hysics ssays volue 6 nube bsoption and ission of Radiation by an toic Oscillato Milan ekovac bstact The theoy of absoption and eission of electoagnetic adiation by an oscillato consisting of the atoic nucleus and one electically chaged paticle is deduced using classical electodynaics In the steady state of an ato eission and absoption of electoagnetic adiation ae equal so the ato is stable In ode to include eactive effects of electoagnetic adiation in the otion equations the Newton equation is odified by adding the adiative eaction foce This pape is an intoduction to the deivation of the basic assuptions of quantu echanics Key wods: absoption atos classical electodynaics electoagnetic adiation eission oscillatos stability steady state INTRODUCTION In 94 JJ Thopson (857 99) poposed a static odel of the ato and in 9 Ruthefod (87 97) poposed a dynaic odel of the ato It was hoped that any atoic phenoena could be explained by Newton s echanics and Maxwell s electodynaics Howeve two big pobles eained unsolved ccoding to Maxwell s electodynaics each acceleated chaged paticle (such as an electon in Ruthefod s odel of the ato) inevitably eits electoagnetic adiation and theefoe collapses into the nucleus The othe poble was the ato s discete spectu which was poved expeientally The theoies of Newton and Maxwell did not povide fo such discontinuity Theoies that cannot explain expeients ae ejected and ightly so ong othe scientists even the foundes of oden physics M lanck () and instein () tied to find satisfactoy answes within the classical continuity theoies but thei attepts wee futile It was obvious that the classical theoies did not contain a pincipal liitation that would pevent thei application down to the level of the ato One of the last effots to apply the classical theoies to the odel of the ato was ade by JH Jeans in the ealy 9s But all pesented aguents could not pevent oden physics fo developing in soe othe diection Howeve the application of classical theoies to the odel of the ato is gaining gound again (4 6) In this pape the poble of electoagnetic adiation and the ato s instability is appoached in tes of Maxwell s electodynaics and Newton s and Coulob s laws The poble of the ato s discete spectu equies two oe classical laws the law of chage and the law of oentu consevation The poble of the ato s discete spectu is elaboated in anothe pape by the sae autho (7) MISSION OF LCTROMGNTIC RDI- TION ission of electoagnetic adiation esults fo electoagnetic fields eitted by acceleated electic chages (8) o geneally fo dynaic electic and agnetic fields (oynting vecto) We view the ato as a syste consisting of a nucleus with chage Q and one paticle with chage q and ass oving in a cicula obit within a adius at an angula velocity Ω This is Ruthefod s planetay odel of the ato (9) (see Fig ) The distance of a paticle fo point x = is x( Ω t) = cos( Ω t + ϕ ) = Xˆ cos( Ω t + ϕ ) () and the distance of a paticle fo point y = is y( Ω t) = sin( Ω t + ϕ ) = Yˆ sin( Ω t + ϕ ) () 6
