Quantum Bound State and Time-Removed Impulse Picture

 In (1), we argued that classically there are two pictures. The impulse picture delta p = Integral dt F does not distinguish between different velocities if p is the same.  In other words, p=m1v1=m2v2 (so time is removed) and v1 and v2 create the same impulse (and quantum mechanically the same interference/diffraction pattern) even though v1 and v2 are physical and different. 

   The second classical picture is the kinetic energy-work one which depends explicitly on time i.e. v dp = Fdx. The two pictures are consistent, but  for the one dimensional reflection-refraction problem, a photon count and photon energy balance equation based on time are not independent. Thus one must go to the time-removed impulse picture in order to have two equations, based on the probability exp(ipx), in order to solve the problem.

   We concluded in (1) that an impulse picture with time removed is necessary to describe interactions in a quantum system in a steady state scenario. Thus this picture should be applied to a bound state. A quantum bound state, however, makes use of a potential V(x) which is associated classically with a kinetic energy at x. Kinetic energy depends on time so how is this described in the impulse picture? We argue that one creates an average of pp based on the impulse probability exp(ipx) a(p)  (a(p) is a weight for p) which has time removed. It is the impulse probability exp(ipx) which allows for interference to occur. As a result, <pp> does not have time present and may include p and -p values which classically would occur during different time intervals.

   Once <pp> is computed in the time-removed picture, however, one divides by 1/2m (in the nonrelativistic case) to create kinetic energy. Thus prms(x), a mathematical average, may be associated with the idea of motion in space and time. pmrs(x), however, may not be considered physical as it is the impulses and exp(ipx) which are physical.

   As argued in (1), however, classically the two pictures should be consistent. As a result, for high energy levels, the description of <pp> in terms of an impulse picture which does not include time, but then adds 1/2m at the end should be equivalent to a picture in which a particle moves either to the left or right. We suggest that at high energy levels, one may actually assign prms(x) as a physical quantity within a tiny dx and follow the particle in time. Even in the time-removed impulse picture, there is velocity i.e. motion in time. It is just not used to calculate interaction averages in a steady state scenario.

   We suggest that to move from the quantum picture to the classical, one switches formalisms i.e drops the impulse- time removed picture and uses the time dependent motion in one direction. These two pictures are consistent at the classical level i.e. the classical time picture is a good approximation to the detailed impulse time removed picture which is now occurring in tiny dx regions.