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1 Quantum mechanical force and derivation of
gravitational constant through quantum theory
Kapil Chandra*
Institute of Pure and Applied Physics, Faraswani, Dabhara, Janjgir – champa, 495688, India
Abstract
We derived a new representation of force through the basic assumption of Bohr’s atom
model, the quantum mechanical force (QMF) which is quantized and is in term of Planck
constant. Thus if we compare the QMF and classical Newton gravitational force, it gives the
universal gravitational constant in term of Planck constant and other conventional constants.
Finally, it has been found that the theoretically calculated value of universal gravitational
constant is comparable to its experimental values and this show the compatibility of gravity with
the quantum theory or vice versa.
PACS 03.65.-w, 04.90.+e, 04.60.-m,
*
[email protected] 2 It is believed that, quantum theory is most promising theory to describe almost physical
phenomena of nature, but gravity is still apart to it. Since quantum theory can explain several
atomic phenomena successfully and mostly tested at atomic scale while gravity governs whole
universe and it has influence at cosmos level. However this was quite a challenge to link
classical gravity with quantum theory. Here we derived the universal gravitational constant
through the quantum theory and this show the quantum mechanical origin of gravity.
On order to derive the universal gravitational constant through quantum theory we used
the quantum theory of atomic model, where many other fundamental forces comes in play,
through this model it is well known that there are some certain levels of energies which are
called the orbitals, where electrons are circulating freely around the nucleus and experiences
certain amount of electrostatic force due to positively charged nucleus whereas electrons remain
negative. While these energy levels are quantized [1] as Bohr claimed however, we assume that
the force experienced by these orbital’s electron may also be quantized and this force can be
derived through the central postulate of Bohr’s atom model [1, 2]. It stated that, angular
momentum of orbitals electrons are quantized and it was represented as in following
mathematical expression,
(1)
where, m is the mass of circulating electrons with supposed velocity c in certain radius of orbital
r and h is the Planck’s constant. Empirically this equation can be written as following,
2
2
2
(2)
clearly, this is a force balance equation, the first term represent the force which is only depends
on the distance between nucleus and orbital or orbital radius r and it is quantized as well
however we can say that orbitals are certain level of forces such as Bohr’s model suggested the
possibilities of certain energy levels, hereafter this new representation of force is called the
Quantum Orbital Force or Quantum Mechanical Force (QMF), if it is satisfied then one can say,
the Bohr’s postulate of quantization of linear momentum of orbitals electrons can be deduced by
comparing the QMF and centripetal force or vice versa.
The Eq. (2) is very crucial and might be very important in some other interests, when we
substitute
2 , it will give the quantum force carried by wave on incident surface, i.e. in the
case of electromagnetic waves, this gives the quantum pressure of electromagnetic wave [3-5]
but its discussion is disregarding here and will be discussed in detail somewhere else.
Now, the Eq. (1) can be rearranged when we separated the constants and variables,
3 (3)
one can seen that, the mass of the particle m moving in any orbit of radius r is a constant, here
we omitted the numerical constant 2 . The protons are found in nucleus and the electron are in
electronic orbitals however this equation can be applied for either particles, if we replace the r
and m with radius of nucleus and mass of proton
and thus one can get,
2.2
10
42
(4)
where,
1.2 10 15
and
1.67 10 27
and used constant has their usual
accepted corresponding values, here it’s remarkable; from computation we found the numerical
value of first term is similar to the second term.
Now, we replace all the concerning values for electron in Eq. (4), we will get
,
where
5.3 10 10 and
9.1 10 31 , the first atomic orbital radius for hydrogen
like and mass of electron respectively and we noted that both side are not equal, however we
used an additional constant to equalize both side, suppose that is , thus it will turn out now
and from computation we got
4.5
10
since we know the value of other
used constants, however Eq. (4) can be written as following,
(5)
the first and second terms of this equation are stands here for the representation of nuclear force
and electrostatic force respectively since we used in and , by analogy of electrostatic force,
we assume that gravitational force also exist there and it also play role to hold together the
protons and electrons however Eq. (5) will turn out as following,
(6)
where,
is the mass of particle and
is distance between proton and electron or electrons
orbital’s radius and let say k is any numerical constant, it is similar to what we used the constant
in case of electrostatic force to equalize both sides of equation as shown in Eq. (5) and (6) and
since its value has been determined theoretically, following same way we will determine the
numerical value of used constant k. The used representation is following,
(7)
we have difficulties with it to determine the value of constant k since we don’t know the value of
and
separately, however we assume that the both side of equation must be equal and this
can be done easily when we take numerically
1, thus the numerical value of k can be
determined when we substitute the corresponding values of used other constants and we will get,
4 2.2
10
(8)
Now, the Eq. (6) is a general form of the Eq. (1) which is derived through the force
balance Eq. (2), but this equation has the representations for many other forces such as first,
second and third term are stands for nuclear, electrostatic and gravitational force respectively.