Milan ekovac 4 q Ω p ( t) = px ( t) + py ( t) = (7) c The esult (7) is known as the Lao foula fo adiated powe () an invaiant of the Loentz tansfoation (4) The aveage powe of one dipole in the x o y axis x o y eitted in one electon otation cycle is (56) x 4 q Ω = y = (8) πε c The total aveage powe : y is the su of x and Figue The paticle of ass and chage q otating in a cicle of adius with velocity v and angula fequency Ω in the field of cental chage Q (Ruthefod s planetay odel of the ato) The adiative eaction foce f R is accoding to Wheele and Feynan 9 out of phase with the acceleation v which eans that it is pependicula to the adius vecto and opposite to the velocity v This foce contibutes to the absoption of the adiant enegy and does not allow an electon to be oe acceleated and thus contibutes to the stability of the ato whee Xˆ + Yˆ = () is the aplitude of foced haonic otion and ϕ is a phase angle (which explains that at the oent t = ϕ /Ω pio to the beginning of obsevation t = the distance x has eached its axiu value) The adius vecto of the electon oving in the plane is = [cos( Ω t + ϕ ) i + sin( Ω t + ϕ ) j ] (4) The total eitted powe of electoagnetic adiation p (t) of an electon in the ato is the su of the oentay powe of electoagnetic adiation p x (t) and p y (t) of two dipoles () at ight angles to each othe along the x and y axes of Catesian coodinates; ie q Ω p t = Ω t + (5) 4 x ( ) cos ( ϕ ) c q Ω p t = Ω t + (6) 4 y ( ) sin ( ϕ ) c 4 q Ω = x + y = (9) c We view the ato as an electoechanical oscillatoy syste We assue that the ato has at least one stable state Suppose the x and y coponents of the diving foce f acting on the electon in an ato oscillate sinusoidally with aplitude ˆF at paticula fequency ω: f ( ω t) = Fˆ cos( ωt + ϕ ) () x f f ( ω t) = Fˆ sin( ωt + ϕ ) () y f f ( ω t) = f ( ω t) i + f ( ω t) j () x y f ( ω t) = f + f = F () ˆ x y whee ϕ f is a phase angle (which explains that at the oent t = ϕ f /ω pio to the beginning of obsevation t = the foce f x eached its axiu value) coect calculation ust include the eaction of the electoagnetic adiation on the otion of the souce (7) So besides the Coulob foce (qq/4πε ) which is actually the centipetal foce (v /) and the othe extenal foces thee is anothe foce acting on an electon ie the adiative eaction foce (8 ) ccoding to J Wheele and R Feynan who take up the poposition put long ago (9) by Tetode () that the act of eission should be soehow associated with the pesence of an absobe this foce is 6
bsoption and ission of Radiation by an toic Oscillato R τ f = (4) t whee τ is the chaacteistic tie () which will be defined late Wheele and Feynan ade a athe geneal deivation of the law of adiative eaction Consequently expession (4) is geneally accepted as coect fo a slowly oving paticle subjected to abitay acceleation Hence the total foce acting on the electon is the su of the Coulob foce othe extenal foces and the adiative eaction foce The echanis of electoagnetic adiation is coplex () and not sufficiently explained (4) In this aticle we conside only one ato so no statistical echanics (56) can be applied Howeve a question aises now of how to include the adiative eaction foce in the equation of otion Thee ae two possibilities a) The su of the adiative eaction foce f R and the extenal foce f ext is a single diving foce f = f R + f ext and the equation of otion is v = f + f (4a) R b) The diving foce is only the extenal foce f ext and the equation of otion is the baha Loentz equation of otion ext ( v τ v) = f ext (4b) s is well known the baha Loentz equation geneated so-called unaway solutions (7) so we opt fo the fist possibility So the equation of otion based on Newton s second law of otion which includes the esistive foce and the estoing foce (8 ) is x x +b + kx = Fˆ cos( ωt + ϕ f ) o afte dividing by and eaanging (5) is the decay constant (4) also called the half-width o line beadth (5) and k Ω = (8) is the angula fequency at which the siple haonic oscillato