The Electrostatic Nature of Gravitation

Transcription

1 The Electostatic Natue of Gavitation 1 National Space Agency of Ukaine, Kyiv, Ukaine V. K. Gudym 1 & E. V. Andeeva Applied Physics Reseach; Vol. 5, No. 6; 013 ISSN E-ISSN Published by Canadian Cente of Science and Education Institute of Semiconducto Physics of the NASU, Kyiv, Ukaine Coespondence: V. K. Gudym, National Space Agency of Ukaine, Kyiv 01010, Ukaine. vgudym@mail.u Received: August 5, 013 Accepted: Septembe 30, 013 Online Published: Novembe 5, 013 doi: /ap.v5n6p93 URL: Abstact In wok on the basis of static model of a stuctue of atoms it is shown that the gavitational and electostatic inteactions have the same natue. The electostatic inteaction is the inteaction of electic chages at small distances, wheeas the gavitational inteaction is the inteaction of electic chages at lage distances. Keywods: the gavitation natue, a binomial potential, static model of atom, the scatteing of electons by potons 1. Intoduction At the pesent time, the gavity emains the most puzzling phenomenon of the Natue. The paadox consists in that the gavitation involves all subjects existing in the wold, fom the Univese itself and to its micocomponents. Howeve, no knowledge of the physical essence of the gavitation and a elation to othe known inteactions is available. Thee exist a lot of opinions on this question. We pesent one of the most chaacteistic thoughts. The answe of Cave ead to the question: Do you have any thoughts about gavitation? was we basically have no clue what it is. It doesn t fit with any of the othe theoies (ead, 001). We believe that the ovecoming of difficulties in the compehension of the gavitational inteaction equies to eject completely the planetay model of the stuctue of atoms and molecules. Instead of the planetay model, we popose a model based on the binomial potential of the electon-poton inteaction (Gudym & Andeeva, 007, 008). In essence, we popose a static model of an atom, in which an electon and a poton and, in the geneal case, the nucleus and atomic electons, undego the action of two oppositely diected foces. One of the foces is esponsible fo the attaction of electic chages, and anothe one epulses them. These foces ae equal in modulus at some distance fom the nucleus which can be called the size of an atom. oeove, the foces of epulsion in ou model with the binomial potential ae shot-ange foces. Theefoe, the inteaction between an atom as a whole and emote chages outside of the atom is mainly attactive, because the shot-ange epulsive foces at lage distances ae significantly less than the attactive ones. Hence, the electically neutal atom, molecule, o any maco body, while inteacting with electic chages of the othe analogous systems, undego only the electostatic attaction at distances much moe than the size of an atom. Just this is the essence of the poposed mechanism of gavitational inteaction. Since the physical basis of the appeaance of the gavitational inteaction is given by the binomial potential of the inteaction of an electon and a poton, ou main task was to demonstate the eality of the binomial potential by specific examples. To this end, we consideed the classical poblem on the motion of an electon in the field of the binomial potential of a poton and solved the Schödinge equation with this potential (Gudym & Andeeva, 008). Then we calculated the enegies of ten fist atoms of the endeleev Peiodic table within the model of the binomial potential (Gudym & Andeeva, 008). The esults of calculations suppoted ou confidence in the eality of the binomial potential. Then, on the basis of models of electically neutal atoms (Gudym & Andeeva, 008), we developed a model of inteaction of these atoms and fee chages. The study of nea-atomic egions within those models allowed us to suely conclude that any electically neutal atom and any electic chage positioned fom it at any distance attact each othe. 93

