Chapte 35 Boh Theoy of Hydogen CHAPTER 35 HYDROGEN BOHR THEORY OF The hydogen atom played a special ole in the histoy of physics by poviding the key that unlocked the new mechanics that eplaced Newtonian mechanics. It stated with Johann Balme's discovey in 1884 of a mathematical fomula fo the wavelengths of some of the spectal lines emitted by hydogen. The simplicity of the fomula suggested that some undestandable mechanisms wee poducing these lines. The next step was Ruthefod's discovey of the atomic nucleus in 1912. Afte that, one knew the basic stuctue of atoms a positive nucleus suounded by negative electons. Within a yea Neils Boh had a model of the hydogen atom that "explained" the spectal lines. Boh intoduced a new concept, the enegy level. The electon in hydogen had cetain allowed enegy levels, and the shap spectal lines wee emitted when the electon jumped fom one enegy level to anothe. To explain the enegy levels, Boh developed a model in which the electon had cetain allowed obits and the jump between enegy levels coesponded to the electon moving fom one allowed obit to anothe. Boh's allowed obits followed fom Newtonian mechanics and the Coulomb foce law, with one small but cucial modification of Newtonian mechanics. The angula momentum of the electon could not vay continuously, it had to have special values, be quantized in units of Planck's constant divided by 2π, h/2π. In Boh's theoy, the diffeent allowed obits coesponded to obits with diffeent allowed values of angula momentum. Again we see Planck's constant appeaing at just the point whee Newtonian mechanics is beaking down. Thee is no way one can explain fom Newtonian mechanics why the electons in the hydogen atom could have only specific quantized values of angula momentum. While Boh's model of hydogen epesented only a slight modification of Newtonian mechanics, it epesented a majo philosophical shift. Newtonian mechanics could no longe be consideed the basic theoy govening the behavio of paticles and matte. Something had to eplace Newtonian mechanics, but fom the time of Boh's theoy in 1913 until 1924, no one knew what the new theoy would be. In 1924, a Fench gaduate student, Louis de Boglie, made a cucial suggestion that was the key that led to the new mechanics. This suggestion was quickly followed up by Schödinge and Heisenbeg who developed the new mechanics called quantum mechanics. In this chapte ou focus will be on the developments leading to de Boglie's idea.
35-2 Boh Theoy of Hydogen THE CLASSICAL HYDROGEN ATOM With Ruthefod's discovey of the atomic nucleus, it became clea that atoms consisted of a positively chaged nucleus suounded by negatively chaged electons that wee held to the nucleus by an electic foce. The simplest atom would be hydogen consisting of one poton and one electon held togethe by a Coulomb foce of magnitude F e = e2 2 p F e F e e (1) (Fo simplicity we will use CGS units in descibing the hydogen atom. We do not need the engineeing units, and we avoid the complicating facto of 1/4πε 0 in the electic foce fomula.) As shown in Equation 1, both the poton and the electon attact each othe, but since the poton is 1836 times moe massive than the electon, the poton should sit nealy at est while the electon obits aound it. Thus the hydogen atom is such a simple system, with known masses and known foces, that it should be a staightfowad matte to make detailed pedictions about the natue of the atom. We could use the obit pogam of Chapte 8, eplacing the gavitational foce GMm/ 2 by e 2 / 2. We would pedict that the electon moved in an elliptical obit about the poton, obeying all of Keple's laws fo obital motion. Thee is one impotant point we would have to take into account in ou analysis of the hydogen atom that we did not have to woy about in ou study of satellite motion. The electon is a chaged paticle, and acceleated chaged paticles adiate electomagnetic waves. Suppose, fo example, that the electon wee in a cicula obit moving at an angula velocity ω as shown in Figue (1a). If we wee looking at the obit fom the side, as shown in Figue (1b), we would see an electon oscillating up and down with a velocity given by v=v 0 sin ωt. Fo an electon in a cicula obit, pedicting the motion is quite easy. If an electon is in an obit of adius, moving at a speed v, then its acceleation a is diected towad the cente of the cicle