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 eergy of a hydroge atom, we offered the hypothesis that this physical quatity correspods to the decrease i the electro s rest mass eergy It is ot possible to establish the groud state eergy of a hydroge atom without quatum mechaics However, for the atom s stability oly, this ca be explaied eve without usig quatum mechaics Our discussio reveals that there exists a off-limit boudary r c withi the electro iside a hydroge atom I Itroductio Accordig to the Rutherford atomic model, oe or more electros orbit aroud the ucleus If we assume that electros are movig i circles aroud the atomic ucleus, the we kow that a electro must emit electromagetic waves through that acceleratio ad will fall ito the ucleus after a period of about - secods Classical mechaics caot curretly be used to explai the stability of a atom, ad we cosider that this problem was solved for the first time by the appearace of Bohr s classical quatum theory By assumig quatum coditios, Bohr derived the orbit radius of a hydroge atom He further explaied that there is a miimum value of the total eergy of a hydroge atom, ad that electro is ot absorbed ito the atomic ucleus I this paper, though, we cosider the potetial eergy of a hydroge atom ad attempt to explai the stability of a hydroge atom However, stability i this paper is used ot to refer to the stability of a atom that has bee successfully described by Bohr, but istead refers to the coditio of a electro ot fallig ito the atomic ucleus II Electro eergy as described accordig to classical mechaics Let us review the eergy of a electro iside a hydroge atom Let us suppose that a atomic
ucleus is at rest because it is heavy, ad cosider the situatio where a electro (electric charge e, mass m) is orbitig at speed v alog a orbit (radius r) with the atomic ucleus as its ceter A equatio describig the motio is as follows: mv e = (II) r 4πε r From this, we obtai: e mv = 4πε r (II) Meawhile, the potetial eergy of the electro is: e V() r = 4πε r (II3) Sice the right side of (II) is / times the potetial eergy, (II) idicates: V r = (II4) () mv Therefore, the total electro eergy: = + () (II5a) E mv V r = K (II5b) Also, the total eergy of the electro is equal to half its potetial eergy E = V () r (II6) The reaso for the differece i potetial eergy ad kietic eergy (K) i (II4) is thought to be the photo eergy ω released by the electro Accordigly, we ca establish the followig law of eergy coservatio V() r + K + ω =, V() r (II7) III A absolute defiitio of the total eergy of a hydroge atom Referrig to classical quatum theory ad (II5b), the relatioship betwee the total eergy ad kietic eergy of a electro iside a hydroge atom is: E 4 me = 4πε (IIIa)
B E = (IIIb) K, =,, = (IIIc) Here, is a pricipal quatum umber I this case, the total eergy of a hydroge atom has a egative value Whe describig the total eergy of a hydroge atom accordig to classical mechaics, we must kow the atom s potetial eergy ad kietic eergy However, discussig the eergy of a hydroge atom accordig to quatum mechaics, we are oly cocered with the icrease or decrease i total eergy Additioally, the classical quatum radius r is represeted as follows: 4πε r = me (IIIa) = a, = B,, (IIIb) Here, the value = is the Bohr radius a B, which correspods to the groud state of the hydroge atom Accordig to classical quatum theory, the total eergy ad the potetial eergy of a hydroge atom are cosidered to be zero whe the electro is separated from the atomic ucleus by a distace of ifiity ad remais at rest i that locatio The total eergy of (IIIa) is the value obtaied from this perspective I classical quatum theory, we emphasize the differece i eergy, ot the absolute eergy However, accordig to quatum mechaics textbooks, the eigevalue of the eergy of a hydroge atom as obtaied from the Dirac equatio, which is a relativistic wave equatio, is as follows [] E 3 k 4 4 γ γ = 4 (III3) It is importat to ote that eergy here is defied o a absolute scale Because Z = i the case of a hydroge atom, γ = e c, (γ is the fie structure costat) If we igore for the third term of this equatio ad defie it as a approximatio, (III3) ca be writte as follows E 4 me = 4πε = + E (III4a) (III4b) Moreover, E of (A5) defies a absolute quatity, which icludes the electro s rest mass eergy (See Appedix) 3