2 Applied Physics Reseach Vol. 5, No. 6; 013 Hence, the foces of attaction acting between physical bodies (gavity foce) ae a esult of the electostatic inteactionn of electically neutal atoms with electicc chages of othe, also electically neutal, atoms.. Substantiation of the Binomial Potential As a basiss of the binomial potential of the inteaction of an electon and a poton, we took the indisputablee fact that the electons in an atom ae located at some distances fom the nucleus. In this case, the inteaction potential fo the electon and the poton in a hydogen atom can be pesented by the function e Г V x, (1) whee the fist and second tems on the ight-hand side epesent, espectively, the Coulomb inteaction andd ou hypothetical inteaction which counteacts the Coulomb attaction. To detemine the constant Г and the exponent x, we constucted a system of two algebaic equations coesponding to the equilibium (gound) state of a hydogen atom accoding to the scheme in (Bon, 1946) ): e Г E 0, x x1 0 0 Hee, Е 0 is the gound state enegy of a hydogen atom, 0 is its equilibium adius, and e is the electon chage. The second elation in () is the sum of the foces acting on the electon in the gound state. The solution (Appendix A) of system () fo Г and x gave the following esults: Г= = CGSE units and x= (3) Geneally speaking, within the model of binomial potential, we eplace, in fact, the postulate about some motion of an electon aound the nucleus, which is extaneous to classical mechanics, by a moe natual hypothesis about the pesence of a epulsive potential. oeove, a paticula attention should be given to the cicumstance that the poposed binomial potential (1) does not contadict the foundations of quantum mechanics (Gudym & Andeeva, 008) and open a way of classical mechanics to many-electon systems. e x Г. () Figue 1. Inteaction enegy of an electon and a poton the distance between them (1), the Coulomb component (), the hypothetical component (3) In Figue 1, we pesent the dependence of the inteaction enegy of an electon and a poton inn the binomial-potential model on the distance between them. It is obvious that the Coulomb component dominates in the total inteaction enegy aleady at distances of the ode of magnitude of two equilibium adii (~ cm). But at small distances, appoximately at half of the equilibium adius (~ cm), the epulsive component becomes dominant. Hence, the positive addition to the Coulomb tem in (1) epesents a shot-ange 94

3 Applied Physics Reseach Vol. 5, No. 6; 013 foce. Appaently, just in this way it is possible to explain the fact that no deviations fom the Coulomb law wee found at maco distances. 3. The Two-Body Poblem with Binomial Potential On the next stage of ou study, it was necessay to veify the eality of the binomial potential (1). With this pupose, we consideed the two-body poblem with the binomial potential and solved the equation (Gudym & Andeeva, 007). E m m e Г, (4) whee m is the electon mass, Е is the total enegy of the system, М is the angula momentum, and is the deivative with espect to time. The solution (Appendix B) of Equation (4) fo a tajectoy has the fom P 1 sink, (5) E mг mг mг whee 1, P, k. (6) 4 me me In the geneal case, Function (5) epesents two types of motion, finite and infinite ones, by depending on the sign of the enegy in Equation (4). The fome is the motion of a bound electon in a hydogen atom, and the latte coesponds to the scatteing of an electon by a poton. 3.1 Hydogen Atom Let us conside fistly a finite motion. The solution of (4) diffes fom the well-known solution of the Keple poblem with the Coulomb potential only by the pesence of the coefficient k of the agument. In the geneal case, the pesence of this coefficient leads to the appeaance of a cicula motion of the peihelion. It follows fom elations (6) that if the positive tem in (1) is dopped, then k = 1, and the tajectoy coesponds to a conical coss-section with the focus at the oigin of coodinates. The nonzeo positive tem makes a value of the coefficient k to be diffeent fom 1. This means that the tajectoy of motion will be unclosed in the geneal case. Only sepaate values of the coefficient k will satisfy the conditions of closedness. In othe wods, if the electon with any enegy in the Coulomb field can move only along a closed obit, its motion along a closed obit in potential (1) can be ealized only at stictly definite values of the enegy (Bon,1946). By solving Equation (4) unde the condition E < 0, we succeeded to obtain the disceteness of enegy levels, Balme fomula, and Planck s constant and elation (Gudym & Andeeva, 013, 007). 3. Scatteing of Electons by Potons If the total enegy (4) is positive, then the motion tajectoy of an electon in the cental field of a poton will be an unclosed cuve, whose beginning and end ae at infinity. Since the motion is infinite in this case, it is convenient to intoduce the so-called impact paamete ρ instead of the constant angula momentum : me. (7) To descibe the tajectoy of motion of the electon scatteed by the poton, we substitute (7) in elations (6) and pass to the Catesian coodinates x cos, y sin (8) whee the values of should be detemined by Fomula (5). In Figue, we show the tajectoies of motion of scatteed electons with enegies of 400, 188, and 40 ev calculated by (8). It is seen that if the motion occus in the Coulomb potential field, no scatteing of electons with such enegies is obseved. 95