However if we replace the r by
it gives the numerical amount of QMF for nucleus and this
will be equal to the nuclear force, therefore the amount of force will be,
(9)
substituting values of all constants and , we equated the nuclear force 2.19 10 , but we
know that the nuclear force is basically due to mutual massive particle exchange and it is a non
central force thus we replace
and followed by
thus we will get,
1.8 10
here we used additional numerical constant 4
and this will be justified
somewhere else, and this represents the non central since it is independent of distance or nuclear
radius and we found magnitude of force is similar in both case.
In similar manner the second term of Eq. (6) which represents the electrostatic interaction
between the proton’s and electron’s charge and the force can be calculated while replacing r with
, the first atomic radius for Bohr’s atom in Eq. (2), here we retain the used numerical constant
α which equalize the QMF with electrostatic force but this electrostatic force can also be
determined form classical electrodynamics however the value of can be calculated if,
2
where, ,
and thus,
2
(10)
2
are charges of proton and electron respectively and
0
is permittivity of vacuum,
(11)
this is a well know quantity in atomic physics, its fine structure constant [6] and a coupling
constant for electrostatic interaction [7]. This constant can also be determined when we take first
two term of Eq. (6) i.e.
can be equated by
, is an alternative equation to equate it however nucleus radius
1836
where
1836
.
From this description it has been found that the used constant is a coupling constant
and it showed the relative force strength of electrostatic force to nuclear force, if we follow this
line, the used constant k be relative strength of gravitational force to strong nuclear force because
of it was used similar way to what we used constant in case of electrostatic force and thus,
5 (12)
rearranging it, we will get,
(13)
but we already know the k from Eq. (8) and in case, suppose if we don’t have the value of G, the
Newton’s universal gravitational constant, however this can be theoretically calculated by
following equation,
(14)
if we insert
this equation will be turn out,
(15)
while substituting corresponding values of all used constants and
we computed the universal
2
2
gravitational constant
4.596 10
whereas the experimental value is
6.674 10
one can conclude that the calculated and experimental values of
universal gravitational constant [8] are quite comparable, this showed, the derived equation is in
good agreement with the experimental findings.
In sum, we derived a new representation of force, the quantum mechanical force through
the basic postulate of Bohr’s atom model. This suggested that the orbitals are certain quantum
level of forces and this force is only function of orbital’s radius. While comparing the classical
gravitational force and quantum mechanical force it gives us the gravitational constant in term of
other familiar constant. It has been reported the theoretically calculated value of universal
gravitational constant is comparable to its experimental values and these finding suggested the
compatibility of macroscopic theory, the gravitation with the microscopic theory, the quantum
theory and it may help to unify the gravity with quantum mechanics.
Acknowledgements we thank to H.S.Tewari for his excellent explanation of atomic
model and his critical comments on preparation of this manuscript. We would like to also pay
thanks to Aditya for proof reading to this manuscript.
REFERENCES:
[1]
[2]
[3]
[4]
[5]
Bohr N., philo. mag. 25, 1 (1913). Cox P.A., Introduction to Quantum Theory and Atomic Structure (Oxford University
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[7]
[8]
Haken H., Wolf H.C., The Physics of Atoms and Quanta (Springer-Verlag, Heidelberg,
2005) pp. 116-117 Kaku M., Quantum Field Theory (Oxford University Press, Oxford, 1993) pp. 5-6
Mohr, P.J., Taylor, B.N., Nowell, D.B., Rev. Mod. Phys., 80, 633 (2008)