From this fact, i this paper, the total eergy i absolute terms, E ab, for a hydroge atom is defied as below E = E + K + V r (III5a) ab, ( ) = E + V( r ) (III5b) = E + E, =,,, E < (III5c) Here, E ab, is the total eergy as defied i absolute terms whe the pricipal quatum umber is To agree with o the left side, is added to K ad r o the right side Whe defied o a absolute scale, the total eergy of a hydroge atom is less tha the electro s rest mass eergy IV True ature of potetial eergy of a hydroge atom The case of a sigle electro, at rest i free space, is cosidered Accordig to Eistei, the oly eergy of a electro i this state is its rest mass eergy, or E, which is m e c [] Although a particle caot exist i a certai locatio with zero mometum accordig to the ucertaity priciple, we shall set that problem aside for the momet ad proceed with our cosideratio If this electro absorbs photo eergy, the photo eergy absorbed is trasferred ito kietic eergy of the electro We kow that the followig Eistei s relatioship is true for the total eergy E ad the mometum p of the electro that begis movig: E = c p + E (IV) Of course, E > E i this case However, what happes if this electro at rest is attracted to the ucleus of a hydroge atom, a proto, ad is draw ito the atom? The electro emits a photo from itself without absorbig exteral eergy, ad at the same time gais a amout of kietic eergy equivalet to the photo eergy Eve if the electro which was at rest begis movig i free space, ad eve if it is absorbed ito a atom, the startig poit of the electro s eergy for either case is its rest mass eergy The electro s eergy E for the former state is E >E ad for the latter state is E < E Meawhile, i order to maitai the law of eergy coservatio i the latter case, a eergy source is eeded to supply the icreased kietic eergy ad released photo eergy Accordig to the explaatio based o classical mechaics i Chapter II of this paper, the 4
reductio comes from potetial eergy However, potetial eergy has o fudametal substace ad is cosidered a cocept itroduced by classical mechaics to maitai the law of eergy coservatio Meawhile, the eergy gaied by the hydroge atom, as obtaied usig the Dirac equatio, icludes the electro s rest mass eergy The eergy here is measured o a absolute scale Because E < E i this case, Eistei s relatioship does ot apply to a electro i this state However, the eergy of a hydroge atom, obtaied through classical quatum theory or the Schrödiger equatio, is egative This ca be cosidered to be a measuremet o a absolute scale which excludes the electro s rest mass eergy However, it appears that we have failed to otice this exclusio Let us imagie a electro trasported a ifiite distace from the proto of a hydroge atom ad placed at rest The eergy of a hydroge atom i this state, as obtaied from classical quatum mechaics or the Schrödiger equatio, is zero This zero eergy is sometimes described as the total eergy ad sometimes described as the potetial eergy However, the Dirac equatio predicts that the potetial eergy for this case will be zero but that the total eergy will be m e c Whe describig the total eergy of a hydroge atom o a absolute scale, the eergy i this case equals the electro s rest mass eergy If we defie the sigle photo eergy ad kietic eergy ow beig released as E, the eergy trasfer i the iitial state could be writte as follows [ E + V( r )] + K + ω = ( E + E ) E E (IVa) = E (IVb) Because the photo is emitted exterally, if we omit that eergy here, the total eergy of a hydroge atom ca be expressed as: E = [ E + V( r )] + K (IV3a) ab, = ( E + E) E (IV3b) = E + E (IV3c) E, which correspods to the potetial eergy V( r ) of a hydroge atom from Eq (IV3b), is thought to be the reductio i the electro s rest mass eergy The potetial eergy of a hydroge atom is usually related to the distace betwee the proto ad the electro However, whe cosiderig that Eq (III4b), which predicts the eergy of a hydroge atom, icludes the electro s rest mass eergy, it is possible to surmise that the potetial 5
eergy of a hydroge atom is related to the eergy of the electro This paper predicts that the rest mass eergy of the electro is the source of the kietic eergy obtaied by the electro ad of the photo eergy emitted by the electro Let us summarize the poits covered util ow Uder classical mechaics, the followig equatio is true V() r + K + ω =, V() r (II7) Meawhile, from the perspective of this mauscript, if we represet the reductio i rest mass eergy of the electro as -ΔE, the the followig equatio is true Δ E + K + ω = (IV4) Thus, i our discussio, i dealig with the potetial eergy of a hydroge atom, we offer the hypothesis that this physical quatity correspods to the reductio of the electro s rest mass eergy: Vr () = ΔE (IV5) If this relatioship is accepted to be true, the it is possible for potetial