4 Applied Physics Reseach Vol. 5, No. 6; 013 Figue. Tajectoies of motion of electons withh enegies of 400, 188 and 400 ev scatteed by a poton. Calculations ae pefomed by fomula (8) fo impact paametes ρ: 1) ; ) ; 3) ; 4) cm; 5) a tajectoy of motion of an electon in the Coulomb potential The deviation angle χ of a paticle moving nea the scatteing cente can be pesented as whee the angle φ 0 is given by the integal taken between the position of the paticle which is the closest to the scatteing cente and the infinity (Lavich, 1969). The value of min is a oot of the adicand. With egad fo Relation (7), Fomula (10) takes the fom By substituting potential (1) in Fomula (11) and caying out the integation, we obtain 0b 0 min E accos E 0 min d me U() d. 1 U ( ) 1 1 E Hee, the lette b in the index of the angle φ 0b, means that the angle is detemined within the binomial-potential model (1). Thus, we obtained the analytic fomula fo the deviation angle χ fo the electon scatteed by the poton as a function of its kinetic enegy E and the impact paamete ρ unde the assumption that they inteact by thee law with the binomial potential (1). If we set Γ = 0 in Fomula (1), which coesponds to the Coulomb law fo f of the electon-poton inteaction (Levich, 1969) we get E( E ) 1 4 e,(9) (10) (11) (1) 96

5 Applied Physics Reseach Vol. 5, No. 6; 013 0k accos Hee, the lette k in the index of the angle φ 0k, indicates that the deviation angle is detemined fo the Coulomb scatteing. Hence, the diffeence of the fomulas deived with the help of the Coulomb and binomial potentials consists, basically, only in the pesence of the coefficient k in (1). In the geneal case, Fomulas (1) and (13) togethe with (8) allow one to calculate the deviation angle χ fo a scatteed electon as a function of its kinetic enegy and the impact paamete in both cases whee the electon inteacts with the poton by the binomial law (1) and the Coulomb law. In the expeimental studies of the pocess of scatteing, the so-called effective scatteing coss-section dσ is used. u Since elation (1) establishes the bijective inteelation between the deviation angle χ and the impact paamete ρ, the effective scatteing coss-section efeed to a solid angle element dθ can be pesented as (Levich, 1969) d d d. (15) d 1 4E 1 4 e E E sin (13) (14) Figue 3. Compaison of the calculated effective coss-sections fo electons with enegies of 400 and 188 ev with expeimental values: points expeimen (Hofstadte, 1956); cuves calculated data Let us intoduce the notation k0 ( )/. Thee Fomula (1) yields 4 e ctg 4E E (16) 97

6 Applied Physics Reseach Vol. 5, No. 6; 013 By substituting the esult of diffeentiation of Fomula (16) in the fomula fo the effective scatteing coss-section (15), we obtain 4 e 1 d d. (17) 16E 4 sin By the fom, this fomula epeats exactly the Ruthefod fomula (Levich, 1969). But fomula (17) diffes basically fom the Ruthefod one in the definition of the angle χ with eagad fo coefficient (14): (18) In Figue 3, we compae the cuve calculated by Fomula (17) fo the effective scatteing coss-section of electons with enegies of 400 and 188 ev with the expeimental data fom wok (Hofstadte, 1956). Geneally speaking, it is the unexpected esult. Indeed, it is commonly accepted (Hofstadte, 1956; Levich, 1969) that the Ruthefod fomula must not wok in the egion of elativistic enegies. Howeve, the whole sequence of above-pefomed mathematical opeations coesponds exactly to the deivation of the Ruthefod fomula fo the Coulomb potential and indicates that the uses of the binomial and Coulomb potentials yield the same fomula fo the effective scatteing coss-section. The situation becomes diffeent, if we calculate the tajectoies of motion of an electon. It follows fom Figue that the esults of calculations of the tajectoies of motion of high-enegy electons in the Coulomb field ae unphysical. On the contay, the tajectoies of motion of electons with any enegy within the binomial-potential model ae quite eal. Thus, we manage to deduce a single fomula simila to the well-known Ruthefod one fo the calculation of the tajectoies of motion and the scatteing angles of electons by potons fo the entie ange of enegies of the scatteed electons fom seveal ev up to ulta elativistic enegies and with impact paametes down to cm. It is woth sepaately noting that ou fomulas do not equie any modification within the othe theoies fo diffeent enegies (Hofstadte, 1956; Levich, 1969), which is necessay in the Ruthefod appoach. Hence, we may conclude that the binomial potential of the inteaction of an electon and a poton eflects moe exactly the chaacte of foces acting between an electon and a poton, than the Coulomb potential. 4. Static odels fo Ten Fist Atoms of the endeleev Peiodic Table To suppot the eality of the binomial potential, we poposed static on it to basis models of ten fist atoms of the endeleev Peiodic table within ou model in the fom of geometic plane o volumetic stuctues, whee the nucleus is located at the cente of the stuctue, and the electons ae positioned at its vetices (Figue 4, Gudym & Andeeva, 008). To veify the validity of such models, we calculated the electon inteaction enegy fo each of those atoms. As an example, we pesent the fomulas fo some atoms. Helium atom: E 4e e A 0 0, (19) R R whee E 0 is the electon inteaction enegy; is the distance of electons fom the nucleus; R is the distance between electons; and A is the magnetic inteaction constant (A = , Gudym & Andeeva, 008). Beyllium atom: E 8e 8e 4e e e 4A A A R 1 R 1 (0) whee E 0 is the electon inteaction enegy; R is the size of a homb edge; 1 and ae the distances of oppositely located electons fom the nucleus which ae elated to the homb edge as follows: 1 = Rsinα and = Rcosα; and α is the angle between an edge and the majo diagonal. To veify the validity of such models, we calculated the electon inteaction enegy fo each of those atoms. As an example, we pesent the fomulas fo some atoms. 98

