The Quantum Oscillatory Modulated Potential I The Hydrogen Atom

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Joural of Moder Physics 3 597-63 http://dx.doi.org/.436/jmp..378 Published Olie July (http://www.scirp.

1 Joural of Moder Physics Published Olie July ( The Quatum Oscillatory Modulated Potetial I The Hydroge Atom Waldemar Woley Filho Istituto de Física Uiversidade Federal de Goiás Goiâia Brazil wwoley@yahoo.com.br Received April 5 ; revised May ; accepted Jue 5 ABSTRACT I this work we are presetig a modified Coulomb potetial fuctio to describe the iteractio betwee two microscopic electric charges. I particular cocerig the iteractio betwee the proto ad the electro i the hydroge atom. The modified potetial fuctio is the product of the classical Coulomb potetial ad a oscillatory fuctio depedet o a quatized phase factor. The oscillatory fuctio picks up oly selected poits alog the Coulomb potetial creatig potetial wells ad barriers aroud the ucleus of the atom. The ew potetial reveals us ew features of the hydroge atom. Searchig for a maer to determie the phase factor we are usig the cocept of the de Broglie particle wavelike behavior ad the quatum aalogue of the virial theorem for describig the boud motio of a particle i a cetral force field. This procedure is a kid of feedback actio where we are makig use of well established cocepts of the quatum mechaics aimig to determie the phase factor of the ew iteractio potetial. Keywords: Potetial; Atom; Electro; Oscillator Fuctio. Itroductio This work is the first of a series of four papers for which we are revisitig reexamiig ad questioig some poits about the foudatios of the atomic theory developed mostly betwee the years 9-93 by may distiguished physicists: Niels Bohr Erwi Schroediger Werer Heiseberg Alberl Eistei Max Bor Louis de Broglie ad may others. What motivate us to produce this work are i our opiio some uclear ad questioable poits that we believe still exist particularly cocerig the applicability of the classical Coulomb potetial for describig the iteractio betwee two microscopic electric particles. For istace i the hydroge atom the Coulomb potetial used by Bohr ad years later i the quatum mechaics (Schroediger) to describe may of the kow properties of the curret atomic model is a icreasig smooth fuctio of the distace betwee the proto ad the electro. Beig so how ca the electro possibly kow that certai positios aroud the ucleus of the atom are more likely to be occupied tha others? [] What is the real cause for this behavior for the electro? It is difficult to imagie the electro makig regular probabilistic choice to occupy certai particular positios i the viciity of the ucleus of the atom based solely i the characteristics preseted by the Coulomb potetial. The observed discrete values of the total eergy of the electro suggest us the existece of potetial wells ad barriers i the form of closed shells surroudig the ucleus of the atom. The questio of how ad why the electro is able to jump from oe state of equilibrium to aother emittig or receivig eergy radiatio (light) could be better explaied if we ca accept the idea that the electro is tuellig through these potetial barriers. The stability of the atom is a iterestig questio that deserves further explaatio. The de Broglie wavelegth associated to the wavig ature of a microscopic charged particle is a attribute of the mass as postulated by Louis de Broglie i 93 or is it a attribute of the charge of the particle? Aother iterestig questio is cocered to the actual movemet of the electro aroud the ucleus of the atom. Is the electro orbitig the ucleus of the atom as predicted by Bohr theory? Is it behavig like a cloud of egative charge aroud the ucleus of the atom? Or yet is it oscillatig radially i quatum wells like a harmoic oscillator? The Coulomb potetial belogs to what is kow as classical physics. Sice log ago it is a fact well kow that it correctly describes the eergy iteractio betwee two puctual macroscopic electric charges. Also it is The stability of the atom will be treated i the secod paper of this series. This subject will be cosidered i the third paper ad for a ucharged particle possessig a electric dipole momet the subject will treated i the fourth paper of the series. Copyright SciRes.

