Quantum-Classical Connection for Hydrogen Atom-like Systems

Transcription

1 Quatum-Classical Coectio for Hydroge Atom-like Systems Debapriyo Syam ad Arup Roy Departmet of Physics, Barasat Govermet College, 0 KNC Road, Barasat, Kolkata , Idia syam.debapriyo@gmail.com Departmet of Physics, Scottish Church College Urquhart Square, Kolkata , Idia aryscottish@gmail.com Abstract The Bohr-Sommerfeld quatum theory specifies the rules of quatizatio for circular ad elliptical orbits for a oeelectro hydroge atom like system. This article illustrates how a formula coectig the pricipal quatum umber ad the legth of the major axis of a elliptical orbit may be arrived at startig from the quatum mechaical descriptio ad how i the limit whe is large oe gets the expected classical result. Keywords: Quatum, hydroge atom, quatum mechaics, oe-electro atoms, classical mechaics Itroductio Accordig to the o-relativistic versio of quatum mechaics, the eergy of the electro i a hydroge-like atom depeds oy o the pricipal quatum umber (apart from the atomic umber Z of the ucleus.) Agai, i o-relativistic classical mechaics, the eergy of such a particle, which describes a elliptical path that keeps the ucleus at oe focus, depeds oy o the size (a) of the major axis. As oe expects to recover the classical result i the large limit of quatum mechaics ( the correspodece priciple ), there ought to be a coectio betwee ad a. This coectio will be examied i this article. We begi by recallig the saliet features of the two approaches: The classical orbits uder iverse-square type force ( Kepleria orbits ): The followig quatities remai coserved (Raa ad Joag, 99; Syge ad Griffith, 959):. Eergy, which depeds oy o the size of the major axis: Ze E = () a. Agular mometum vector;. Ruge-Lez vector, which gives the directio of the major axis. Hydroge-like atom i quatum mechaics (Ghatak ad Lokaatha, 984): The wave fuctio (igorig spi) of the system : Ψ ( r, θ, φ) = NR ( r) Y ( θ, φ) () m lm 60

2 where N is a ormalizatio costat, while Y lm / l+ m m (l + )( l m)! m / d l imφ ( θ, φ) = ( )..[( ).. ( ) ] ( 0) 4 ( )! x x e m () l l+ m π l + m l! dx x stadig for cosθ ; ad Y m * θ, φ) = ( ) Y l m ( θ, φ) (m < 0 ) (4) lm(, The radial part of the wave fuctio reads L l+ + l l l+ R ( r) = e.. L ( ) (5) ( ) beig a associated Laguerre polyomial (Ghatak ad Lokaatha, 984) with = ηr µ E η = (7) E beig the eergy (E< 0) ad µ the reduced mass of the system. The quatum umbers, l, m have their usual meaigs. Eigevalue of eergy (E) of boud states is give by 4 µ Z e E = (8) while l + [ ] k l k+ ( + l)! L+ l ( ) = ( ). (9) ( l k)!(l + + k)! k! k= 0 We are lookig for the classical limit of the quatum mechaical solutio. This requires that we take a large value for. (Also large Z, which, however, is a costat for a give system.) But somethig else is also to be take ito accout. I classical mechaics all three compoets of agular mometum have well-defied values. There is also a vector (Ruge-Lez vector) which poits i the directio of the major axis; we shall come to this later. As the operators that correspod to the compoets of agular mometum (orbital agular mometum) do ot commute with each other, oy oe compoet (z-compoet, by covetio) of agular mometum ad the magitude of the agular mometum have, simultaeously, defiite values i quatum mechaics. Now, writig the orbital agular mometum as l, we have ˆ ˆ ˆ lx + ly + lz = (ˆ l ) = l( l + ) (0) + l (6) 6

3 where  stads for the expectatio value of the operator Â. Now with If lˆ z = ± l, the ˆ l z = l, ad l ˆ z = m ( - l m l ) () l ˆ lˆ = 0 () x = y ˆ ˆ l + l x y = l () i.e. ( l ) + ( Δl ) l (4) Δ, x y = Δl x, Δl y beig the stadard deviatios of l x, l y ; so that the relative ucertaity i the directio of the agular mometum is small. Coversely, if l ˆ = 0, the ˆ ˆ l + l = l( l + ) ; the directio of the agular mometum is ow totally ucertai. Clearly the classical situatio correspods to l ˆ z = ± l. For cocreteess we shall take lˆ l z =. As a matter of fact, we shall cosider oy large values of l. Notice ow that m = Y lm ( θ, φ) l ( x ) z lˆ z = l implies ad, for large values of l, this quatity is practically equal to zero except whe x = 0 or θ = π /, i.e.o the XY plae (Paulig ad Wilso, 95; Roy, 00). The electro therefore speds most of its time o the XY plae: the first hit of the emergece of a circular or a elliptical orbit i the classical limit. Coectio betwee ad the legth of the major axis We shall ext fid the most probable value of r for the electro; this requires fidig the value of r (or the values of r) at which R r has a local maximum (maxima.) The cases l =, -,, 4 ca be hadled aalytically. (Abel s theorem rules out the possibility of solvig equatios of degree higher tha four by algebraic techiques (Hall ad Kight, 969). The case l = - (5) x y Here R r e (6) 6