and has a magnitude a = v2 (2) Using Equation 1 fo the electic foce and Equation 2 fo the acceleation, and noting that the foce is in the same diection as the acceleation, as indicated in Figue (2), Newton's second law gives F =ma e 2 2 =mv2 (3) One facto of cancels and we can immediately solve fo the electon's speed v to get v 2 =e 2 /m, o v electon = e (4) m The peiod of the electon's obit should be the distance 2π tavelled, divided by the speed v, o 2π/v seconds pe cycle, and the fequency should be the invese of that, o v/2π cycles pe second. Using Equation 4 fo v, we get fequency of electon in obit = v 2π = e 2π m (5) Accoding to Maxwell's theoy, this should also be the fequency of the adiation emitted by the electon. p v 0 e e p v = v sin(ωt) 0 In ou discussion of adio antennas in Chapte 32, we saw that adio waves could be poduced by moving electons up and down in an antenna wie. If electons oscillated up and down at a fequency ω, they poduced adio waves of the same fequency. Thus it is a pediction of Maxwell's equations that the electon in the hydogen atom should emit electomagnetic adiation, and the fequency of the adiation should be the fequency at which the electon obits the poton. a) electon in cicula obit Figue 1 The side view of cicula motion is an up and down oscillation. b) side view of cicula obit
35-3 Electomagnetic adiation caies enegy. Thus, to see what effect this has on the electon s obit, let us look at the fomula fo the enegy of an obiting electon. Fom Equation 3 we can immediately solve fo the electon's kinetic enegy. The esult is 1 2 mv2 = e2 2 electon kinetic enegy (6) The electon also has electic potential enegy just as an eath satellite had gavitational potential enegy. The fomula fo the gavitational potential enegy of a satellite was potential enegy of an eath satellite = GMm (10-50a) whee M and m ae the masses of the eath and the satellite espectively. This is the esult we used in Chapte 8 to test fo consevation of enegy (Equations 8-29 and 8-31) and in Chapte 10 whee we calculated the potential enegy (Equations 10-50a and 10-51). The minus sign indicated that the gavitational foce is attactive, that the satellite stats with zeo potential enegy when = and loses potential enegy as it falls in towad the eath. We can convet the fomula fo gavitational potential enegy to a fomula fo electical potential enegy by compaing fomulas fo the gavitational and electic foces on the two obiting objects. The foces ae F gavity = GMm 2 ; F electic = e2 2 Since both ae 1/ 2 foces, we can go fom the gavitational to the electic foce fomula by eplacing the p a v F e e constant GMm by e 2. Making this same substitution in the potential enegy fomula gives PE = e2 electical potential enegy of the electon in the hydogen atom (7) Again the potential enegy is zeo when the paticles ae infinitely fa apat, and the electon loses potential enegy as it falls towad the poton. (We used this esult in the analysis of the binding enegy of the hydogen molecule ion, explicitly in Equation 18-15.) The fomula fo the total enegy E total of the electon in hydogen should be the sum of the kinetic enegy, Equation 6, and the potential enegy, Equation 7. E total = kinetic enegy + potential enegy = e2 2 E total = e2 2 e2 total enegy of electon (8) The significance of the minus ( ) sign is that the electon is bound. Enegy is equied to pull the electon out, to ionize the atom. Fo an electon to escape, its total enegy must be bought up to zeo. We ae now eady to look at the pedictions that follow fom Equations 5 and 8. As the electon adiates light it must lose enegy and its total enegy must become moe negative. Fom Equation 8 we see that fo the electon's enegy to become moe negative, the adius must become smalle. Then Equation 5 tells us that as the adius becomes smalle, the fequency of the adiation inceases. We ae lead to the pictue of the electon spialing in towad the poton, adiating even highe fequency light. Thee is nothing to stop the pocess until the electon cashes into the poton. It is an unambiguous pediction of Newtonian mechanics and Maxwell's equations that the hydogen atom is unstable. It should emit a continuously inceasing fequency of light until it collapses. Figue 2 Fo a cicula obit, both the acceleation a and the foce F point towad the cente of the cicle. Thus we can equate the magnitudes of F and ma.