oscillates also called its natual fequency (6) (to distinguish it fo the angula fequency ω at which it ight be foced to oscillate in steady state by a diving foce f) Using () and () we get x ˆ y = X Ωsin( Ω t + ϕ ˆ ) = YΩcos( Ω t + ϕ) (9) x = X Ω Ω t + y = Y Ω Ω + ˆ cos( ϕ) ˆ sin( t ϕ) x = X y = Y Ω Ω + ˆ Ω sin( Ω t + ϕ) ˆ cos( t ϕ) Using () (4) (4) and () we get R () () f = τ Ω [sin( Ω t + ϕ ) i cos( Ω t + ϕ ) j ] () whee the aplitude of the adiative eaction foce is ˆ R τ F = Ω () The solution of diffeential equation (5) in the steady state is () So fo () (6) (9) and () we get x x ˆ F + Γ + Ω x = cos( ωt + ϕ f ) whee b is the daping constant () k is the sping constant () (6) Xˆ [( Ω Ω )sin ϕ ΓΩ cos ϕ ]sin Ωt Xˆ [( Ω Ω )cos ϕ + ΓΩ sin ϕ ]cos Ωt Fˆ Fˆ + sinϕ f sinωt cosϕ f cosωt = (4) b Γ = (7) The electon oscillates in the steady state of the ato at a paticula fequency ω of the diving foce f 64
Milan ekovac even if this fequency is diffeent fo the natual fequency Ω of the undaped oscillations (7) So in the case of steady state and (4) becoes Ω = ω (5) Xˆ [( ω Ω )sin ϕ Γω cos ϕ ] Fˆ sin + ϕ f sin ωt Xˆ [( ω Ω )cos ϕ + Γω sin ϕ ] Fˆ + cos ϕ f cos ωt = (6) Figue The total aveage powe eitted in one cycle of paticle otation and the total aveage powe absobed in one cycle of paticle otation vesus the fequency of extenal foce ; the natual fequency of a paticle is Ω and the width of an oscillato is Γ quation (6) is valid fo evey value of t only when the coefficients of the linealy independent tiefunctions sin ωt and cos ωt each equal zeo So fo aplitude Xˆ we get y 4 ˆ q ω F = πε c ( ω Ω ) + Γ ω and the total aveage eitted powe is () ˆ ˆ F X = ( ω Ω ) + Γ ω and fo a phase (ϕ ϕ f ) (89) Γω tan( ϕ ϕ ) = f ω Ω (7) (8) whee = + x y 4 ˆ q ω F = 6 πε c ( ω Ω ) + Γ ω = 4 ω ( ω Ω ) + Γ ω () whee (ϕ ϕ f ) is the phase angle between the diving foce f x (ω t) and the distance x(ω t) [o the phase angle between the diving foce f y (ω t) and the distance y(ω t)] ie the phase angle between the diving foce f and the adius vecto If we substitute in (8) fo ˆX fo (7) then the aveage powe (8) of one dipole in an x o y axis x o y eitted in one cycle of electon otation in steady state (Ω = ω) is (see Figs and ) x 4 ˆ q ω F = πε c ( ω Ω ) + Γ ω (9) Fˆ q = c () is the aveage powe eitted in one cycle of electon otation if the fequency ω of the extenal foce is appoaching ie ω BSORTION OF LCTROMGNTIC R- DITION bsoption of electoagnetic adiation (4) of one dipole in the x axis esults fo the wok of f x dx = f x v x dt = f x x/dt = p x dt done on the chage q by the 65
bsoption and ission of Radiation by an toic Oscillato lso the aveage powe y absobed in one cycle of the diving foce f y is like x in (6): y ˆ F Γω = ( ω Ω ) + Γ ω (7) The total aveage absobed powe in one cycle of electon otation is the su of x and y : (4) Fˆ = x + y = Γω ( ω Ω ) + Γ ω (8) Figue Detail of Fig diving foce f x : 66 t t t x pxdt = f x dt t () t [ Fˆ cos( ωt ϕ )][ Xˆ sin( t ϕ )] dt = + t f Ω Ω + So in the sae syste of an ato as the one entioned above the aveage powe x absobed in one cycle of diving foce f x with fequency ω [because of (5) we can also take the integal to t = π/ω instead of t = π/ω] using (7) and () in steady state (Ω = ω) is x t = π / ω Fˆ ω cos( ωt + ϕ f )sin( ωt + ϕ) ω = πt = ( ω Ω ) + Γ ω ˆ F ω = π ( ω Ω ) + Γ ω t = π / ω + t = f + cos( ωt ϕ )sin( ωt ϕ ) dt The solution of this equation is ˆ F ω = sin( ϕ ϕ ) (5) x f ( ω Ω ) + Γ ω Using (8) and sin(ϕ ϕ f ) = tan(ϕ ϕ f )/[ + tan (ϕ ϕ f )] / we get x ˆ F Γω = ( ω Ω ) + Γ ω dt (6) (4) The su of the total eitted and absobed aveage powes (oveall electoagnetic spectu if not only ω but othe coponents of fequency in the Fouie seies of diving foces ae pesent) in the steady state (ω = Ω) is zeo (see Fig ): = + = (9) (by the steady state ie by ω = Ω) By using () (8) and (9) we get and by using () and (4) (8) is = Ω q Γ = (4) c Ω ω ( ω Ω ) + Γ ω (4) 4 RMTRS OF N TOMIC OSCILL- TOR We obseve an electon on one stationay cicula obit The centifugal and centipetal foces of the electon in this obit ae in equilibiu If thee is any sall distubance in such a syste the electon becoes exposed to additional adial oscillations with a sall aplitude nea the stationay cicula obit ie nea the equilibiu position Such an aplitude is salle than the adius of an electon obit So thee is one electoechanical oscillato with a estoing foce and paaetes such as the sping constant (foce constant) natual fequency daping constant half-width and chaacteistic tie lthough we do not know the size of the electon we will deteine
Milan ekovac all these paaetes In this aticle it is not necessay to know the size of the electon no the size of the nucleus We assue that the size of the electon and the size of the nucleus ae uch salle than the distance between the (a) Sping constant and natual fequency The next thee elations (all esulting fo the sae equation v = Ω) ae valid fo the unifo cicula otions of an electon on the adius with angula fequency Ω and linea velocity v: v ( Ω v) = (4) Ω v Ω ( v ) = (4) v( Ω ) = Ω (44) We assue that f is the su of all adial foces acting on the electon nea the equilibiu position The equilibiu position is on the cicle of adius If the electon is oved eithe to one side o to the othe side away fo the equilibiu position the foce f etuns it to the equilibiu position This foce is called the estoing foce (4) The sall agnitude of the estoing foce df is found to be diectly popotional to the distance d (d being the dislocation of the electon fo the equilibiu position on the adius ): df kd = (45) We assue that in a nea-equilibiu position thee adial foces ae acting on the electon: F F C d v = = centipetal (adial) foce (46) = qq Coulob s law (foce) 4πε = (47) and the estoing foce f Thus we can wite v qq f + + 4πε = (48) o using (44) + Ω + qq 4πε = (49) f The diffeential of the foce f (Ω ) is ie using (49) f f d[ f ( Ω )] = d + dω Ω qq d f d ( ) d [ ( Ω )] = Ω + + Ω Ω πε (5) (5) The diffeential dω of angula fequency Ω accoding to (4) is Ω Ω v dω ( v ) = dv + d = dv d (5) v The absolute value of the linea velocity v in steady state is constant so dv = (5) Fo (5) by using (4) (5) and (5) we get ie qq v d[ f ( Ω )] = Ω + + Ω d (54) πε d f qq d [ ( Ω )] = Ω + πε (55) In copliance with (45) the estoing foce is df = kd and if we set this equal to (55) we get the sping constant k: k qq = Ω πε (56) So the natual fequency (8) of an electon oving in cicula atoic obits is 67
bsoption and ission of Radiation by an toic Oscillato Ω = k = Ω qq (57) πε In the equilibiu state f = and accoding to (48) we have v = qq 4πε (58) Because of (4) (ie v/ = Ω) fo (58) it follows that qq Ω = 4πε (59) So fo (57) and using (59) we get Ω = Ω qq = qq = Ω (6) πε 4πε (b) half-width and daping constant Relation (6) eans that absoption and eission ae equal in case of the condition Ω = Ω ny cicula otion satisfies this condition So accoding to (4) Γ is also Ω q Γ = (6) c and accoding to (4) is also In the steady state (ω = Ω = Ω ω