2 598 W. W. FILHO well kow that the behavior of a particle i the micro-world of the quatum physics is ot the same as that i the macro-world of classical physics. What we wat to say is that the classical Coulomb potetial reveal us oly the macro-behavior of the iteractio betwee electric particles This make us to suspect the existece of a differet iteractio potetial capable of describig more accurately the physics of the atoms ad also be able to aswer the questios formulated above. We believe that the choice of ay potetial fuctio that proposes to be cadidate to describe the iteractio betwee two microscopic charges i some way eeds to carry by itself the seeds of the quatum physics of the atom that is the quatizatio of the agular mometum ad eergy both cocepts itimately related to the Heiseberg ucertaity priciple. Besides i some way the ew potetial must be resposible to the wavelike ature of the particles.. The Hydroge Atom I this paper tryig to aswer the questios formulated above let us postulate that i the hydroge atom the iteractio eergy betwee the proto (ucleus of the atom) ad the electro cosidered as spiless particles ca be expressed by the product of the Coulomb potetial ad oe oscillatory fuctio writte as Ze Uosc. r= cos r. 4π r The parameter with dimesio of iverse of distace represets a radial-space frequecy. We believe that the phase factor r of the oscillatory fuctio is the key poit to aswer the questios formulated preously. As we will see soo this phase factor is directly related to the Heiseberg ucertaity priciple ad the quatizatio of the agular mometum. For fixed values of the parameter ( ) the fuctio cos r oscillates betwee zero ad oe while the fuctio Uosc. r oscillates betwee zero ad the classical Coulomb potetial eergy. The Equatio () tell us that whe cos r the ew potetial fuctio becomes equal to the classical Coulomb potetial. () Ze Ucou. r =. () 4π r However we eed to observe that if we make = we have Uosc. r = Ucou. r for all values of the distace r betwee the proto ad the electro i the hydroge atom. This implies that the liear mometum of the electro p = is ull. That is the electro is at rest i the LFR (Laboratory Frame of Referece). This coditio strogly cotraries oe of the basilar priciples of the quatum mechaics the Heiseberg ucertaity priciple rpr. That is if we make the ucertaity of the liear mometum pr = = we have r ad the radial positio of the electro becomes completely udetermied. O the other had if we make r = we have p = r makig the kietic eergy of the electro ifiitely large [-4]. This clearly shows the iadequacy of the Coulomb potetial to describe the iteractio betwee the proto ad the electro i the hydroge atom except at particular poits of the space i the viciity of the ucleus of the atom. The phase factor of the oscillatory fuctio selects oly specific poits of the Coulomb potetial makig it to satisfy the Heiseberg ucertaity priciple oly at poits where the coditio r = π is satisfied. This costitutes a quatized relatio betwee the ormal ad the reciprocal space. Lookig for the determiatio of the parameter we will make use of the quatum-mechaical aalogue of the virial theorem ad the de Broglie relatio betwee the liear mometum of the particle ad the wave umber k. With this procedure we are makig use of well kow quatum cocepts to determie the phase factor of the ew potetial. The quatum aalogue of the virial theorem ca be set i the form [4] T = r. Ur (3) ad the required de Broglie relatio is p = k (4) To facilitate the applicatio of Equatio () it turs out to be coveiet that the parameter may be expressed i terms of kow quatities such as the charge ad mass of the electro ad some physical costats. With the use of Equatios () ad (3) the average kietic eergy of the electro i the hydroge atom is foud to be expressible as p Ze = cos r sir (5) m 4π r With the use of Equatio (4) it follows that cos Zme k = r si r. 4 r (6) We will select the desired particular poits alog the Coulomb potetial observig that every time the product r satisfies the quatizatio relatio r = π =3 (7) the Coulomb potetial becomes equal to the oscillatory potetial ad the Equatio (6) may be writte as Copyright SciRes.