35-4 Boh Theoy of Hydogen Enegy Levels By 1913, when Neils Boh was tying to undestand the behavio of the electon in hydogen, it was no supise that Maxwell's equations did not wok at an atomic scale. To explain blackbody adiation and the photoelectic effect, Planck and Einstein wee led to the pictue that light consists of photons athe than Maxwell's waves of electic and magnetic foce. To constuct a theoy of hydogen, Boh knew the following fact. Hydogen gas at oom tempeatue emits no light. To get adiation, it has to be heated to athe high tempeatues. Then you get distinct spectal lines athe than the continuous adiation spectum expected classically. The visible spectal lines ae the H α, H β and H γ lines we saw in the hydogen spectum expeiment. These and many infa ed lines we saw in the spectum of the hydogen sta, Figue (33-28) epoduced below, make up the Balme seies of lines. Something must be going on inside the hydogen atom to poduce these shap spectal lines. The question is, why does the electon in hydogen emit only cetain enegy photons? The answe is Boh's main contibution to physics. Boh assumed that the electon had, fo some eason, only cetain allowed enegies in the hydogen atom. He called these allowed enegy levels. When an electon jumped fom one enegy level to anothe, it emitted a photon whose enegy was equal to the diffeence in the enegy of the two levels. The ed 1.89 ev photon, fo example, was adiated when the electon fell fom one enegy level to anothe level 1.89 ev lowe. Thee was a bottom, lowest enegy level below which the electon could not fall. In cold hydogen, all the electons wee in the bottom enegy level and theefoe emitted no light. When the hydogen atom is viewed in tems of Boh s enegy levels, the whole pictue becomes extemely simple. The lowest enegy level is at -13.6 ev. This is the total enegy of the electon in any cold hydogen atom. It equies 13.6 ev to ionize hydogen to ip an electon out. Viewing the light adiated by hydogen in tems of Einstein's photon pictue, we see that the hydogen atom emits photons with cetain pecise enegies. As an execise in the last chapte you wee asked to calculate, in ev, the enegies of the photons in the H α, H β and H γ spectal lines. The answes ae E Hα E Hβ = 1.89 ev = 2.55 ev E Hγ = 2.86 ev (9) Figue 3 Enegy level diagam fo the hydogen atom. All the enegy levels ae given by the simple fomula E n = 13.6/n 2 ev. All Balme seies lines esult fom jumps down to the n = 2 level. The 3 jumps shown give ise to the thee visible hydogen lines. 0.544.850 1.51 3.40 H α H β H γ n = 5 n = 4 n = 3 n = 2 5 5 5 5 5 3.65 10 3.70 10 3.75 10 wavelength 3.80 10 3.85 10 H40 H30 H20 H15 H14 H13 H12 H11 H10 H9 Figue 33-28 Spectum of a hydogen sta 13.6 n = 1
35-5 The fist enegy level above the bottom is at 3.40 ev which tuns out to be ( 13.6/4) ev. The next level is at 1.51 ev which is ( 13.6/9) ev. All of the enegy levels needed to explain evey spectal line emitted by hydogen ae given by the fomula E n = 13.6 ev n 2 (10) whee n takes on the intege values 1, 2, 3,... These enegy levels ae shown in Figue (3). Execise 1 Use Equation 10 to calculate the lowest 5 enegy levels and compae you answe with Figue 3. Let us see explicitly how Boh's enegy level diagam explains the spectum of light emitted by hydogen. If, fo example, an electon fell fom the n=3 to the n=2 level, the amount of enegy E 3 2 it would lose and theefoe the enegy it would adiate would be E 3 2 =E 3 E 2 = 1.51 ev ( 3.40 ev) = 1.89 ev = enegy lost in falling fom n = 3 to n = 2 level (11) which is the enegy of the ed photons in the H α line. Execise 2 Show that the H β and H γ lines coespond to jumps to the n = 2 level fom the n = 4 and the n = 5 levels espectively. Fom Execise 2 we see that the fist thee lines in the Balme seies esult fom the electon falling fom the thid, fouth and fifth levels down to the second level, as indicated by the aows in Figue (3). All of the lines in the Balme esult fom jumps down to the second enegy level. Fo histoical inteest, let us see how Balme's fomula fo the wavelengths in this seies follows fom Boh's fomula fo the enegy levels. Fo Balme's fomula, the lines we have been calling H α, H β and H γ ae H 3, H 4, H 5. An abitay line in the seies is denoted by H n, whee n takes on the values stating fom 3 on up. The Balme fomula fo the wavelength of the H n line is fom Equation 33-6 λ n = 3.65 10 5 cm n 2 n 2 4 (33-6) Refeing to Boh's enegy level diagam in Figue (3), conside a dop fom the nth enegy level to the second. The enegy lost by the electon is ( E n E 2 ) which has the value E n E 2 = 13.6 ev n 2 13.6 ev 2 2 enegy lost by electon going fom nth to second level This must be the enegy E H n caied out by the photon in the H n spectal line. Thus EH n = 13.6 ev 1 4 1 n 2 = 13.6 ev n2 4 4n 2 (12) We now use the fomula λ = 12.4 10 5 cm ev E photon in ev (34-8) elating the photon's enegy to its wavelength. Using Equation 12 fo the photon enegy gives λ n = 12.4 10 5 cm ev 13.6 ev λ n = 3.65 10 5 cm which is Balme's fomula. n2 n 2 4 4n 2 n 2 4