Γ ) accoding to (8) tan(ϕ ϕ f ) = ie ϕ π ϕ = (65) f The adius vecto and the diving foce f ae at ight angles to each othe (see Fig ) Using () () and () and by ω = Ω and ϕ f = ϕ π/ the diving foce f is f = ˆF[sin( Ω t + ϕ ) i cos( Ω t + ϕ ) j ] (66) copaison of the adiative eaction foce () and the diving foce (66) shows that the diving foce f and the adiative eaction foce f R in the steady state ae two paallel foces If thee ae no othe extenal foces the adiative eaction foce is the only acting foce (44) So we get ˆ = = Ω (67) F FˆR τ Using () (7) and ω = Ω = Ω we get (45) and using (4) and (68) τ ccoding to (7) and (68) Γ = Ω (68) q τ = (69) c = Ω ω ( ω ) Ω + Γ ω (6) b τ = Ω (7) Using (6) we can show (6) in the steady state as Finally in the equilibiu state (Ω = Ω ) fo (56) and (6) we get o the foce constant (4) is also qq k = 4πε (6) k = Ω = Ω (64) Q Γ = 6 πε q c (c) Oscillato enegy and fequency The total enegy of a haonic oscillato is (4647) ˆ ˆ X Y k (7) = Ω = Ω = (7) 68
Milan ekovac and using (58) and (6) we get qq 8πε = v = (7) o T κ = = = (78) T Now we have all the paaetes of the ato as an electoechanical oscillato We can now discuss othe inteesting elations We can quite feely select any of the states in an ato as the state of efeence ll the vaiables in that state will be witten with an undelined sybol The peiod of one cycle of an electon is π T = = Ω ν ccoding to (6) and (74) ν (74) ν = ( / ) (75) = κ (79) whee κ is any positive eal nube Obviously in any state of efeence κ = (/) / = This eans that κ is a natual nube and always equals one Fo (75) and (79) we get and fo (78) ν ν = (8) κ T = = κ (8) The poduct of the peiod T and enegy of an oscillato we call the echanical action and denote as = T ccoding to (6) (7) (7) and (74) we get and (76) becoes π v = κ (8) qqπ = = π Ω = = π (76) 4ε T v If we divide (76) by v we get π = /v the action/oentu of the paticle It esebles de Boglie s basic postulate fo the atte wave (48) λ = h/v whee h is lanck s constant Since an electon oves in cicles accoding to de Boglie s hypothesis the electon wave ust be a cicula standing wave with π = nλ and should be nh/ But fo ou consideation hee we do not need the atte wave o lanck s constant Fo the state of efeence we have fo (76) qqπ = T = = π v (77) 4ε If we select the state of efeence in one ato then is the efeent echanical action fo that ato It is the value of the fixed aount; ie it is a constant fo that ato We denote the quotient of two echanical actions as κ and accoding to (76) and (77) it is xpession (8) esebles Boh s quantu condition (49) (πv = nh) κ is a eal nube that can also be any natual nube n The efeent echanical action is a constant as h is in Boh s condition But we cannot affi now that κ is a natual nube This affiation is confied in Ref 7 whee electoagnetic popeties of the ato ae included quations (74) (8) and (8) give ν = κν = (8) κ The elation (8) = κ ν esebles lanck s quantu hypothesis (5) ( = nhν) which is vey ipotant fo quantu physics Thee was a lot of discussion on how to explain in tes of physics the eaning of soe quantities in the elation = nhν Howeve in ou elation (8) all of the quantities ae copletely clea in tes of physics The basic physical diffeence between lanck s elation = nhν and (8) is the eaning of fequency ν ccoding to lanck s elation = nhν the fequency ν eans the fequency of an electoagnetic wave ν e In elation (8) = κ ν the fequency ν eans the fequency 69