eergy, which did ot exist whe the electro was at rest, to decrease V Discussio I the cetral field iside a hydroge atom, the amout of reduced potetial eergy ca be thought to be equivalet to the sum of icrease i the kietic eergy of a electro ad the eergy of photos emitted by the electro Accordigly, we ca establish the followig law of eergy coservatio Δ V() r + Δ K + ω = (V) Accordig to (IVa), half decrease i rest mass eergy was released outside the atom as photo eergy, while the other half was coverted ito the electro s kietic eergy Whe each of photo eergy ad electro s kietic eergy reach m e c /, the electro caot obtai more kietic eergy tha this, ad it is also uable to decrease its potetial eergy Thus, the value V(r) must satisfy the followig iequality (V) e V() r Therefore, there exists a miimum value of potetial eergy, whereupo the followig relatioship is established e = e (V3) 4πε re 6
The locatio that satisfies this relatioship is the distace of closest approach r e, which idicates how close the electro comes to the ceter of the atom From (V3), r e is the followig value e r = (V4a) e 4πε e = r (V4b) c Here, r c is the classical electro radius Thus, it becomes clear that a electro is ot absorbed i a atomic ucleus Furthermore, the distace r c agrees with the distace of closest approach of α -particle i Rutherford scatterig [4,5] Eve though we could explai the reaso why a electro is ot absorbed ito a ucleus without waitig for the appearace of Bohr s classical quatum theory, we remaied oblivious to this fact for the past cetury VI Coclusio I cosiderig the potetial eergy of a hydroge atom, we offered the hypothesis that this physical quatity correspods to the decrease i the electro s rest mass eergy Thus, Vr () = ΔE The stability of a hydroge atom was first explaied accordig to Bohr s classical quatum theory However, it becomes clear that withi the electro iside a hydroge atom, there exists a off-limits boudary r c Furthermore, this distace of closest approach correspods to the classical electro radius (See Table ) Distace from ucleus a BB r Photoic eergy (total) ω Kietic eergy K Potetial eergy Vr () = ΔE Total eergy E + V() r + K Rest mass eergy E + V() r E E c E + E + E Table Eergy of electros at distaces, a B B ad rc from a atomic ucleus 7
This value is differet from the value predicted by classical quatum theory, or the Bohr radius, but the atom s stability ca be explaied eve by our approach However, this does ot ecessarily mea that our discussio has casts doubt oto quatum mechaics Our discussio shows oly that it is possible to explai the stability of a hydroge atom accordig to theories other tha quatum mechaics The coclusio of this paper is a commo sese coclusio based o classical mechaics logic Eve so, it has bee itetioally highlighted because this fact is ot commoly kow by all curret physicists Ackowledgemet Chapter II was borrowed ad traslated from the Japaese laguage textbook of Dr H Ezawa I wish to express my gratitude to Dr H Ezawa Appedix Oe of the importat relatioships i the Special Theory of Relativity is as follows E = c p + E (A) Here, E is the total eergy of a object or a particle Gasiorowicz discusses the relativistic aalog of Schrödiger for a boud (scalar) electro iside a hydroge atom, which does iclude the rest mass eergy of the electro i a attractive, cetral potetial [3] This equatio is + = + E Ze ψ ψ ψ c 4πε cr (A) which is the operator versio of (A) whe a potetial is icluded, ( E V) = c p + E (A3) The solutio by solvig for this (A) did ot agree with the actual eergy level of the hydroge atom The reaso proposed is that electros are / spi particles ad do ot follow the Klei-Gordo equatio However, as a remaiig problem, the left side of (A3) is as follows 8
E V = ( K + V) V (A4a) = K (A4b) Thus, K E, > or ( p / m) > ( ), but this kid of iequality should ormally ot be possible Here, let us surmise that E of (A3) is defied ot as the E of (II5b) but istead as: E = E K (A5) By substitutig this E ito (A3) ad cosiderig the relatio to (II4), we obtai: ( ) E + K = c p + E (A6) This equatio is idetical to Eistei s relatioship I the ed, the total eergy E of (A3) is the eergy as defied by (A5) E of (A3) icludes the electro s rest mass eergy ad is defied o a absolute scale Refereces [] L I Schiff, QUANTUM MECHANICS (McGRAW-HILL BOOK COMPANY, New York, 968) [] A Eistei, Techical Joural 5,6 (946) [3] S Gasiorowicz, Quatum Physics (Wiley Iteratioal Editio,3) [4] E Rutherford, PhilMag 37,537 (99) [5] S Weiberg, The Discovery of Subatomic Particles (Revised Editio, Cambridge Uiversity Press, 3) 9