3 W. W. FILHO 599 or k Zme = 4π r k Zme r = r. 4π r (8) (9) The Equatios (7) ad (9) become cosistet if we set = k ad Zme = r 4π what permit us to rewrite the Equatio (9) as πzme r = r. 4π () () Comparig the Equatios (7) ad () we ca idetify the space-frequecy as a quatized quatity give by πzme =. () 4π I the hydroge atom we have Z = ad i its groud state =. The quatity 4 π me may be idetified as the first Bohr radius r = a =.59 Å of the electro. The Equatio () may be rewritte as π =. (3) a This implies that for two atomic boud particles such as the proto ad the electro i the hydroge atom the average value of the liear mometum of the electro is also a quatized quatity me p =. ((4)) 4π Usig Equatio (3) the ew potetial fuctio may be rewritte as e Ur= cos r. (5) 4π r We are amig this potetial by Quatum Oscillatory Modulated Potetial-QOMP; because it is quatized it oscillates with k-space frequecy ad it is partially modulated by the classical Coulomb potetial. Usig the coditio give by Equatio (7) ad the Equatios () ad (5) the average total eergy of the electro with respect to the ucleus is T U = e Ze 4π r 4π r = Ze. 4π r (6) Takig the average value r as give by Equatio () the eergy of the electro becomes quatized ad may be writte as 4 me E = 4π (7) as would be expected it is the same relatio as obtaied with Bohr theory ad also with the solutio of the time-idepedet Schroediger wave equatio for the hydroge atom whe the Coulomb potetial is used. But differetly from these two theories that i differet circumstaces make explicit use of the quatizatio of the agular mometum to obtai the total eergy of the electro i this work the same result is achieved without makig explicit use of this cocept. Implicitly the quatizatio of the agular mometum is cotaied i the phase factor of the oscillatory fuctio Equatio (7). The Figure shows a graphic of the potetial U r. As show i the figure the electro i the hydroge atom occupies the first well with positio ad total eergy satisfyig the Equatios () ad (7) respectively. The Figure shows the graphics of U r for = ad 3. The values of the eergies show i the graphic determied by the positios of the electro are respectively 3.6 ev 3.4 ev ad.5 ev. The same results are predicted by Equatio (7) [45]. The ew potetial preset us with several iterestig features: ) Its wavig character shows a sequece of periodic wells ad barriers with variable depth ad height; ) The first well is at the positio most likely to be occupied by the electro i the groud state of the atom; 3) The quatizatio of the eergy may be attributed to the existece of cocetric quatum shells formed by wells ad barriers created by the iteractio of the ucleus of the atom ad the electro; 4) Accordig to this potetial the electro boud to the ucleus ca oly jumps from oe quatum well to aother by tuelig through fiite potetial barriers. The tuelig effect (a purely quatum mechaical pheomeo) ow appears aturally as ecessary coditio for emissio or absorptio of eergy (light) by the atom; 5) Aother ew iterestig aspect of the oscillatory potetial observed i the Figure ad predicted by Equatios (3) ad (4) is the fact that the de Broglie wavelegth associated to the electro is equal to the first Bohr radius of the electro i the hydroge atom. Thus the average distace betwee the proto ad the electro for ay excited state of the atom may be writte as r = r. (8) Or equivaletly i terms of the de Broglie wavelegths we may write (9) =. Copyright SciRes.

6 W. W. FILHO Figure. The Coulomb potetial U(r) Coul ad the quatum oscillatory modulated potetial U (r) showig the positio ad eergy of the electro i the groud state of the hydroge atom.

4 6 W. W. FILHO Figure. The Coulomb potetial U(r) Coul ad the quatum oscillatory modulated potetial U (r) showig the positio ad eergy of the electro i the groud state of the hydroge atom. The positios R ad R are classical turig poits for the electro. Figure. The Coulomb potetials U(r) Coul the quatum oscillatory modulated potetials U (r) ad the eergy levels ad positios of the electro i the hydroge atom for = ad 3. This result is completely differet from that predicted by the quatizatio rule π r = for each allowed orbit for the electro as postulated by Louis de Broglie to explai the quatizatio of the agular mometum of the electro i the Old Quatum Theory [56]. Figure shows that the averaged positios occupied by the electro i the first three quatum wells of U r U r ad U3 r are respectively r =.53 Å r =4 r ad r 3 =9 r. I this work we are arguig that the Coulomb potetial is ot totally appropriated to describe correctly the potetial iteractio betwee two microscopic electric charges. However as is well kow the solutio of the time idepedet Schroediger wave equatio predicts correctly the quatizatio of the agular mometum ad eergy of the electro whe Coulomb potetial is used i the Hamiltoia of the protoelectro system. This occur because i the solutio of the radial part of the Schroediger differetial wave equatio without oticig it we are makig use the quatizatio rule r = π for the phase factor of the Copyright SciRes.