35-6 Boh Theoy of Hydogen It does not take geat intuition to suspect that thee ae othe seies of spectal lines beyond the Balme seies. The photons emitted when the electon falls down to the lowest level, down to -13.6 ev as indicated in Figue (4), fom what is called the Lyman seies. In this seies the least enegy photon, esulting fom a fall fom -3.40 ev down to -13.6 ev, has an enegy of 10.2 ev, well out in the ultaviolet pat of the spectum. All the othe photons in the Lyman seies have moe enegy, and theefoe ae fathe out in the ultaviolet. It is inteesting to note that when you heat hydogen and see a Balme seies photon like H α, H β o H γ, eventually a 10.2 ev Lyman seies photon must be emitted befoe the hydogen can get back down to its gound state. With telescopes on eath we see many hydogen stas adiating Balme seies lines. We do not see the Lyman seies lines because these ultaviolet photons do not make it down though the eath's atmosphee. But the Lyman seies lines ae all visible using obiting telescopes like the Ultaviolet Exploe and the Hubble telescope. 0.544.850 1.51 n = 5 n = 4 n = 3 Anothe seies, all of whose lines lie in the infa ed, is the Paschen seies, epesenting jumps down to the n = 3 enegy level at -1.55 ev, as indicated in Figue (5). Thee ae othe infa ed seies, epesenting jumps down to the n = 4 level, n = 5 level, etc. Thee ae many seies, each containing many spectal lines. And all these lines ae explained by Boh's conjectue that the hydogen atom has cetain allowed enegy levels, all given by the simple fomula E n =( 13.6/n 2 )ev. This one simple fomula explains a huge amount of expeimental data on the spectum of hydogen. Execise 3 Calculate the enegies (in ev) and wavelengths of the 5 longest wavelength lines in (a) the Lyman seies (b) the Paschen seies On a Boh enegy level diagam show the electon jumps coesponding to each line. Execise 4 In Figue (33-28), epeated 2 pages back, we showed the spectum of light emitted by a hydogen sta. The lines get close and close togethe as we get to H 40 and just beyond. Explain why the lines get close togethe and calculate the limiting wavelength. 3.40 n = 2 Figue 4 The Lyman seies consists of all jumps down to the 13.6eV level. (Since this is as fa down as the electon can go, this level is called the gound state.) 0 Figue 5.278 n = 7 The Paschen seies.378 6.544 n = 5 consists of all jumps.850 n = 4 down to the n = 3 level. These ae all in the infa ed. 1.51 n = 3 13.6 n = 1
35-7 THE BOHR MODEL Whee do Boh's enegy levels come fom? Cetainly not fom Newtonian mechanics. Thee is no excuse in Newtonian mechanics fo a set of allowed enegy levels. But did Newtonian mechanics have to be ejected altogethe? Planck was able to explain the blackbody adiation fomula by patching up classical physics, by assuming that, fo some eason, light was emitted and absobed in quanta whose enegy was popotional to the light's fequency. The eason why Planck's tick woked was undestood late, with Einstein's poposal that light actually consisted of paticles whose enegy was popotional to fequency. Blackbody adiation had to be emitted and absobed in quanta because light itself was made up of these quanta. By 1913 it had become espectable, fustating pehaps, but espectable to modify classical physics in ode to explain atomic phenomena. The hope was that a deepe theoy would come along and natually explain the modifications. What kind of a theoy do we constuct to explain the allowed enegy levels in hydogen? In the classical pictue we have a miniatue sola system with the poton at the cente and the electon in obit. This can be simplified by esticting the discussion to cicula obits. Fom ou ealie wok with the classical model of hydogen, we saw that an electon in an obit of adius