7 Applied Physics Reseach Vol. 5, No. 6; 013 Helium atom: E 0 4e e A, R R whee E 0 is the electon inteaction enegy; is the distance of electons fom the nucleus; R is the distance between electons; and A is the magnetic inteaction constant (A = , Gudym & Andeeva, 008). ( 19) Figue. 4 Static models of atoms; ed filled cicle atomic nucleus; blue filled cicle electon Beyllium atom: E 04 8e 8 e 4 e e R e 4 A A R 1 1 whee E 04 is the electon inteactionn enegy; R is the size of a homb edge; 1 and ae the distances of oppositely located electons fom the nucleus which ae elated to the homb edge as follows: 1 = Rsinα andd = Rcosα; and α is the angle between an edge and the majo diagonal. Oxygen atom: E 08 48e 16e 6 6e 6e 3e sin 1 sin 6 3 1e e 1A 6A 6A 3A R R 1 1 sin 1 sin 6 3 whee E 08 is the electon inteaction enegy in an oxygen atom; 1 is the distance of electons located at vetices 99 1 A 1 A (0), (1)

8 Applied Physics Reseach Vol. 5, No. 6; 013 of the pyamid base fom the nucleus; is the distance of electons located at vetices of the pyamids fom the nucleus; R is the length of a lateal edge of the pyamid. The values of R, 1 and ae connected by the elations: 1 = Rsinα and = Rcosα, whee α is the angle between a lateal edge and the pyamid base. The esults of calculations of the binding enegies of electons in each of the poposed models of atoms ae given in Table 1. Table 1. Compaison of the theoetical and expeimental values of binding enegies of electons in atoms. Elements H He Li Be B C N O F Ne exp E, ev theo Eo, % Thus, the possibility to constuct the static models of atoms by the same pinciple with the same constants testifies, in ou opinion, to the eality of the binomial potential. 5. A odel fo the Study of Nea-Atom Regions odels fo the study of the inteaction of electically neutal atoms with fee chages wee constucted on the basis of the models poposed by us (Gudym & Andeeva, 008), fo neutal atoms (Figue 4). Fo example, if we imagine that the hydogen atom consists of a poton and an electon located at the distance 0 fom it, then the model of the inteaction fo such a system with a fee electon e 0 o a poton p 0 can be pesented in the fom shown in Figue 5. The electon inteaction enegy E is as follows: e e e A 0 0 R R E. () whee A is the constant of the magnetic inteaction of two electons (Gudym & Andeeva, 008); R is the distance between the electon e 0 o the poton p 0 and the atomic nucleus; 0 is the equilibium adius of a hydogen atom; 0 R 0Rcos (3) and α is the angle between the fixed coodinate axis and the diection R to the fee electon e 0 o the poton p 0. The inteaction enegy E of the poton of the hydogen atom is given by the fomula e e e E R R 0 0 ( ' ) Figue 5. odel of inteaction of a neutal hydogen atom with tial chages: e 0 -electon and p 0 -poton 100

