Chapter 17 The Kepler Problem: Planetary Mechanics and the Bohr Atom


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1 Chapte 7 The Keple Poblem: Planetay Mechanics and the Boh Atom Keple s Laws: Each planet moves in an ellipse with the sun at one focus. The adius vecto fom the sun to a planet sweeps out equal aeas in equal time. The peiod of evolution T of a planet about the sun is elated to the majo axis A of the ellipse by 3 T = k A whee k is the same fo all planets. 7. Planetay Obits: The Keple Poblem Intoduction Since Johannes Keple fist fomulated the laws that descibe planetay motion, scientists endeavoed to solve fo the equation of motion of the planets. In his hono, this poblem has been named The Keple Poblem. When thee ae moe than two bodies, the poblem becomes impossible to solve exactly. The most impotant theebody poblem at the time involved finding the motion of the moon, since the moon inteacts gavitationally with both the sun and the eath. Newton ealized that if the exact position of the moon wee known, the longitude of any obseve on the eath could be detemined by measuing the moon s position with espect to the stas. In the eighteenth centuy, Leonhad Eule and othe mathematicians spent many yeas tying to solve the theebody poblem, and they aised a deepe question. Do the small contibutions fom the gavitational inteactions of all the planets make the planetay system unstable ove long peiods of time? At the end of 8th centuy, Piee Simon Laplace and othes found a seies solution to this stability question, but it was unknown whethe o not the seies solution conveged afte a long peiod of time. Heni Poincaé poved that the seies actually diveged. Poincaé went on to invent new mathematical methods that poduced the moden fields of diffeential geomety and topology in ode to answe the stability question using geometic aguments, athe than analytic methods. Poincaé and othes did manage As stated in An Intoduction to Mechanics, Daniel Kleppne and Robet Kolenkow, McGawHill, 973, p 4. //9 7 
2 to show that the theebody poblem was indeed stable, due to the existence of peiodic solutions. Just as in the time of Newton and Leibniz and the invention of calculus, unsolved poblems in celestial mechanics became the expeimental laboatoy fo the discovey of new mathematics. 7. Reducing the TwoBody Poblem into a OneBody Poblem We shall begin ou solution of the twobody poblem by showing how the motion of two bodies inteacting via a gavitational foce (twobody poblem) is mathematically equivalent to the motion of a single body with a educed mass given by mm μ = m + m (7..) that is acted on by an extenal cental gavitational foce. Once we solve fo the motion of the educed body in this equivalent onebody poblem, we can then etun to the eal twobody poblem and solve fo the actual motion of the two oiginal bodies. The educed mass was intoduced in Section.7 of these notes. That section used simila but diffeent notation fom that used in this chapte. Conside the gavitational foce between two bodies with masses m and m as shown in Figue 7.. Figue 7. Gavitational foce between two bodies. Choose a coodinate system with a choice of oigin such that body has position and body has position (Figue 7.). The elative position vecto pointing fom body to body is =. We denote the magnitude of by =, whee is the distance between the bodies, and ˆ is the unit vecto pointing fom body to body, so that = ˆ (7..) //9 7 
3 Figue 7. Coodinate system fo the twobody poblem. The foce on body (due to the inteaction of the two bodies) can be descibed as mm F ˆ, = F, = G ˆ. (7..3) Recall that Newton s Thid Law equies that the foce on body is equal in magnitude and opposite in diection to the foce on body, F = F,, Newton s Second Law can be applied individually to the two bodies: F,. (7..4) d = m, (7..5) dt F d = m dt,. (7..6) Dividing though by the mass in each of Equations (7..5) and (7..6) yields F d, = dt m F d, = dt m, (7..7). (7..8) Subtacting the expession in Equation (7..8) fom that in Equation (7..7) gives //9 73