had a total enegy E() given by E() = e2 2 total enegy of an electon in a cicula obit of adius (8 epeated) If the electon can have only cetain allowed enegies E n = 13.6/n 2 ev, then if Equation (8) holds, the electon obits can have only cetain allowed obits of adius n given by E n = e2 2 n (13) The n ae the adii of the famous Boh obits. This leads to the athe peculia pictue that the electon can exist in only cetain allowed obits, and when the electon jumps fom one allowed obit to anothe, it emits a photon whose enegy is equal to the diffeence in enegy between the two obits. This model is indicated schematically in Figue (6). Execise 5 Fom Equation 13 and the fact that E 1 = 13.6 ev, calculate the adius of the fist Boh obit 1. [Hint: fist convet ev to egs.] This is known as the Boh adius and is in fact a good measue of the actual adius of a cold hydogen atom. [The answe is 1 =.529 10 8 cm =.529A.] Then show that n =n 2 1. Figue 6 The Boh obits ae detemined by equating the allowed enegy E n = 13.6 n 2 to the enegy E n = e 2 2 n fo an electon in an obit of adius n. The Lyman seies epesents all jumps down to the smallest obit, the Balme seies to the second obit, the Paschen seies to the thid obit, etc. (The adii in this diagam ae not to scale, the adii n incease in size as n 2, as you can easily show by equating the two values fo E n.) 3 2 1 Lyman seies Balme seies Paschen seies
35-8 Boh Theoy of Hydogen Angula Momentum in the Boh Model Nothing in Newtonian mechanics gives the slightest hint as to why the electon in hydogen should have only cetain allowed obits. In the classical pictue thee is nothing special about these paticula adii. But eve since the time of Max Planck, thee was a special unit of angula momentum, the amount given by Planck's constant h. Since Planck's constant keeps appeaing wheneve Newtonian mechanics fails, and since Planck's constant has the dimensions of angula momentum, pehaps thee was something special about the electon's angula momentum when it was in one of the allowed obits. We can check this idea by e expessing the electon's total enegy not in tems of the obital adius, but in tems of its angula momentum L. We fist need the fomula fo the electon's angula momentum when in a cicula obit of adius. Back in Equation 4, we found that the speed v of the electon was given by v= e (4 epeated) m Multiplying this though by m gives us the electon's linea momentum mv mv = me =e m m (14) The electon's angula momentum about the cente of the cicle is its linea momentum mv times the leve am, as indicated in the sketch of Figue (7). The esult is L= mv= e m =em whee we used Equation 14 fo mv. (15) The next step is to expess in tems of the angula momentum L. Squaing Equation 13 gives L 2 =e 2 m o = L2 e 2 (16) m Finally we can eliminate the vaiable in favo of the angula momentum L in ou fomula fo the electon's total enegy. We get total enegy of the electon E = e2 2 = e 2 2 L 2 e 2 m = e2 2 e 2 m L 2 = e4 m 2L 2 (17) In the fomula e 4 m/2l 2 fo the electon's enegy, only the angula momentum L changes fom one obit to anothe. If the enegy of the nth obit is E n, then thee must be a coesponding value L n fo the angula momentum of the obit. Thus we should wite E n = e4 m (18) 2 2L n v L = mv m Figue 7 Angula momentum of a paticle moving in a cicle of adius.
35-9 At this point, Boh had the clue as to how to modify Newtonian mechanics in ode to get his allowed enegy levels. Suppose that angula momentum is quantized in units of some quantity we will call L 0. In the smallest obit, suppose it has one unit, i.e., L 1 =1 L 0. In the second obit assume it has twice as much angula momentum, L 2 =2L 0. In the nth obit it would have n units L n =nl 0 quantization of angula momentum Substituting Equation 19 into Equation 18 gives E n = e4 m 2L 2 0 (19) 1 n 2 (20) as the total enegy of an electon with n units of angula momentum. Compaing Equation 20 with Boh's enegy level fomula E n = 13.6 ev 1 n 2 (10 epeated) we see that we can explain the enegy levels by assuming that the electon in the nth enegy level has n units of quantized angula momentum L 0. We can also evaluate the size of L 0 by equating the constant factos in Equations 10 and 20. We get e 4 m = 13.6 ev (21) 2 2L 0 Conveting 13.6 ev to egs, and solving fo L 0 gives e 4 m 2L 0 2 = 13.6 ev 1.6 10 12 egs ev With e = 4.8 10 10 esu and m =.911 10 27 gm in CGS units, we get L 