9 Applied Physics Reseach Vol. 5, No. 6; 013 Figue 6. Inteaction enegy of a neutal hydogen atom with tial chages (1, 3-electon and, 4-poton) the distance between them fo angles: 1, -9 o, 3, o Thus, by using Fomulas () and (') and setting values of R and α, we have the possibility to calculate the total enegy of the hydogen atom fee chage system fo evey point, whee the extaneous electon e 0 o poton p 0 is located in the nea-atom egion. The sign and the modulus of the calculated enegy will indicate the chaacte of the inteaction of the atom with a fee chage. If the sign is positive, then the inteaction is epulsive; fo the negative sign, the inteaction is attactive. In Figue 6, we demonstatee how the enegy of the neutal atom fee chage system vaies as a function of the distance of the fee chage fom the nucleus. It is clea that, by stating fom infinity and to distances of appoximately 10-8 cm, only the attactive foces act between a neutal hydogen atom and a fee chage independent of the sign of the fee chage fo all diections. Figue 7. odel of inteaction of a neutal beyllium atom withtial chages: e 0 -electon and p 0 -poton As anothe example of the model of inteaction of an electically neutal atom with a fee chage, we took a beyllium atom (Figue 7). Within this model, we wite analogously the inteaction enegy of an electon with w a atom beyllium in the fom whee 8e 8e 4e e e 4A A A E 1 1 R1 1 R1 1 4e e A R R (3) 101

10 Applied Physics Reseach Vol. 5, No. 6; R 1 R cos, 4 R R cos, 5 1 R 1R cos, 6 R R cos. (4)( Hee, R is the distance fom the atom cente to the fee chage, α is the angle between the diection fomm the atom cente to the fee chage and a fixed coodinate axis with the oigin at the nucleus (Gudym & Andeeva, 007). Values of R 1, 1, wee taken fom the calculation of the neutal atom. Fo the inteaction enegy of a poton with a beyllium atom, we can wite the fomula E 8e 8e 4e e e 4A A A 1 1 R1 1 R1 1 4e e R R (3 ' ) Figue 8. Inteaction enegy of a neutal beyllium atom with tial chages (1, 3, 5 electons,, 4, 6 potons) the distance between them. In the plane of the atom at angles: 1, 9 o ; 3, o ; 5 and 6 -pependiculaly to the plane of the atom In Figue 8, we pesent the vaiation of the inteaction enegy of fee electic chages with a beyllium atomm as a function of the distance between the fee chage and the atom cente fo diffeent diections to the fee chage. Cuves 1 and give the inteaction enegy of a neutal atom of beyllium with electic chages as a function of the distance fom the atom cente fo the angle α = 9 o. It is obvious that, at distances of the electic chage fom the atom cente moe than cm, the total enegy of the system is negative. Hence, in this egion, the neutal atom and electic chages iespective of thei chage ae attacted. An analogous patten is obseved fo all othe diections to extaneous electic chages (3, 4, 5, 6). Thus, the example with a beyllium atom also indicates that, by stating fom infinity and to distances of ~ cm, the electic chages with any sign undego only the mutual attaction with neutal atoms. An analogous attaction of fee chages and neutal atoms is obseved fo all ten fist atoms of the endeleev Peiodic table. It is obvious fom Figues 6 and 8 that the chaacte of the inteaction of fee chages with neutal atoms at 10

