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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 Press,Oxford, 1996), pp. 58-67 Ohshima, Y.N. et al, Phys.Rev.Lett., 78, 3963 (1997) Ashkin, A., Phys. Rev. Lett., 24, 156 (1970) Laudon, R., Barnett S.M. and Baxter, C., Phy.Rev A, 71, 063802 (2005) 6 [6] [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)