5 W. W. FILHO 6 oscillatory fuctio of the ew potetial. That is implicitly we are makig cos r = ad cosequetly selectig oly the poits of miima of the ew potetial where both iteractio potetials are coicidet see Figure. I these selected poits the Coulomb potetial satisfies the Heiseberg ucertaity priciple. I terms of a mathematical treatmet the quatizatio of the phase factor r allow us to see that the Coulomb potetial is a particular case of the oscillatory potetial by trasformig the Schroediger radial differetial equatio ds d S d d l l cos S 4 = () obtaied with the use of the Quatum Oscillatory Modulated Potetial i the Hamiltoia of the protoelectro system ito the simpler ad familiar radial differetial equatio [45] d d S S l l = 4 S d d () obtaied with the use of the classical Coulomb potetial with the coditio = π or cos =. The solutio of Equatio () is well kow ad it is foud i may books of moder physics literature [4-7]. Its solutio shows that the total eergy of the electro i the hydroge atom is the same as predicted by Equatio (7). Up to this poit we ca also state that the Coulomb potetial is restricted to describe correctly the iteractio betwee two microscopic electric charges oly at particular poits of the space aroud ucleus of the atom poits where the quatizatio relatio r = π ad cosequetly the Heiseberg ucertaity priciple are satisfied. 3. A New Electric Iteractio Microscopic Force The electric force betwee two macroscopic electric poit charges Q ad Q separated by a distace r is give by the well kow classical Coulomb formula QQ F coul. = () 4π r which is perfectly valid as far as we cosider the two charges built up of a very large umber of protos or electros but still cosidered as poit charges. However the electric force betwee two microscopic charges such as proto-proto proto-electro or electro-electro origiated from the ew potetial fuctio has the form of a wave packet writte as d Fr= Ur d r = Ze cos r si r 4π r r F r (3) The forces oscillate with the same k-space frequecies. For the hydroge atom Figure 3 shows a graphic of F r as predicted by Equatio (3). Sice this force is the gradiet of the eergy fuctio U r which presets a miimum at positio r it becomes ull at this poit. We must observe that with the use of the classical Coulomb potetial this situatio ever occurs the iteractio force betwee the proto ad the electro is always attractive. The acceleratio due to the electric force beig ull at this positio elimiates the argumet or defiace agaist the Classical Laws of the Electrodyamic that a electro revolvig the ucleus of the atom lose eergy by radiatio [8-]. The force beig ull at the equilibrium positio r = a meas that the electro caot be orbitig the ucleus of the atom as predicted by Bohr theory. Thus accordig to the QOMP the agular mometum of the electro i its groud state is ull i agreemet with the predictio of the Schroediger radial differetial wave Equatio (). O the other had the electro caot be a static physical object due to the Heiseberg ucertaity priciple. The agular momet of the electro beig ull implies that the movemet of the electro iside a quatum well must be radial boucig back ad forth cofied by two potetial barriers. This argumet is reiforced by observig that i Figure 3 the iteractio force betwee the proto ad the electro is repulsive to the left of the positio r ad attractive to the right of this positio. That is the electro is always beig pushed towards the equilibrium positio actig like a harmoic oscillator. However beig accelerated towards the equilibrium positio agai the electro should lose eergy by radiatio ad the problem of stability of the atom cotiues to be uexplaied. The stability of the atom will be object of a deeper discussio i the secod paper of this series of work. Figure 4 shows the behavior of the electric forces F r for = ad 3. The graphics show that the wavig character of iteractio electric force decreases whe icreases i agreemet with the Bohr correspodece priciple. The reasos for this behavior are two: ) The destructive iterferece of the wavig patter of the idividual charges protos ad electros occur because whe macroscopic charges 3 are built up these particles do ot occupy the same positio i space ad the wavig patter of the force must be destroyed by a iterferece process; ) The actual measuremets of the iteractio 3 A macroscopic charge of μc = 6 C cotais about 3 electroic charges. Copyright SciRes.