11 Applied Physics Reseach Vol. 5, No. 6; 013 inta-atomic distances is significantly moe complicated than that obseved at distances lage than cm. But the consideation of this poblem is outside of the pesent wok. 6. Conclusion The pupose of the pesent wok was to claify the natue of the gavitational inteaction. To this end, we developed a method fo the calculation of a numeical value of the inteaction enegy of the atomic system with tial chages located at vaious points of the nea-atom egion. Then we consideed the nea-atom egions fo ten fist elements of the endeleev Peiodic table. On this way, the analysis of electostatic fields pesent in the nea-atom egions fo electically neutal atoms allows us to undestand the eason fo the appeaance of the attaction between electically neutal bodies. Fo the complete compehension of the mechanism of appeaance of the gavitational inteaction, we need to conside this diffeence not only nea a nucleus o at the bounday of an atom, but also at lage distances fom the atom. The diffeence between the Coulomb and binomial potentials nea a nucleus esults in the dominance of the epulsive component of the binomial potential at small distances as compaed with the attactive Coulomb component. This makes an atom to be a stable system. Hee, we should like to make a emak in ode that the eade will not identify the stability of an obit and the stability of an atom. When we conside the fome in the sense of the Betand theoem, we mean only a one-electon system of the hydogen-atom type. But if we wite geneally about the stability of an atom, we mean many-electon systems. Accoding to ou static model (Figue 4), electons and the nucleus of such many-electon system ae in the stable equilibium position with espect to one anothe. In othe wods, each electon is located in the own potential well, whee thee is no obital motion, as distinct fom the othe old models of atoms. Hence, any connection between the stability of obits and the stability of an atom has no physical sense. Let us now conside (qualitatively) how the atio of attactive and epulsive foces will be changed, as the distance fom a nucleus will incease. In this case, the epulsive foce deceases faste than the attactive Coulomb one. Theefoe, they become equal in modulus at some distance fom the nucleus. This distance gives the size of atom. Fo geate distances fom the atom, the epulsive foce will be less in modulus that the attactive Coulomb one. It is essential that the epulsive foce does not disappea completely. As the distance fom the atom inceases, both foces decease, and the dominance of the Coulomb component will incease moe and moe. Thus, the esidual electic field exists outside the limits of each electically neutal atom. The inteaction of such fields is esponsible fo the attaction between electically neutal atoms, molecules, and, geneally, macobodies. All this is illustated in Figues 1, 6, and 8. At the pesent time, many theoies of gavitation ae available. But thei common shotcoming is that they cannot join the gavity with any othe inteaction known in physics. But we succeeded, fo the fist time, to find such geneal citeion that allows us to connect the gavity with the electostatic inteaction. At this point, ou theoy made a significant step fowad on the way of cognition of the natue of gavitation as compaed with the othe available theoies elated to the gavity. Thus, we have succeeded to stictly pove that the gavitational and electostatic inteactions have the same natue, so that the gavitational inteaction is not caused by some unique field ceated by electically neutal bodies. In ou opinion, the gavity is elated to the well-known electostatic field and possesses, like all electic fields, the popeties of Faaday-axwell foce fields. We note that ou conclusion agees completely with the esults of the Logunov s elativistic theoy of gavitation (Logunov, 1987). Hence, the attactive foces existing between electically neutal physical bodies (gavitational foces) aise as a esult of the electostatic inteaction of electically neutal atoms with the electic chages of othe, also electically neutal, atoms. The equivalence of the Coulomb and Newton laws can be also shown by compaing the dimensions of the quantities in the fomulas of these laws. Let us conside the equalities 3 3 e L L L E eg (5) T L T L T Whee γ is the gavitational constant; is the mass; L is the length; and T is the time. In (5), the second and thid equalities pesent, espectively, the Coulomb and Newton laws. The substitution of 103

12 Applied Physics Reseach Vol. 5, No. 6; 013 the coesponding dimensions (fouth and fifth equalities) leads us to the same enegy unit in the CGSE system, which demonstates once moe the equivalence of the Coulomb and Newton laws. This connection follows easily fom the following easoning. As is known, all electically neutal bodies in the Natue eveal always the mutual attaction. In ou wok, we have shown that all neutal atoms and molecules (see Figues 5 8) manifest also the mutual attaction to the emote electic chages iespective of whethe these chages ae positive o negative, i.e., to electons and nuclei on the same basis. In othe wods, we have shown that the attactive foces between electically neutal bodies bea the electostatic chaacte. Ealie, the attactive foces between bodies wee called the gavity foces. Hence, we can suely asset that both electostatic and gavitational foces have the same natue, i.e., they ae equivalent. Geneally speaking, the notion gavitational inteaction establishes only the fact of attaction of bodies. Since we have found that the attaction between bodies appeas as a esult of the electostatic inteaction, we have the fim basis to identify the gavity with the electostatic inteaction. Refeences Bon,. (1946). Atomic Physics. New Yok: Hafne. Gudym, V. K., & Andeeva, E. V. (007a). otion of an electon in the field of a binomial potential of a poton. The Old and New Concepts of Physics, 4(4), Gudym, V. K., & Andeeva, E. V. (007b). Binomial potential of the electon-poton inteaction as an altenative to the Coulomb law. Jounal of Suface Investigation, 1(), Gudym, V. K., & Andeeva, E. V. (007c). Povekhn. 4, 1. Gudym, V. K., & Andeeva, E. V. (008a). The enegy of the binding of multielectonic stuctues in the field of binomial potential. The Old and New Concepts of Physics, 5(3), Gudym, V. K., & Andeeva, E. V. (008b). Povekhn. 10, 86. Gudym, V. K., & Andeeva, E. V. (013). Quantum mechanics of a hydogen atom: A new look at poblems and a new appoach to thei explanation. Physics Essays, 6(), Hofstadte, R. (1956). Electon scatteing and nuclea stuctue. Rev. od. Phys., 8, Levich, V. G. (1969). Couse of Theoetical Physics. Vol. 1, oscow: Fizmatgiz, p. 1. Logunov, A. A. (1987). Lectues on elativity theoy and gavitation: oden analysis of the poblem. oscow: Nauka. ead, C. (001). Ameican Spectato, Sep/Oct 34 Issue 7, 68. Zhinov, N. I. (1980). Classical echanics, Posveschenie, oscow, 1980 (in Russian). Appendix A. The solution of system of the Equations () By educing this system to the fom e Г E0, x 0 0 e Г E0, x 0 0 And by adding tem-by-tem, we obtain the equality It can be easily tansfomed to the elation 0 0 e x Г 0 () x e x Г 0 (A1) x 0 0 e e E0 x. (A) 104