bsoption and ission of Radiation by an toic Oscillato of echanical otation of an electon in the atoic obit t the state of efeence (whee κ = ) accoding to (8) we have = ν (84) Notwithstanding the siilaity between (84) and instein s photon equation (5) ( = hν e ) we note that ν is not the fequency of electoagnetic oscillations in a vacuu ν e (the fequency of light) but the fequency of echanical otation of an electon in the ato s state of efeence and this is a significant diffeence The physical connection between these two fequencies deands a detailed electoagnetic analysis which is ade in Ref 7 /ν accoding to (6) (6) (7) and (74) can be shown as = 4ε qq ν (85) Fo a definite syste (whee q and Q ae deteinate and fixed) /ν is constant We can show (85) as qq 4ε 4 = = 9 J s (89) The geatest ionization potentials of 4 He+ 7 Li++ 9 Be +++ 4 and 5 B++++ ae 544 V 44 V 765 V and 9965 V espectively If we select these states as the states of efeence and calculate the echanical actions accoding to (89) we get the sae esult fo all of the eleents: = 9 4 J s (see Table I) It sees that the echanical action in such states of efeence is constant fo all eleents Still it does not ean that is a fundaental constant because depends on ou fee choice In diffeent states of efeence has a diffeent value Setting = v / and = qq/8πε both expessed fo (7) sepaately equal to (88) we get and qq v v = = (9) ε κ κ 4 4ε κ π qq = = κ (9) qq = ν 4ε Setting = κ ν fo (8) equal to (86) we get qq q Q ν = = κ 4ε ε κ If we put (87) in (86) we get (86) (87) Setting (8) equal to (88) and using (84) we get = νν (9) Using (87) and (88) fo the fequency diffeence and the enegy diffeence of any two stationay states chaacteized by κ and κ (κ > κ ) we get q Q ν = ν ν = ν = ε κ κ κ κ (9) q Q = ε κ (88) The geatest ionization potential of hydogen (55) ( H ) is V i = 5978 V We select that state as the state of efeence of the hydogen ato The total enegy of a haonic oscillato in that state is = ev i = 786 8 J So accoding to (88) we can now calculate the echanical action in the state of efeence (κ = q = Q = e): and q Q = = = (94) ε κ κ κ κ quations (9) and (94) eind us of the well-known Boh expession of the atoic adiant fequency (54) and the adiant enegy (55) But thee is a fundaental diffeence between the fequencies ν of the cicula otion of the electon and the fequency of adiated electoagnetic enegy ν e by Boh The explanation of this diffeence is the subject of anothe aticle (7) 7
Table I: The Quantity of Oscillatos Based on the Hydogen to Sybol V i(h) = 5978V* (Ref 5) (expeiental) H κ = H κ = H κ = H κ = 4 4 He + κ = 7 ++ Li κ = 9 Be +++ 4 κ = Be ++++ 5 κ = Milan ekovac Z toic nube 4 5 V i Ionization potential (calculated H expeiental) 5978* 9948 5885 84987 544 44 765 9965 V = qq /8πε qv i ; 4 q = - e; Q = Ze 59e e 477e 847e 65e 76e e 6e τ τ = q / c 67e 4 67e 4 67e 4 67e 4 67e 4 67e 4 67e 4 67e 4 s/ad Ω Ω = Ω = ( qq/4πε ) / 4e+6 56e+5 5e+5 645e+4 65e+7 7e+7 66e+7 e+8 ad/s T T = π/ω 5e 6 e 5 4e 5 974e 5 8e 7 69e 7 95e 8 66e 8 s ν ν = /T 657e+5 8e+4 4e+4 e+4 6e+6 59e+6 5e+7 64e+7 Hz Γ Γ = τω 7e+ 67e+8 47e+7 6e+6 7e+ 867e+ 74e+ 669e+ ad/s b b = τω 974e 5e 4e 8e 4 56e 9 79e 9 5e 8 69e 8 kg/s k k = qq/4πε = Ω 55e+ 4e+ e+ 79e 49e+4 6e+5 98e+5 97e+5 kg/s = Ω / = k / = v / 8e 8 545e 9 4e 9 6e 9 87e 8 96e 7 49e 7 545e 7 J Fˆ R Fˆ = τω R e 4 66e 6 974e 8 e 8 68e 58e 8e 667e N F C F C = qq /4πε 8e 8 54e 9 e 9 e e 6 667e 6 e 5 55e 5 N = F q / c e 9e 5 65e 8 6e 9 e 8 85e 6 8e 5 6e 4 W ˆR = = F /Γ ˆR 466e 8 8e 7e 7e 99e 6 4e 5 9e 4 79e 4 W Γ Γ = / 7e+ 67e+8 47e+7 6e+6 