4 F F d d d = =. (7..9) m m dt dt dt,, Using Newton s Thid Law as given in Equation (7..4), Equation (7..9) becomes F d + =. (7..), m m dt Using the educed mass μ, as defined in Equation (7..),, μ = m + m (7..) Equation (7..) becomes F, d = μ dt d F, = μ dt (7..) whee F is given by Equation (7..3)., Ou esult has a special intepetation using Newton s Second Law. Let μ be the educed mass of a educed body with position vecto = ˆ with espect to an oigin O, whee ˆ is the unit vecto pointing fom the oigin O to the educed body. Then the equation of motion, Equation (7..), implies that the body of educed mass μ is unde the influence of an attactive gavitational foce pointing towad the oigin. So, the oiginal twobody gavitational poblem has now been educed to an equivalent onebody poblem, involving a educed body with educed mass μ unde the influence of a cental foce F ˆ,. Note that in this efomulation, thee is no body located at the cental point (the oigin O ). The paamete in the twobody poblem is the elative distance between the oiginal two bodies, while the same paamete in the onebody poblem is the distance between the educed body and the cental point. 7.3 Enegy and Angula Momentum, Constants of the Motion The equivalent onebody poblem has two constants of the motion, enegy E and the angula momentum L about the oigin O. Enegy is a constant because thee ae no //9 74
5 extenal foces acting on the educed body, and angula momentum is constant about the oigin because the only foce is diected towads the oigin, and hence the toque about the oigin due to that foce is zeo (the vecto fom the oigin to the educed body is antipaallel to the foce vecto and sinπ = ). Since angula momentum is constant, the obit of the educed body lies in a plane with the angula momentum vecto pointing pependicula to this plane. In the plane of the obit, choose pola coodinates (, θ ) fo the educed body (see Figue 7.3), whee is the distance of the educed body fom the cental point that is now taken as the oigin, and θ is the angle that the educed body makes with espect to a chosen diection, and which inceases positively in the counteclockwise diection. Figue 7.3 Coodinate system fo the obit of the educed body. Thee ae two appoaches to descibing the motion of the educed body. We can ty to find both the distance fom the oigin, t ( ) and the angle, θ ( t), as functions of the paamete time, but in most cases explicit functions can t be found analytically. We can also find the distance fom the oigin, ( θ ), as a function of the angle θ. This second appoach offes a spatial desciption of the motion of the educed body (see Appendix 7.A). The Obit Equation fo the Reduced Body Conside the educed body with educed mass given by Equation (7..), obiting about a cental point unde the influence of a adially attactive foce given by Equation (7..3). Since the foce is consevative, the potential enegy with choice of zeo efeence point U ( ) = is given by U () Gm m =. (7.3.) The total enegy enegy is E is constant, and the sum of the kinetic enegy and the potential //9 75
6 E v Gm m = μ. (7.3.) The kinetic enegy tem, μv /, has the educed mass and the elative speed v of the two bodies. As in Chaptes 5 and 7, we will use the notation v = v ˆ ˆ ad + vtanθ, d (7.3.3) v = v =, dt whee v = d dt and vtan = ( dθ / dt). Equation (7.3.) then becomes ad / d dθ G m m E μ = + dt dt. (7.3.4) The magnitude of the angula momentum with espect to the cente of mass is dθ dt = μ tan = μ. (7.3.5) L v We shall explicitly eliminate the θ dependence fom Equation (7.3.4) by using ou expession in Equation (7.3.5), dθ L =. (7.3.6) dt μ The mechanical enegy as expessed in Equation (7.3.4) then becomes d L G mm dt E = μ + μ. (7.3.7) Equation (7.3.7) is a sepaable diffeential equation involving the vaiable function of time t and can be solved fo the fist deivative d / dt, as a d L G m m = E + dt μ μ. (7.3.8) Equation (7.3.8) can in pinciple be integated diectly fo t (). In fact, in doing the integal no fewe than six cases need to be consideed, and even then the solution is of the fom t ( ) instead of t ( ). These integals ae pesented in Appendix 7.E. The function //9 76