13 Applied Physics Reseach Vol. 5, No. 6; E0 0 1 x e (A3) By substituting the numeical values of E 0, 0, and e fom handbooks, we obtain that х = with the eo in the fifth decimal point ( ). This coesponds to the accuacy of the elevant constants E , e , (A4) With this value of х, the second equation of system () becomes e 0 (A5) Thus, the above-pesented solution of system () fo Г and x gives Г= (CGSE units) and x = (3) We note that the special attention should be given to the value of x equal to. This is of impotance because this value was obtained as a esult of the exact solution of the system of two equations coesponding to the gound state of a hydogen atom. We emphasize that only such value of x fo the binomial potential allows us to exactly integate the equations of motion. In othe wods, this esult makes the binomial potential to be applicable to othe poblems of atomic physics. Appendix B. The solution of the Equation (4) m e Г E (4) m In the geneal case, this equation has the fom m E U (B1) m This yields d d t EU dt m m EU m m (B) Using the well-known consevation law of the angula moment m, (B3) We wite d dt m Substituting dt to the last elation, we obtain o d dt. (B4) m d meu. (B5) This fomula pesents the geneal solution of the posed poblem to detemine the connection between the adius-vecto and the angle of its otation. That is, we obtained the equation fo tajectoies. e Substituting the binomial potential U to the last fomula, we aive at the expession 105

14 Applied Physics Reseach Vol. 5, No. 6; 013 d e me The integation was pefomed with the use of the tabula fomula As a esult, we have Intoducing the designations a b c b 4ac (B6) d 1 a b acsin. (B7) a me m acsin m m e me m 4. (B8) E mг mг mг 1, P, and k, (B9) 4 me me We obtain the fomula fo a motion tajectoy in the fom P 1 sink. (B10) Geneally speaking, the eade can have some doubts as to the eality of the equation of obits (5). At the pesent time, it is commonly accepted that this poblem is solved in a finite fom and descibes the closed tajectoies 1 k only with the fields popotional to and, accoding to the Betand theoem. But this is not the case, since an impotant thing is omitted. The Betand theoem is valid only if we conside the closed tajectoies unde abitay initial conditions, i.e., unde any initial values of the enegy and the angula momentum (Zhinov, 1980). Unde definite initial conditions, the closed tajectoies can be obtained with othe potentials. Ou exact solution of this poblem with the binomial potential indicates that if the Coulomb law is supplemented 1 by a positive tem popotional to, then the two-body poblem can be also solved in a finite fom. But it should be mentioned that, in ode that the tajectoies be closed in ou case, we must set the coesponding initial conditions, i.e., a cetain collection of values of the enegy and the angula momentum. In Appendix 1, we gave attention to the value of the exponent. If x, we cannot obtain the exact solution of Equation (4). Thus, we have found one moe potential, fo which the two-body poblem admits a finite solution. Copyights Copyight fo this aticle is etained by the autho(s), with fist publication ights ganted to the jounal. This is an open-access aticle distibuted unde the tems and conditions of the Ceative Commons Attibution license ( 106