7e+ 867e+ 74e+ 669e+ ad/s p p = v = Fˆ R /Γ 99e 4 996e 5 664e 5 498e 5 99e 4 598e 4 797e 4 996e 4 kg /s = T = Ω/ = ( qqπ/4ε ) / e 4 66e 4 994e 4 e e 4 e 4 e 4 e 4 J s = T = ( qqπ/4ε ) / e 4 e 4 e 4 e 4 e 4 e 4 e 4 e 4 J s κ κ = T/ T 4 = κ T = πv = κ e 4 66e 4 994e 4 e e 4 e 4 e 4 e 4 J s = (νν ) / = κ ν = ν/κ = 8e 8 545e 9 4e 9 6e 9 87e 8 96e 7 49e 7 545e 7 J q Q /(ε κ ) /ν /ν = (qq/4ε ) / 9e 85 9e 85 9e 85 9e 85 957e 85 5e 84 8e 84 598e 84 J s ν ν = q Q /(ε κ ) = ν/κ 657e+5 8e+4 4e+4 e+4 6e+6 59e+6 5e+7 64e+7 /s v v = qq/(4ε κ ) = v/κ 9e+6 9e+6 79e+5 547e+5 48e+6 656e+6 875e+6 9e+7 /s = 4ε κ /(πqq) = κ 59e e 477e 847e 65e 76e e 6e The expeiental value of the ionization potential calculated fo spectoscopic data of hydogen H( κ = ) V i = 5978 V The ionization potentials of H( κ = ) H( κ = ) H( κ = 4) 4 + He ( κ = ) 7 ++ Li ( κ = ) 9 +++ ++++ 4 Be ( κ = ) and 5 Be ( κ = ) ae calculated hee in such a way that the quotient of echanical actions of each ato κ = T/ T becoes a natual nube Without additional citeia κ can theoetically be any ational nube Unit The enegy in (94) and the enegy of the electoagnetic wave c 8π γ n n e( ) (95) in Ref 7 ae identical (the ato-stuctue coefficient γ is nealy constant; n and n ae positive integes) So q Q /ε = c /8π γ and we get qq = π e π Z qq γ = ε c = (96) whee Z = /ε c = 767 Ω We call /γ the eleentay action and denote it as e In the case of hydogen ( qq = e ) this action is inial e = 59 7
bsoption and ission of Radiation by an toic Oscillato 4 J s It is 8 ties less than the efeent echanical action accoding to (89) and 46 ties less than the lanck constant h The eleentay action e = / πz e is a univesal physical constant 5 CONCLUSION n electic chage eits electoagnetic enegy wheneve it is acceleating Thus an electon that otates aound the nucleus with a constant centipetal acceleation constantly eits electoagnetic enegy Consequently its enegy should diinish gadually This would lead to a gadual eduction in the diensions of its obit so that the electon would finally fall into the nucleus (56) Howeve at the sae tie thee is a pocess in the ato woking in the opposite diection Geneally an electon in the ato is also absobing electoagnetic adiation Indeed it is the adiative eaction foce that by eission of electoagnetic adiation in a steady state contibutes to the absoption of electoagnetic adiation in the ato This eans that in the ato s steady state this absoption is equal to the eission of electoagnetic adiation and the ato eains stable In this aticle the ato is teated as an electoechanical oscillato ll the paaetes of this oscillato ae deteinate: chaacteistic tie τ daping constant b sping constant k half-width Γ and natual fequency Ω ission of electoagnetic adiation in the ato s steady state was a fundaental aguent against applying classical electodynaics to it ccoding to this aticle such objections no longe hold gound On the basis of echanical consideations this aticle lays the foundations fo a deduction of lanck s quantu hypothesis instein s photon equation Boh s quantu condition and de Boglie s hypothesis wheeas the details of quantization ae given by the sae autho in anothe aticle (7) by including the othes electoagnetic consideation Received 8 Mach Résué La théoie de l absoption et de l éission de la adiation électoagnétique d un oscillateu coposé du noyau atoique et d une paticule électiqueent chagée est déduite en utilisant l électodynaique classique n état stationnaie d un atoe l éission et l absoption