7 t () can then, in pinciple, be substituted into Equation (7.3.6) and can then be integated to find θ ( t). Instead of solving fo the position of the educed body as a function of time, we shall find a geometic desciption of the obit by finding ( θ ). We fist divide Equation (7.3.6) by Equation (7.3.8) to obtain dθ L dθ dt μ = = d d L dt E μ + μ Gm m. (7.3.9) The vaiables and θ ae sepaable; L d μ dθ = L E μ + μ Gmm ( L/ ) d = μ L E + Gmm μ (7.3.). Equation (7.3.) can be integated to find the adius as a function of the angle θ ; see Appendix 7.A fo the exact integal solution. The esult is called the obit equation fo the educed body and is given by ε cosθ = (7.3.) whee L = (7.3.) μ Gm m is a constant (known as the semilatus ectum) and ε = + μ EL ( Gm m ) (7.3.3) //9 77
8 is the eccenticity of the obit. The two constants of the motion in tems of L= ( μ Gm m ) Gmm E = ( ε ). and ε ae (7.3.4) An altenate deivation of Equation (7.3.) is given in Appendix 7.F. The obit equation as given in Equation (7.3.) is a geneal conic section and is pehaps somewhat moe familia in Catesian coodinates. Let x= cosθ and y = sinθ, with = x + y. The obit equation can be ewitten as = + ε cosθ. (7.3.5) Using the Catesian substitutions fo x and y, ewite Equation (7.3.5) as Squaing both sides of Equation (7.3.6), ( ) / x + y = + ε x. (7.3.6) x + y = + ε x + ε x. (7.3.7) Afte eaanging tems, Equation (7.3.7) is the geneal expession of a conic section with axis on the x axis, ( ε ) x ε x y + = (7.3.8) (we now see that the dotted axis in Figue 7.3 can be taken to be the x axis). Fo a given >, coesponding to a given nonzeo angula momentum as in Equation (7.3.), thee ae fou cases detemined by the value of the eccenticity. Case : When ε =, E = E min < and =. Equation (7.3.8) is the equation fo a cicle, x y + = (7.3.9) Case : When < ε <, E < E < and Equation (7.3.8) descibes an ellipse, min y + Ax Bx= k (7.3.) //9 78
9 whee A > and k is a positive constant. ( Appendix 7.C shows how this expession may be expessed in the moe taditional fom involving the coodinates of the cente of the ellipse and the semimajo and semimino axes.) Case 3: When ε =, E = and Equation (7.3.8) descibes a paabola, x y =. (7.3.) Case 4: When ε >, E > and Equation (7.3.8) descibes a hypebola, whee A > and k is a positive constant. y Ax Bx= k (7.3.) 7.4 Enegy Diagam, Effective Potential Enegy, and Obits of Motion The enegy (Equation (7.3.7)) of the educed body moving in two dimensions can be eintepeted as the enegy of a educed body moving in one dimension, the adial diection, in an ective potential enegy given by two tems, U L Gm m μ =. (7.4.) The total enegy is still the same, but ou intepetation has changed; d L G m m E = K + U = μ + dt μ, (7.4.) whee the ective kinetic enegy K associated with the onedimensional motion is K d = μ dt. (7.4.3) The gaph of U as a function of = /, whee as given in Equation (7.3.), is shown in Figue 7.4. The uppe cuve (ed, if you can see this in colo) is popotional to L /( μ ) /. The lowe blue cuve is popotional to Gm m / /. The sum U is epesented by the geen cuve. The minimum value of U is at =, as will be shown analytically below. The vetical scale is in units of U. ( ) //9 79
10 Figue 7.4 Gaph of ective potential enegy. Wheneve the onedimensional kinetic enegy is zeo, K =, the enegy is equal to the ective potential enegy, E L Gm m μ = U =. (7.4.4) Recall that the potential enegy is defined to be the negative integal of the wok done by the foce. Fo ou eduction to a onebody poblem, using the ective potential, we will intoduce an ective foce such that U U F d F d B B, B, A = = A A (7.4.5) The fundamental theoem of calculus (fo one vaiable) then states that the integal of the deivative of the ective potential enegy function between two points is the ective potential enegy diffeence between those two points, B du U, B U, A d d A = (7.4.6) Compaing Equation (7.4.6) to Equation (7.4.5) shows that the adial component of the ective foce is the negative of the deivative of the ective potential enegy, //9 7 