0 = 1.05 10 27 gm cm 2 sec (22) which tuns out to be Planck's constant divided by 2π. This quantity, Planck's constant divided by 2π, appeas so often in physics and chemisty that it is given the special name h ba and is witten h h 2π h "h ba" (23) Using h fo L 0 in the fomula fo E n, we get Boh's fomula E n = e4 m 2h 2 1 n 2 (24) whee e 4 m/2h 2, expessed in electon volts, is 13.6 ev. This quantity is known as the Rydbeg constant. [Remembe that we ae using CGS units, whee e is in esu, m in gams, and h is eg-sec.] Execise 6 Use Equation 21 to evaluate L 0. Execise 7 What is the fomula fo the fist Boh adius in tems of the electon mass m, chage e, and Planck's constant h. Evaluate you esult and show that 1 =.51 10 8 cm =.51A. (Answe: 1 =h 2 /e 2 m.) Execise 8 Stating fom Newtonian mechanics and the Coulomb foce law F=e 2 / 2, wite out a clea and concise deivation of the fomula E n = e4 m 1 2h 2 n 2 Explain the cucial steps of the deivation. A day o so late, on an empty piece of pape and a clean desk, see if you can epeat the deivation without looking at notes. When you can, you have a secue knowledge of the Boh theoy. L 0 = 2π h = 6.63 10 27 gm cm/sec 2π = 1.05 10 27 gm cm sec
35-10 Boh Theoy of Hydogen Execise 9 An ionized helium atom consists of a single electon obiting a nucleus containing two potons as shown in Figue (8). Thus the Coulomb foce on the electon has a magnitude F e = e 2e 2 = 2e2 2 Figue 8 Ionized helium has a nucleus with two potons, suounded by one electon. 2e e a) Using Newtonian mechanics, calculate the total enegy of the electon. (You answe should be e 2 /. Note that the is not squaed.) b) Expess this enegy in tems of the electon's angula momentum L. (Fist calculate L in tems of, solve fo, and substitute as we did in going fom Equations 16 to 17.) c) Find the fomula fo the enegy levels of the electon in ionized helium, assuming that the electon's angula momentum is quantized in units of h. d) Figue out whethe ionized helium emits any visible spectal lines (lines with photon enegies between 1.8 ev and 3.1 ev.) How many visible lines ae thee and what ae thei wavelengths?) Execise 10 You can handle all single electon atoms in one calculation by assuming that thee ae z potons in the nucleus. (z = 1 fo hydogen, z = 2 fo ionized helium, z = 3 fo doubly ionized lithium, etc.) Repeat pats a), b), and c) of Execise 9 fo a single electon atom with z potons in the nucleus. (Thee is no simple fomula fo multi electon atoms because of the epulsive foce between the electons.) DE BROGLIE'S HYPOTHESIS Despite its spectacula success descibing the specta of hydogen and othe one-electon atoms, Boh's theoy epesented moe of a poblem than a solution. It woked only fo one electon atoms, and it pointed to an explicit failue of Newtonian mechanics. The idea of coecting Newtonian mechanics by equiing the angula momentum of the electon be quantized in units of h, while successful, epesented a bandaid teatment. It simply coveed a deepe wound in the theoy. Fo two centuies Newtonian mechanics had epesented a complete, consistent scheme, applicable without exception. Special elativity did not ham the integity of Newtonian mechanics elativistic Newtonian mechanics is a consistent theoy compatible with the pinciple of elativity. Even geneal elativity, with its concepts of cuved space, left Newtonian mechanics intact, and consistent, in a slightly alteed fom. The famewok of Newtonian mechanics could not be alteed to include the concept of quantized angula momentum. Boh, Sommefield, and othes tied duing the decade following the intoduction of Boh's model, but thee was little success. In Pais, in 1923, a gaduate student Louis de Boglie, had an idea. He noted that light had a wave natue, seen in the 2-slit expeiment and Maxwell's theoy, and a paticle natue seen in Einstein's explanation of the photoelectic effect. Physicists could not explain how light could behave as a paticle in some expeiments, and a wave in othes. This poblem seemed so inconguous that it was put on the back bune, moe o less ignoed fo nealy 20 yeas. De Boglie's idea was that, if light can have both a paticle and a wave natue, pehaps electons can too! Pehaps the quantization of the angula momentum of an electon in the hydogen atom was due to the wave natue of the electon. The main question de Boglie had to answe was how do you detemine the wavelength of an electon wave?
