and space are linked in px-p*p2/m t, but in a quantum resonance things seem to be different. Finally, we suggest that since it is possible to have exp(ipx+p*p2/m t) states combine to form an overall new exp(ikx + Et) where E is the average energy and k the root mean square momentum, it may be that exp(ipx + ip*p/2m t) for a quantum particle is also of the same form. In other words, a quantum particle may have internal motion and components which lead to the exp(ipx + ip*p/2m t) result with p being a root mean square momentum of internal components and p*p/2m an average energy. In a previous note (3), it was suggested Einstein’s energy-momentum equation E*E = p*p + m*m where c=1 leads to a backward and forward motion as a particle moves forward with average velocity v. It was suggested that such motion might be modeled by exp(ipx + ip*p/2m t)