de la adiation électoagnétique sont égales donc l atoe est stable fin d intége les effets éactifs de la adiation dans l équation du ouveent l équation de Newton est odifiée en ajoutant la foce de éaction adiative L aticle pésente une intoduction à la déduction des popositions de base de la écanique quantique ndnotes The question as to whethe the ays of light ae quantized o the quantu effect oiginates only inside the atte is indeed just the fist and toughest dilea the whole quantu theoy is faced with and the answe to that question is still to diect its futhe developent (In the oiginal: In de Tat ist die Fage ob die Lichtstahlen selbe gequantelt sind ode ob die Quantenwikung nu in de Mateie stattfindet wohl das este und schweste Dilea vo das die ganze Quantentheoie gestellt ist und dessen Beantwotung ih est die weitee ntwicklung weisen wid Lectue Das Wesen des Lichts 8 Octobe 99) It is useful to note that the longest chaacteistic tie τ (τ = e / c ) fo chaged paticles is fo electons and that its value is τ = 66 4 sec 4 This is of the ode of tie taken fo light to tavel 5 Only fo phenoena involving such distances o ties will we expect adiative effects to play a cucial ole () This stateent shows us that classical analyses ae used with systes fo which ass appoaches the electon ass and chage q appoaches the electon chage e The esults and the figues in the text ae geneated by Wolfa Reseach Matheatica coutesy of Systeco Zageb Coatia The adii of the obits of hydogen ae coputed by eans of the ionization potential (57) V i accoding to the elativistic foula = qq 4 πε qvi [ + /( + qvi / c )] 7
Milan ekovac Refeences M lanck Wege zu hysikalischen kenntnis (Velag von S Hizel in Leipzig 944) p 96 H Wichann Quantu hysics (McGaw- Hill New Yok 97) p 6 G Taube lbet instein s Theoy of Geneal Relativity (Globus/Zageb 984) pp 4 4 H Sallhofe Natufosch 45a 6 (99) 5 VM Siulik and IYu Kivsky dv ppl Cliff lgeba 7 5 (997) 6 Ide nn Fond L de Boglie 7 () 7 M ekovac hys ssays 5 4 () 8 FK Richtye H Kennad and T Lauitsen Intoduction to Moden hysics (McGaw-Hill New Yok 955) p 9 DC Giancoli hysics nd edition (entice- Hall nglewood Cliffs NJ 988) p 878 H Hänsel and W Neuann hysik (Spektu kad Vel Heidelbeg Belin 995) p 8 W Schpolski tophysik Teil I (VB Deutsche Velag de Wissenschaften Belin 979) pp 7 5 LD Landau and M Lifschitz Lehbuch de Theoetischen hysik Babd II Klassische Feldtheoie 6th edition (kadeie-velag Belin 97) p 5 JD Jackson Classical lectodynaics nd edition (John Wiley & Sons New Yok 975) p 659 4 L age and NI das lectodynaics nd pinting (Van Nostand New Yok 945) p 8 5 Ref p 7 6 H Czichos HÜTT Die Gundlagen de Ingenieuwissenschaften (Spinge-Velag Belin Heidelbeg 989) p B 98 7 Ref p 78 8 J Wheele and R Feynan Rev Mod hys 7 57 (945) p 6 9 Ref p 784 Ref p 8 H Tetode Z hys 7 (9) Ref p 78 Ref 8 pp 58 59 4 Ref p 78 5 V Fok Dokl kad Nauk 6 57 (948) 6 S Watanabe Rev Mod hys 7 6 4 79 (955) 7 Ref p 784 8 Ref 9 p 4 9 Ibid p 78 J Deziski and C Géad Scatteing Theoy of Classical and Quantu N-aticle Systes (Spinge Belin Tokyo 997) p Ref p 8 Ref 9 p 9 Ibid p 6 4 Ref p 799 5 Ibid p 8 6 Ref 9 p 7 Ibid p 4 8 Ibid 9 Ref 6 p B / 4 Ibid p B 4 Ibid 4 Ref 9 p 5 4 Ref p 87 44 Ibid p 78 45 Ibid p 87 46 Ref 6 p B 7 47 Ref p 66 48 L de Boglie nn hys (95) 49 Ref 9 p 88 5 Ibid p 869 5 Ibid 5 Ibid p 88 5 Ref 8 p 67 54 Ref 9 p 88 55 Ibid p 88 56 J Oea hysik (Cal Hanse Velag München Wien 987) pp 46 46 57 Ref 8 p 67 Milan ekovac Dives-Contol O Box 5 HR- Zageb Coatia e-ail: ilanpekovac@divesh 7