The Quantum Mechanical Nature of the Law of Universal Gravitation and the Law of Coulomb s Interactions

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1 The Quantum echanical Natue of the Law of Univesal Gavitation and the Law of Coulomb s Inteactions Fayang Qiu Laboatoy of olecula Engineeing, and Laboatoy of Natual Poduct Synthesis, Guangzhou Institute of Biomedicine and Health, The Chinese Academy of Sciences, Guangzhou, 50663, China qiu_fayang@gibh.ac.cn Abstact With the intoduction of a complex numbe descibing a matte, the law of univesal gavitation and the Coulomb s law of electostatic inteactions can be deived fom the supeposition of matte waves. oe specifically, matte inteactions can be descibed via the supeposition of the associated de Boglie waves and thei natue is detemined by the quantum numbe l of the supeposing waves. Key Wods Univesal Gavitation, Coulomb's Inteaction, Quantum echanics, atte Wave, atte Repesentation Intoduction Cuently, the theoy of quantum mechanics is consideed to be the foundation of moden science and technology. Howeve, the natue of the univesal gavitation and Coulomb s electostatic inteactions ae not well undestood o intepeted by the fundamental pinciples of quantum mechanics. Thus, the natue of both the univesal gavitation and Coulomb s inteaction emains empiical. We believe that the univesal gavitation and Coulomb s inteaction can only be undestood on the basis of coect epesentation of matte stuctues. In this pape, we wish to epot ou initial wok on this impotant topic. The Wave Packet Stuctue of a atte Ou knowledge to the existence and behavio of a matte is obtained though the inteactions with it. If we do not know how to inteact with a matte, we simply do not know its existence. Theefoe, matte inteactions ae of paamount impotance in ou knowledge of a matte. The question is: How does one matte inteact with anothe? What govens the inteaction? To answe these questions, we must fist know the stuctue of the matte.

2 In quantum mechanics, the state of a paticle moving in a foce field defined by the potential function V is descibed by the SchÖdinge equation: h μ ψ + Vψ = Eψ () whee h is the Planck constant h ove π, E the total enegy, μ the educed mass and ψ the associated wave function of the system. If the mass of the foce field is much lage than that of the paticle, then μ can be substituted by the est mass of the paticle. Obviously, the motion of the paticle is descibed with espect to the cente of the foce field. So the kinetic and potential enegies ae the outcome of the inteactions between the paticle and the foce field. Now, conside a system that contains only one paticle. In this case, the paticle is fee and we must use the paticle itself as the fame of efeence to descibe its motion. Thus, the potential enegy and the total enegy must be zeo. Hence the kinetic enegy must also be zeo. As a esult, the SchÖdinge equation becomes h ψ = 0 () Rewite this equation, we have Wite then ( h ψ ) = 0 (3) ψ ' = h ψ (4) ψ ' = 0 (5) which is the well-known Laplace equation. This simple deivation indicates that the desciption of the motion of a fee paticle is the Laplace equation. It is shown in many standad textbooks that equation (5) has a geneal non-zeo solution in the spheical pola coodinates: ψ ' = ( a l + a l )P m l ( cos θ )e im φ (6)

3 whee a and a ae constants that can be detemined unde pope bounday conditions; both m and l ae quantum numbes, m l and l = 0,,,... ; P l m (cosθ ) is the associated Legende polynomials. On the basis of ou knowledge, ψ ' 0, then we have to set a = 0 to satisfy this physical equiement. Thus, ψ ' = a l P l m (cos θ )e im φ (7) Combining (7) with (4) and witing a = f, we obtain ψ = l m f Pl (cos θ ) e h imφ (8) It can be seen fom equation (8) that ψ is popotional to the est mass of the paticle, which is in ageement with ou knowledge. Thus, it can be undestood that ψ is the wave function descibing the matte waves associated with the paticle of est mass. Obviously, the wave function has a quantum stuctue. The absolute magnitude, ψ, is the intensity of the wave and the contous of ψ vs.,θ and φ ove all pemitted l and m values in equation (8) descibe the stuctue of the wave packet associated with the paticle. Since the solution of the Laplace equation is independent of the magnitude of the mass o the size of the paticle, the change of the paticle mass does not have any effect on the stuctual featues of the associated matte waves, though the intensity of the associated matte waves will change with the magnitude of the mass. Hence the wave packet stuctue of any fee object, fom an electon to a planet, must have the same stuctual featues descibed by the solution of the Laplace equation. The Supeposition of atte Waves: ψ ψ Accoding to de Boglie, matte can only be detected though the inteaction with thei associated waves. Thus, the inteaction between two objects occus when thei associated waves supepose. Whethe the inteaction foce is attactive o epulsive depends on the outcome of the wave supeposition. If the waves supepose constuctively, the two objects will attact each othe; if the supeposition is destuctive, thei inteaction foce will be epulsive.