11 F du d = (7.4.7) The ective potential enegy descibes the potential enegy fo a educed body moving in one dimension. (Note that the ective potential enegy is only a function of the vaiable and is independent of the vaiable θ ). Thee ae two contibutions to the ective potential enegy, and the total adial component of the foce is F d d L Gm m (7.4.8) d d μ = U = Thus thee ae two foces acting on the educed body, with an ective centifugal foce given by F = F + F, centifugal, gavity, (7.4.9) F d L L = = d μ μ 3,centifugal (7.4.) and the conventional gavitational foce F Gm m =. (7.4.), gavity With this nomenclatue, let s eview the fou cases pesented in Section 7.3. Case : Cicula Obit E = E min The lowest enegy state, enegy, E zeo since = ( U ) min min E = K + U E min, coesponds to the minimum of the ective potential. When this condition is satisfied the ective kinetic enegy is. The condition K d = μ = dt (7.4.) implies that the adial velocity is zeo, so the distance fom the cental point is a constant. This is the condition fo a cicula obit. The condition fo the minimum of the ective potential enegy is = du L G m m 3 d = μ +. (7.4.3) //9 7 
12 We can solve Equation (7.4.3) fo, L =, (7.4.4) Gmm epoducing Equation (7.3.). Case : Elliptic Obit Emin < E < When K =, the mechanical enegy is equal to the ective potential enegy, E = U, as in Equation (7.4.4). Having d / dt = coesponds to a point of closest o futhest appoach as seen in Figue 7.4. This condition coesponds to the minimum and maximum values of fo an elliptic obit, E L Gm m μ = (7.4.5) Equation (7.4.5) is a quadatic equation fo the distance, + Gm m L E μe = (7.4.6) with two oots E E μe Gmm Gmm L = ± + Equation (7.4.7) may be simplified somewhat as /. (7.4.7) Gm m LE E μ( Gmm) = ± + / (7.4.8) Fom Equation (7.3.3), the squae oot is the eccenticity ε, ε = + μ EL ( Gm m ), (7.4.9) //9 7 
13 and Equation (7.4.8) becomes Gm m E ( = ±. (7.4.) ε ) A little algeba shows that L / μ Gmm = ε LE + μ( Gmm) = LE/ μ( Gmm ) Gm m = E L / μgmm. (7.4.) Substituting the last expession in (7.4.) into Equation (7.4.) gives an expession fo the points of closest and futhest appoach, ( ε ). (7.4.) ε = ± The minus sign coesponds to the distance of closest appoach, ε and the plus sign coesponds to the distance of futhest appoach, Case 3: Paabolic Obit E = min = (7.4.3) + ε max =. (7.4.4) The ective potential enegy, as given in Equation (7.4.), appoaches zeo ( U ) when the distance appoaches infinity ( ). Since the total enegy is zeo, when the kinetic enegy also appoaches zeo, K. This coesponds to a paabolic obit (see Equation (7.3.)). Recall that in ode fo a body to escape fom a planet, the body must have a total enegy E = (we set U = at infinity). This escape velocity condition coesponds to a paabolic obit. Fo a paabolic obit, the body also has a distance of closest appoach. This distance pa can be found fom the condition //9 73
14 L Gmm E = U = =. (7.4.5) μ Solving Equation (7.4.5) fo yields L = = ; (7.4.6) pa μ Gmm the fact that the minimum distance to the oigin (the focus of a paabola) is half the semilatus ectum is a wellknown popety of a paabola. Case 4: Hypebolic Obit E > When E >, in the limit as the kinetic enegy is positive, K >. This coesponds to a hypebolic obit (see Equation (7.3.)). The condition fo closest appoach is simila to Equation (7.4.5) except that the enegy is now positive. This implies that thee is only one positive solution to the quadatic Equation (7.4.6), the distance of closest appoach fo the hypebolic obit The constant ε hyp =. (7.4.7) + is independent of the enegy and fom Equation (7.3.3) as the enegy of the educed body inceases, the eccenticity inceases, and hence fom Equation (7.4.7), the distance of closest appoach gets smalle. 7.5 Obits of the Two Bodies The obit of the educed body can be cicula, elliptical, paabolic o hypebolic, depending on the values of the two constants of the motion, the angula momentum and the enegy. Once we have the explicit solution (in this discussion, ( θ ) ) fo the educed body, we can find the actual obits of the two bodies. Choose a coodinate system as we did fo the eduction of the twobody poblem (Figue 7.5). //9 74