35-11 An analogy with photons might help. Thee is, howeve, a significant diffeence between electons and photons. Electons have a est mass enegy and photons do not, thus thee can be no diect analogy between the total enegies of the two paticles. But both paticles have mass and cay linea momentum, and the amount of momentum can vay fom zeo on up fo both paticles. Thus photons and electons could have simila fomulas fo linea momentum. Back in Equation 34-13 we saw that the linea momentum p of a photon was elated to its wavelength λ by the simple equation λ = h p de Boglie wavelength (34-13) De Boglie assumed that this same elationship also applied to electons. An electon with a linea momentum p would have a wavelength λ =h/p. This is now called the de Boglie wavelength. This elationship applies not only to photons and electons, but as fa as we know, to all paticles! With a fomula fo the electon wavelength, de Boglie was able to constuct a simple model explaining the quantization of angula momentum in the hydogen atom. In de Boglie's model, one pictues an electon wave chasing itself aound a cicle in the hydogen atom. If the cicumfeence of the cicle, 2π did not have an exact integal numbe of wavelengths, then the wave, afte going aound many times, would eventually cancel itself out as illustated in Figue (9). But if the cicumfeence of the cicle wee an exact integal numbe of wavelengths as illustated in Figue (10), thee would be no cancellation. This would theefoe be one of Boh's allowed obits shown in Figue (6). Suppose (n) wavelengths fit aound a paticula cicle of adius n. Then we have nλ =2π n (25) Using the de Boglie fomula λ =h/p fo the electon wavelength, we get n h p =2π n (26) Multiplying both sides by p and dividing though by 2π gives n 2π h =p n (27) Now h/2π is just h, and p n is the angula momentum L n (momentum times leve am) of the electon. Thus Equation 27 gives nh =p n =L n (28) Equation 28 tells us that fo the allowed obits, the obits in which the electon wave does not cancel, the angula momentum comes in intege amounts of the angula momentum h. The quantization of angula momentum is thus due to the wave natue of the electon, a concept completely foeign to Newtonian mechanics. Figue 9 De Boglie pictue of an electon wave cancelling itself out. Figue 10 If the cicumfeence of the obit is an intege numbe of wavelengths, the electon wave will go aound without any cancellation. Figue 10a--Movie The standing waves on a cicula metal band nicely illustate de Boglie s waves
35-12 Boh Theoy of Hydogen When a gaduate student does a thesis poject, typically the student does a lot of wok unde the supevision of a thesis adviso, and comes up with some new, hopefully veifiable, esults. What do you do with a student that comes up with a stange idea, completely unveified, that can be explained in a few pages of algeba? Einstein happened to be passing though Pais in the summe of 1924 and was asked if de Boglie's thesis should be accepted. Although doubtful himself about a wave natue of the electon, Einstein ecommended that the thesis be accepted, fo de Boglie just might be ight. In 1925, two physicists at Bell Telephone Laboatoies, C. J. Davisson and L. H. Geme wee studying the suface of nickel by scatteing electons fom the suface. The point of the eseach was to lean moe about metal sufaces in ode to impove the quality of switches used in telephone communication. Afte heating the metal taget to emove an oxide laye that accumulated following a beak in the vacuum line, they discoveed that the electons scatteed diffeently. The electon gun metal had cystallized duing the heating, and the peculia scatteing had occued as a esult of the cystallization. Davisson and Geme then pepaed a taget consisting of a single cystal, and studied the peculia scatteing phenomena extensively. Thei appaatus is illustated schematically in Figue (11), and thei expeimental esults ae shown in Figue (12). Fo thei expeiment, thee was a maked peak in the scatteing when the detecto was located at an angle of 50 fom the incident beam. Davisson pesented these esults at a meeting in London in the summe of 1927. At that time thee was a consideable discussion about de Boglie's hypothesis that electons have a wave natue. Heaing of this idea, Davisson ecognized the eason fo the scatteing peak. The atoms of the cystal wee diffacting electon waves. The enhanced scatteing at 50 was a diffaction peak, a maximum simila to the eflected maxima we saw back in Figue (33-19) when light goes though a diffaction gating. Davisson had the expeimental evidence that de Boglie's idea about electon waves was coect afte all. detecto Reflected maximum θ = 50 θ electon beam nickel cystal Figue 11 Scatteing electons fom the suface of a nickel cystal. tansmitted maximum Figue 33-19 Lase beam impinging on a diffaction gating. Figue 12 Plot of intensity vs. angle fo electons scatteed by a nickel cystal, as measued by Davisson and Geme. The peak in intensity at 50 was a diffaction peak like the ones poduced by diffaction gatings. (The intensity is popotional to the distance out fom the oigin.)