6 6 W. W. FILHO Figure 3. The iteractio oscillatory force F (r) the quatum modulated potetial U (r) ad the Coulomb potetial U(r) Coul i the groud state of the hydroge atom. The figure also shows that the force is ull at the equilibrium positio r. Figure 4. The Coulomb force F(r) Coul ad the iteractio oscillatory forces F (r) betwee the proto ad the electro i the hydroge atom for = ad 3. force betwee two macroscopic particles are usually performed usig large ioized atoms for which the escaped electros come from outer shells of the atoms makig the ew electric force to approximate quickly to the classical Coulomb force as idicated i Figure Coclusio We ca summarize this work as follows: For the hydroge atom the oscillatory potetial fuctio is able to predict correctly the values of the total eergy ad positio of the electro i the atom without the ecessity of solvig the time-idepedet Schroediger wave equa- tio. I additio it shows several peculiarities of the hydroge atom such as: The existece of quatum wells ad barriers formig closed shells aroud ucleus of the atom. These wells costitute the most probable positios for the electro to occupy whe movig aroud the ucleus of the atom; The emissio or absorptio of eergy by the atom happes whe the electro moves from oe state of equilibrium to aother by tuelig through these potetial barriers created by the oscillatory potetial; The ew potetial shows that the electric force betwee the proto ad the electro oscillates presetig egatives ad positives peaks the positive peaks represet regios where the iteractio force betwee the proto ad the Copyright SciRes.

7 W. W. FILHO 63 electro is repulsive ad the egative peaks where it is attractive that is the electro is always bee pushed to the equilibrium positio which is a ecessary coditio for the stability of the atom. The ew potetial predicts that de Broglie wavelegth associated to the electro is equal to the radius of Bohr r = a what is quite differet of the coditio postulated by Louis de Broglie for the quatizatio of the agular mometum of the electro i the Bohr Old Quatum Theory. Accordig to the ew potetial the electro is ot orbitig the ucleus of the atom but oscillatig radially cofied by two potetial barriers. Whats more this work presets strog evideces that the Coulomb potetial fuctio is ot adequate to describe the correct iteractio betwee two microscopic electric charges because except at particular poits aroud the ucleus of the atom it is i disagreemet with oe of the basic priciples of quatum mechaics-the Heiseberg ucertaity priciple. The ew potetial is much more iformative tha the classical Coulomb potetial ad more appropriated to accomplish the fuctio for the iteractio betwee two microscopic electric charges. REFERENCES [] R. P. Feyma R. B. Leighto ad M. Sads Lectures o Physics Addiso Wesley Readig 966. [] C. Cohe-Taoudji B. Diu ad F. Laloë Quatum Mechaics Joh Wiley & Sos New York 997. [3] L. Gilder The Age of Etagleme Alfred A. Kopf New York 8. [4] W. W. Filho Mecâica Quâtica Editora da UFG Goiâia. [5] R. M. Eisberg Fudametals of Moder Physics Joh Wiley & Sos Ic. New York 96. [6] P. A. Tipler Moder Physics Worth Publishers Ic. Irvig 986. [7] L. Paulig ad E. B. Wilso Itroductio to Quatum Mechaics McGraw Hill New York 935. [8] W. Hauser Itroductio to the Priciples of Electromagetism Addiso Wesley Publishig Compay Ic. Bosto 97. [9] E. Merzbacher Quatum Mechaics Joh Wiley & Sos Ic. New York 976. [] V. A. Fock Fudametals of Quatum Mechaics Mir Publishers Moscow 98. Copyright SciRes.

True Nature of Potential Energy of a Hydrogen Atom

True Nature of Potential Energy of a Hydrogen Atom True Nature of Potetial Eergy of a Hydroge Atom Koshu Suto Key words: Bohr Radius, Potetial Eergy, Rest Mass Eergy, Classical Electro Radius PACS codes: 365Sq, 365-w, 33+p Abstract I cosiderig the potetial