15 Figue 7.5 Cente of mass coodinate system. The cente of mass of the system is given by R cm m+ m =. (7.5.) m + m Let be the vecto fom the cente of mass to body and cente of mass to body. Then, by the geomety in Figue 7.5, = = be the vecto fom the (7.5.) and hence m + m m ( ) μ R. (7.5.3) = cm = = = m+ m m+ m m A simila calculation shows that μ =. (7.5.4) m Thus each body undegoes a motion about the cente of mass in the same manne that the educed body moves about the cental point given by Equation (7.3.). The only diffeence is that the distance fom eithe body to the cente of mass is shotened by a facto μ / mi. When the obit of the educed body is an ellipse, then the obits of the two bodies ae also ellipses, as shown in Figue 7.6. //9 75
16 Figue 7.6 The elliptical motion of bodies unde mutual gavitation. When one mass is much smalle than the othe, fo example mass is appoximately the smalle mass, m m, then the educed mm mm m + μ = = m m m (7.5.5) The cente of mass is located appoximately at the position of the lage mass, body of mass. Thus body moves accoding to m μ = m (7.5.6) and body is appoximately stationay, μ m = m m (7.5.7) 7.6 Keple s Laws Elliptic Obit Law Each planet moves in an ellipse with the sun at one focus. When the enegy is negative, E <, and accoding to Equation (7.3.3), //9 76
17 ε = + μ EL ( Gm m ) (7.6.) and the eccenticity must fall within the ange ε <. These obits ae eithe cicles o ellipses. Note the elliptic obit law is only valid if we assume that thee is only one cental foce acting. We ae ignoing the gavitational inteactions due to all the othe bodies in the univese, a necessay appoximation fo ou analytic solution. Equal Aea Law The adius vecto fom the sun to a planet sweeps out equal aeas in equal time. Using analytic geomety, the sum of the aeas of the tiangles in Figue 7.7 is given by ( Δθ) ( Δθ) Δ A= ( Δ θ) + Δ = ( Δ θ) + Δ (7.6.) in the limit of small Δ θ (the aea of the small piece on the ight, bounded on one side by the cicula segment, is appoximated by that of a tiangle). The aveage ate of the change of aea, Figue 7.7 Keple s equal aea law. Δ A, in time In the limit as Δt, Δθ, this becomes ( θ) ( θ) Δ t, is given by ΔA Δ Δ Δ = + Δt Δt Δ t. (7.6.3) da dt dθ dt =. (7.6.4) //9 77
18 Note that in this appoximation, we ae essentially neglecting the small piece on the ight in Figue 7.7 Recall that accoding to Equation (7.3.6) (epoduced below as Equation (7.6.5)), the angula momentum is elated to the angula velocity dθ / dt by dθ L = (7.6.5) dt μ and Equation (7.6.4) is then da L =. (7.6.6) dt μ Since L and μ ae constants, the ate of change of aea with espect to time is a constant. This is often familialy efeed to by the expession: equal aeas ae swept out in equal times (see Keple s Laws at the beginning of this chapte). Peiod Law The peiod of evolution T of a planet about the sun is elated to the majo axis the ellipse by 3 T = k A A of whee k is the same fo all planets. When Keple stated his peiod law fo planetay obits based on obsevation, he only noted the dependence on the lage mass of the sun. Since the mass of the sun is much geate than the mass of the planets, his obsevation is an excellent appoximation. Equation (7.6.6) can be ewitten in the fom Equation (7.6.7) can be integated as μ da = L. (7.6.7) dt μ da = L dt (7.6.8) obit T whee T is the peiod of the obit. Fo an ellipse, //9 78
