13. Deriving Newton’s gravitational law from the Heisenberg’s uncertainty principle

13.1 Introduction ...................................................................................................... 89. 89
13.2 Quantum fluctuation. 90 ....................................................................................... 90
13.3 Conclusion . 91....................................................................................................... 91


In this section we are going to see how the Newton’s gravitation and the Heisenberg’s uncertainty principle are two different aspects of the same formula. Both formulae were obtained at different times and by different methods but with the same formula.

13.1 Introduction

The universal gravitational law is physics classic law which describes the gravitational interaction between different bodies with mass.
The Heisenberg’s uncertainty principle establishes the impossibility of letting certain physical magnitude pairs to be known with arbitrary accuracy.
The Heisenberg’s indetermination is applicable to pairs of physical variables, such as energy and time. In this case:
(13.1)
In the limit (minimum product):
(13.2)
Identic equation to the Planck’s energy, 
                                               (13.3)
except for the factor ½ where Ep is the Planck energy and tp is the Planck’s time.
The uncertainty does not arise from the measurement devices but from the mere fact of measuring. Until now the principle could only be measured in the subatomic scale, where the rules of the quantum mechanics are more evident and decisive.
Recently it has been possible to detect the effects of the uncertainty principle in a concrete macroscopic object, in a small drum with half millimeter length [41]. 
 Nonetheless, the principle vanishes when it is applied to the gravity, although from the Newton’s gravitational law the Heisenberg’s uncertainty principle can be deduced as detailed hereafter.
The Newton’s gravitational law is deduced by the observation, whereas the Heisenberg’s indeterminacy relation is deduced from the quantum mechanics postulates.
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13.3 Conclusion

The Heisenberg indeterminacy and the Newton gravitational are two different aspects of the same formula.
The Newton’s gravitational is only valid for masses at rest, hence it fails it some cases since the total energy of the mass (at rest and at motion) is not taken into consideration.

Figure 13.2 Minimum distance at which m is constant.
The Newton’s gravitational and the Heisenberg’s uncertainty principle are the same formula, yet obtained by different methods at different times and expressing different concepts.
In the QM, the reduced Compton wavelength is a representation of the mass and arises in many of the QM equations, being the Schrodinger equation an example.

The reduced Compton wavelength  can be regarded as the minimum distance that we can measure and consider that the mass is m.

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