Heisenberg Uncertainty Reloaded

The actual philosophy is to introduce appropriately the quantum fluctuation information a = a(x ) respecting the average of the observed coordinate (Xg), by the Fe3mman integration rule founded in the ordinary quantum 

It is obvious that the Eqs. (4.536) and (4.537) fulfill the necessary (natural) condition according which the average of the coordinate over the quantum fluctuations recovers the observed quantity of Eq. (4.535), 

The next test is about the validity of the Heisenbeig uncertainty relationship (HUR) itself To this end the standard deviation of coordinate (x) and momentum (p) 

In the same manner, the evaluations for the integrals of Ihe first and second orders of kinetic moment unfold as 

Worth noting is that from the coordinate and momentum dispersions, Eqs. (4.543) and (4.546), it appears that the dependency of Planck constant is restricted only to the latter, whereas the quantum fluctuations are in both present, in a direct and inverse manner, respectively. 

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