19 Aea = da =π ab obit (7.6.9) whee a is the semimajo axis and b is the semimino axis. ( Appendix 7.D deives this esult fom Equation (7.3.).) Thus we have T μ π ab =. (7.6.) L Squaing Equation (7.6.) then yields T 4π μ ab =. (7.6.) L In Appendix 7.B, the angula momentum is given in tems of the semimajo axis and the eccenticity by Equation (B..). Substitution fo the angula momentum into Equation (7.6.) yields T 4π μ ab = μ Gm m a ( ε ). (7.6.) In Appendix 7.B, the semimino axis is given by Equation (B.3.7), which upon substitution into Equation (7.6.) yields T 3 4π μ a =. (7.6.3) μ Gm m Using Equation (7..) fo educed mass, the squae of the peiod of the obit is popotional to the semimajo axis cubed, T 4π a = G m 3 ( + m ). (7.6.4) 7.7 The Boh Atom Numeical values of physical constants ae fom the Paticle Data Goup tables, available fom //9 79
20 Conside the electic foce between two pointlike objects with chages q and q. The foce law is an invese squae law, like the gavitational foce. The diffeence is that the constant Gm m is eplaced by kq q whee 9 k = = N m C (7.7.) 4πε (the constant k is not a sping constant o the Boltzmann constant, meely a eflection of ou finite alphabet). The minus sign in the gavitational inteaction does not appea in the electic inteaction because thee ae two types of electic chage, positive and negative. The electic foce is attactive fo chages of opposite sign and epulsive fo chages of the same sign. Figue 7.8 Coulomb inteaction between two chages. q The foce on the chaged paticle of chage due to the electic inteaction between the two chaged paticles is given by Coulomb s law, qq F = ˆ ( ),, 4πε. (7.7.) Coulomb s Law povides an accuate desciption of the motion of chaged paticles when they ae not bound togethe. We cannot model the inteaction between the electon and the poton when the chaged paticles ae as close togethe as in the hydogen atom, since the Newtonian concept of foce is not welldefined at length scales associated with the size of atoms. Thus we need a new theoy, quantum mechanics, to explain the popeties of the atom. When the chaged paticles ae fa apat they ae essentially fee paticles and the quantum mechanical desciption of the bound system is not necessay. Theefoe the exact same method of solution that was used in the Keple Poblem fo obits of planets applies to the motion of chaged paticles. //9 7 
21 Fo a poton and an electon in a bound system, the hydogen atom, Niels Boh found a semiclassical agument that allows one to use the classical theoy of electic foces to pedict the obseved enegies of the hydogen atom. The following agument does not satisfy the pinciples of quantum mechanics, even though the esult is in easonable ageement with expeimentally detemined popeties of the hydogen atom. We begin ou discussion by ecalling ou esult, Equation (7.4.) fo the enegy of the gavitational system of two bodies when consideed as a single educed body of educed mass μ = mm /( m+ m) moving in one dimension, with the distance fom the cental point denoted by the vaiable, E = K + U (7.7.3) whee the ective potential enegy is L L U = U gavity G μ + = μ mm (7.7.4) and the ective kinetic enegy is K d = μ dt. (7.7.5) We can extend this desciption to the electic inteaction between the electon and poton of the hydogen atom by eplacing the constant Gmm with kqq, whee the chage of the poton is q = e, and the chage of the electon is q = e. The enegy is then given by d d L E = μ + U e = μ +. (7.7.6) dt dt μ 4πε Since the mass of the electon of the poton m p m e = kg is much smalle than the mass 7 =.6767 kg, the educed mass is appoximately the mass of the electon, μ m e. A schematic plot of the ective potential enegy as a function of the vaiable fo the hydogen atom is shown in Figue 7.9. //9 7 
22 Figue 7.9 Hydogen atom enegy diagam. Thee ae two impotant diffeences between the classical mechanical desciption of the possible obits unde a cental foce and a quantum mechanical desciption of the possible states of a hydogen atom. The fist diffeence is that the enegy E of the hydogen atom can only take on discete quantized values, unlike the classical case whee the enegy E can take on a continuous ange of values, negative fo cicula and elliptic bound obits, and positive fo hypebolic fee obits. The second diffeence is that thee ae additional states with the same value of enegy, but which have diffeent discete values of angula momentum, (and othe discete quantum