4 Let ψ and ψ be the wave functions of two paticles with est mass and, and chage Q and Q, espectively. Thus, the intensity of the inteaction between the two paticles will be popotional to the absolute magnitude of both ψ and ψ, o, thei conjugate poduct: ψ ψ. Accoding to equation (8), we have m ψ * ψ l cos, l k P ( l + l + ) l l im φ m im φ ( cos θ ) e P ( θ ) e = (9) To obtain the intensity of an inteaction between two paticles sepaated by a distance, it is necessay to know both the l and the m values involved in the inteaction. If l = l = 0, then m = m = 0. This means that only one type of inteaction with the given quantum numbes is pemitted. This pemitted inteaction is the long ange inteaction due to the fact that it is least sensitive to the change in distance between the two paticles. The magnitude of this special type of inteaction is F l = 0 ψ * ψ l= 0 = G (0) whee G is a popotional constant detemined by the bounday conditions. Equation (0), which is the same as Newton's Law of Univesal Gavitation, is the consequence of the supeposition of a special type of waves associated with the two inteacting paticles. This esult demonstates that Univesal Gavitation has the natue of wave supepositions! Although ψ ψ has been teated as the pobability density of finding a paticle in a designated aea, it is, in fact, the intensity o the foce of the inteaction between two objects. Since this foce is quantized, it povides the gound fo the cuent undestanding of the quantum pobability intepetation. atte Repesentation Since Coulomb s law of an electostatic inteaction has the same stuctual patten of mathematics as Newton s law of univesal gavitation, we believe that Coulomb s law shall have the same quantum mechanical oigin, i.e., the natue of quantum wave supepositions. Howeve, (8) does not contain a chage tem. Theefoe, the desciption of a matte with mass alone is incomplete and it is necessay to modify ou geneal desciption of matte, i.e., matte shall be descibed with both mass and chage. Since the inteaction between mass and chage cannot be obseved expeimentally, those two physical quantities may be combined though a

5 complex numbe: ( + iq), whee Q is the chage associated with the matte of est mass, which can be eithe positive o negative. Thus, afte eplacing with ( + iq), (8) becomes ψ = + iq l m f Pl (cos θ ) e h imφ () As a consequence, the foce of the long ange inteaction between two objects, i.e., the supeposition of thei coesponding matte waves with quantum numbes l = l = 0, and m = m = 0, can be epesented as F ( + iq )( iq ) = G + iq i Q + QQ = G QQ iq i Q = G + G + G () l= 0 ψ * ψ l= 0 The fist tem is the foce of univesal gavitation; the second tem is the foce of Coulomb inteaction; while the thid tem is an imaginay numbe, which may be undestood as the matte field of the entie newly fomed two-paticle system. At this point, it is clea that both the univesal gavitation and the Coulomb inteaction ae the consequence of the supeposition of matte waves. In a micoscopic inteaction, since QQ G << G, it is appopiate to neglect the foce of univesal inteaction; while in a QQ macoscopic system, we haveg >> G. Thus, it is easonable to ignoe the Coulomb inteaction. As we can see, the poposed complex epesentation of a matte will poduce temendous implications to ou undestanding of quantum mechanics. Fo example, it is now easy to undestand why the inteaction of the chages with the same sign is epulsive while the inteaction between the chages of diffeent sign is attactive. Conclusion The intoduction of a complex numbe ( + iq) povides a complete desciption of a matte. We have discoveed that the supeposition of matte waves, i.e., the conjugate poduct of the two wave functions associated, espectively, with the two paticles epesents the foce of an inteaction between the two paticles. With this complex epesentation, we have

6 established the quantum mechanical foundation fo both the law of univesal gavitation and the law of Coulomb inteaction. They ae the outcome of the supeposition of the long-ange matte waves associated, espectively, with the inteacting paticles.

Physics 235 Chapter 5. Chapter 5 Gravitation

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