35-13 Index Symbols 13.6 ev, hydogen spectum 35-4 A Allowed obits, Boh theoy 35-1 Angula momentum Boh model 35-1, 35-8 Planck's constant 35-8 Atoms Classical hydogen atom 35-2 B Balme seies Enegy level diagam fo 35-6 Fomula fom Boh theoy 35-5 Hydogen spectum 35-4 Bell Telephone Lab, electon waves 35-12 Boh model Allowed obits 35-1 Angula momentum 35-1, 35-8 Chapte on 35-1 De Boglie explanation 35-1 Deivation of 35-8 Enegy levels 35-4 Planck's constant 35-1, 35-8 Quantum mechanics 35-1 Rydbeg constant 35-9 Boh obits, adii of 35-7 C CGS units Classical hydogen atom 35-2 Cicula obit, classical hydogen atom 35-2 Classical hydogen atom 35-2 Coulomb's law Classical hydogen atom 35-2 D Davisson & Geme, electon waves 35-12 De Boglie Electon waves 35-11 Fomula fo momentum 35-11 Hypothesis 35-10 Key to quantum mechanics 35-1 Wavelength, fomula fo 35-11 Waves, movie of standing wave model 35-11 E Electomagnetic adiation Enegy adiated by classical H atom 35-3 Electon In classical hydogen atom 35-2 Electon scatteing Fist expeiment on wave natue 35-12 Electon waves Davisson & Geme expeiment 35-12 De Boglie pictue 35-11 Scatteing of 35-12 Enegy Electic potential enegy In classical hydogen atom 35-3 Enegy level 35-1 Kinetic enegy Boh model of hydogen 35-3 Classical hydogen atom 35-3 Total enegy Classical H atom 35-3 Enegy level diagam Balme seies 35-6 Boh theoy 35-4 Lyman seies 35-6 Paschen seies 35-6 F Foce Electic foce Classical hydogen atom 35-2 H h ba, Planck's constant 35-9 Hydogen atom Boh theoy 35-1 Classical 35-2 Hydogen atom, classical Failue of Newtonian mechanics 35-3 Hydogen spectum Balme seies 35-4 Lyman seies 35-6 Of sta 35-4 Paschen seies 35-6 I Infaed light Paschen seies, hydogen specta 35-6 K Kinetic enegy Boh model of hydogen 35-3 Classical hydogen atom 35-3 L Light Hydogen spectum Balme fomula 35-5 Spectal lines, hydogen Boh theoy 35-4 Lyman seies, enegy level diagam 35-6
35-14 Boh Theoy of Hydogen M Maxwell's equations Failue of In classical hydogen atom 35-2 Mechanics Newtonian Classical H atom 35-3 Momentum De Boglie fomula fo momentum 35-11 Movie Standing De Boglie like waves 35-11 N Newtonian mechanics Classical H atom 35-3 Failue of In the classical hydogen atom 35-3 Nucleus Discovey of, Ruthefod 35-1 O Obits Boh, adii of 35-7 Classical hydogen atom 35-2 P Paticle-wave natue De Boglie pictue 35-10 Of electons Davisson and Geme expeiment 35-12 De Boglie pictue 35-10 Paschen seies Enegy level diagam 35-6 Hydogen specta 35-6 Photon Hydogen spectum 35-5 Planck's constant Angula momentum, Boh model 35-8 Boh theoy 35-1 In de Boglie wavelength fomula 35-11 Potential enegy Electic potential enegy In classical hydogen atom 35-3 Q Quantized angula momentum In Boh theoy 35-9 In de Boglie's hypothesis 35-10 Quantum mechanics Boh theoy of hydogen 35-1 R Radiation Radiated enegy and the classical H atom 35-3 Radio waves Pedicted fom the classical hydogen atom 35-2 Ruthefod and the nucleus 35-1 Rydbeg constant, in Boh theoy 35-9 S Satellite motion Classical hydogen atom 35-2 Scatteing of waves Davisson-Geme expeiment 35-12 Spectal lines Hydogen Boh theoy of 35-4 Spectum Hydogen Boh theoy of 35-4 Lyman seies, ultaviolet 35-6 Paschen seies, infaed 35-6 Hydogen sta 35-4 Standing waves De Boglie waves Movie 35-11 Sta Hydogen spectum of 35-4 T Total enegy Classical hydogen atom 35-3 U Unit of angula momentum In Boh theoy 35-9 W Wave De Boglie, standing wave movie 35-11 Electon waves, de Boglie pictue 35-11 Wavelength De Boglie 35-11 X x-ch35 Execise 1 35-5 Execise 2 35-5 Execise 3 35-6 Execise 4 35-6 Execise 5 35-7 Execise 6 35-9 Execise 7 35-9 Execise 8 35-9 Execise 9 35-10 Execise 10 35-10