popeties of the atom, fo example spin of the electon). Chemists identify these discete values of angula momentum by alphabetical labels ( s, p, d, f, g,...) while physicists label them by the obital angula momentum quantum numbes l =,,,.... We shall ty to estimate the enegy levels of the electon in the hydogen atom. We shall begin by assuming that the discete enegy states descibe cicula electon obits about the poton. (Quantum mechanics equies us to dop the notion that the electon can be thought of as a point paticle moving in an obit and eplace the paticle pictue with the idea that the electon s position can only be descibed by pobabilistic aguments.) Despite the unphysical natue of ou hypothesis, ou estimation of the enegy levels of the electon in the atom agee supisingly well with expeiment. The cicula obits coesponded to the situation descibed by This occus when E L e = ( U ) =. (7.7.7) μ 4πε min min //9 7 
23 du L e = 3 d = μ + 4πε. (7.7.8) Solving Equation (7.7.8) fo the adius of the obit, we find 4πε L =. (7.7.9) μ e We also note that the squae of the angula momentum is then μ e L =. (7.7.) 4πε We now make ou semiclassical assumption that the angula momentum assume discete values L can only h L= n. (7.7.) π 34 whee n is an intege, n =,,..., and h = kg m s is the Planck constant. With this assumption that the values L ae discete, Equation (7.7.) becomes nh μ e n = π 4πε (7.7.) whee n denotes the adii of the discete cicula obits, 4πε nh n =. (7.7.3) 4π μe The adii ae quantized, 4πε nh n = =n (7.7.4) 4π μe and equal to integal multiples of the gound state adius 4πεh = (7.7.5) 4π μe //9 73
24 whee, setting μ = me, the electon mass, is 34 ( 6.66 kg m s ) ( )( )( ) = 4π N m C 9.9 kg.6 C = 5.9 m to fou significant figues. The length as a o, pehaps counteintuitively, (7.7.6) is known as the Boh adius and is often given a, the latte notation indicating a nuclea mass of infinity, in which case, μ = me, as in the above calculation. We can substitute Equation (7.7.6) into Equation (7.7.7) and find that the enegy levels ae also quantized and given by E n L e μ e n e = = μn 4πε n 4πε μn n (7.7.7) e e. n πε n = = 4πε 4 Using Equation (7.7.5) fo E n, Equation (7.7.7) fo the enegy levels becomes e e = = = A (7.7.8) 4πε 4πε 4π μ 4πε 4 π μ nh e ( ) h n n whee, with μ = me, the constant A is given by A = π me 4 e h ( 4πε ) ( )( ) ( 34 ( 6.66 kg m s ) π 9.9 kg N m C.6 C = = 8.8 J ) (7.7.9) to fou significant figues. We can expess this enegy in tems of the enegy units of electonvolts ( ev ), whee an electonvolt is the enegy necessay to acceleate an electon with chage e acoss a potential enegy pe unit chage of volt ( volt = 9 9 ev =.6 C J C =.6 J and joule pe coulomb). Thus ( )( ) //9 74
25 ev A = = ( ) ( 9.6 J ) The enegy in Equation (7.7.8) can be witten as 8.8 J.36 ev. (7.7.) E n hc R = (7.7.) n whee R, the Rydbeg constant, is given by 8 (.8 J) 34 8 ( 6.66 kg m s )(.998 m s ) A R = = hc 7.97 m. = (7.7.) and c = m s is the speed of light. The fist few enegy levels ae shown in Figue 7., along with the enegy diffeence between the second and thid levels. The fist thee enegy levels ae E = 3.6 ev, E = 3.39 ev, E 3 =.5 ev. Figue 7. Enegy levels fo an electon in a hydogen atom Emission of light When an electon makes a tansition fom a highe enegy state E i to a lowe enegy state E f, light is emitted. The fequency of the emitted light is given by //9 75
26 f ΔE h E E f i = =. (7.7.3) h Using ou esult as given in Equation (7.7.8) fo the enegy levels, we have f = Rc n f n. (7.7.4) i The wavelength of the emitted light λ is elated to the fequency f of the light by f λ = c. (7.7.5) Thus the invese wavelength of the light is given by f = = R λ c. (7.7.6) nf n 7.7. Example: Calculate the wavelength of the light emitted when an electon in the enegy level n = 3 dops to the enegy level n =. Fom Equation (7.7.6) the wavelength is λ nn = = i f 3, R ni nf m. (7.7.7) The emitted light lies in the visible spectum and appeas ed to the human eye. //9 76
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