Quantum Mechanics and Time versus Space Sampling of Spatial Distributions

Classical mechanics samples space as being uniform i.e. each x point having the same weight. Thus the number of x states in a length L is proportional to L or in 3-space to volume V. The ln of this value then appears in classical entropy calculations. In the case of a particle moving in space the same idea holds, but this means the particle spends equal amounts of time in each dx region. Thus we argue this result of a constant probability in space is based on time sampling. Time sampling is the sampling done for an equilibrium gas. The system does not change on average in time, so it is as if one has a snapshot. Thus one has a constant probability in a length L in time sampling. One may argue that this constant is information so ln(probability(x))=constant1 or P(x)=constant 2.

    The relativistic/nonrelativistic action may be written as A = -Et+px if one uses v=x/t. This Lorentz invariant form allows one to clearly separate t and x with E and p as constants and may also be considered to be information. In such a case, if one ignores time, then px is the spatial contribution to A and this does represent a constant probability in space. Rather there is scaling of space by p. If x=constant/p, then each p has its own unique length, called a wavelength, in which space sampling is the same i.e. creates the same distribution form, but with 1/p as the wavelength. This we argue is space sampling of a spatial distribution and has physical consequences (i.e. two slit interference, measuring small regions etc). One may note that the wavelength=hbar/p depends on the magnitude of p. In actuality, p has a direction as this is space sampling. If px is considered to be information, then ln(probability(x))= px or probability x = exp(ipx). To obtain time sampling, one needs to freeze out motion i.e. exp(ipx)exp(-ipx) (AND situation).

    Thus the manner in which one samples an underlying distribution may give rise to a different distribution. Arguing that a particle spends the same amount of time in each dx region and so the spatial probability is constant completely ignores the form px from the classical action. 

    exp(ipx) is associated with a wavelength, thus x samples are within different lengths for different p’s. This suggests an associated entropy which is not additive because one loses information when adding p1+p2. Probability exp(ip1x)exp(ip2x) = exp(i (p1+p2)x), however, is conserved as is the overall impulse p1+p2 which depends only on the net momentum, whereas p1 entropy depends on a specific value.