Abstract
We report on a proof to the Heisenberg inequalities, for both vector-like and scalar-like variables, that is based on statistical dependence of quantum events on appropriate Venn diagrams. A similar proof is provided for the “energy-velocity” uncertainty principle of Haidar (2010).
Highlights
Physical parameters of a particle, such as momentum p, position r, energy E and time t, are conceivable in quantum mechanics as values of corresponding random variables that we shall denote respectively as P, X, E and T
We report on a proof to the Heisenberg inequalities, for both vector-like and scalar-like variables, that is based on statistical dependence of quantum events on appropriate Venn diagrams
A similar proof is provided for the “energy-velocity” uncertainty principle of Haidar (2010)
Summary
Physical parameters of a particle, such as momentum p, position r, energy E and time t, are conceivable in quantum mechanics as values of corresponding random variables that we shall denote respectively as P, X, E and T Their pertaining probability density functions fP(p), fX(r) , fE(E) and fT (t) satisfy the well known defining condition. Comparison of ,in the context of (2) or (3) with ∥ψ∥2 in (5) indicates that the sense of these closed inequalities appears to be the uniting feature between their physical and mathematical aspects and not their right hand side limits This fact is supported further by Bohr’s reformulation (Fermi, 1961) of the HUP to the situation when △ represents a rather unspecified generalized measure of size, and not necessarily the standard deviation as in the work of Kennard,1927. It should be noted here that G and R can have the same geometrical units in an abstract Venn diagram space (of arbitrary size) despite the fact that they represent different variables in the real physical space
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