4 ad maximizatio of value of (or, rather Ze E =, we get r d gives =. Usig this d r = ) i the classical formula for the eergy E (for a circular orbit) viz. η R r through demadig that ( Rr ) = 0 µ Z e E = 4 (7) this is precisely Bohr s formula. Let us work out the width of the distributio of + α, ad assumig that α <<, we have e e e 4 ( ) e α aroud the maximum. Puttig = Thus the stadard deviatio, Δ, of is oy. The relative width, Δ /, of the distributio becomes smaller with icreasig : The electro is practically cofied to a circle o the XY plae for large ( whe m =l ad l =.) It is iterestig to calculate the eergy associated with radial motio i this case. (Oe does ot expect ay from a strictly classical stad poit.) There are two ways of calculatio: l( l + ) (i) Oe may subtract the eergy, liked to agular motio, from E which is just the µ r total kietic eergy i both classical ad quatum mechaics. Settig l = - ad usig the appropriate value of r viz. r = = we obtai η η E ( E) radial = (9) (8) (ii) Oe may, alteratively, use the ucertaity relatio to first compute ( Δ p ) the calculate (E) radial via (E) radial = r. Thus µ 4 η E ( E ) radial (0) µ ( Δ) Δ pr Δ from Δr = ; ad η l = 6

5 The primary aim of this work is to establish the coectio betwee the size of the major axis (a) ad the pricipal quatum umber (): For a give ucleus (havig charge Ze) the eergy of the system depeds oy o a i classical mechaics while it depeds oy o the pricipal quatum umber i quatum mechaics. Remember that the major axis jois the poit i the orbit where the radial distace of the electro from the force-cetre (ucleus) is least to the poit where the radial distace from the ucleus is most; it is also called the apsidal lie. As r does ot chage for small chages i the directio of the radius vector ear the major axis, the electro speds relatively loger periods of time at the eds of the apsidal lie. Sice there is o aalogue of the Ruge-Lez vector i quatum mechaics (uess oe thiks of superpositio of states or deals with wave-packets), distaces to the two eds of the major axis may be ascertaied by fidig the distaces to the two maxima of R r at which it has the two highest values. 64

6 Fig.. For l =, R r has its maximum value at r = r =. The distributio has a η width of approximately. η Fig.. A three-dimesioal versio of Fig.. Fig.. For l, R r has two or more maxima. The values of r at which two highest magitudes lie at the eds of the major axis. I fact, these correspod to the largest ad the smallest values of r at which its maxima. Note that the umber of maxima of maxima of R r there is always a miimum of Notice that we are relatig maximum time spet to maximum of guess. R r has its R r have R r is ( l). I betwee two successive R r where the fuctio also goes to zero. R r, presumably a sesible For l =, Thus l+ L ( ) + l L = ( )![ ( )] R r e!! [!! (! )!! ] () () The maxima of R r are located at ± = ( ) ± 4 ±, () while there is a miimum at = ( ). Argumets advaced above lead to a legth of the ( + + ) major axis equal to. Goig back to the expressio for the eergy i classical η η mechaics, we fid that such a legth of the major axis leads to the correct value of the eergy. Thus our expectatios are fulfilled for l =. Here l = + L l + l [( )!] [ + ]!( 5)! ( 4)! ( )!! (4) 65

7 Hece the roots (ad miima) of R r are at = ( ) ± ( ) (5) To fid where the maxima of is R r lie, let us put = ) + α (. The equatio satisfied by α α (0 ) α ( 6) = 0 (6) Followig Carda [ Hall ad Kight, loc cit] let us set α = y + z ad require: yz = The y = ( 4) + 4( 4) (0 ) 7 (7) ( 0 ). (0 ) z = ( 4) 4( 4) (8). 7 Whe is large ad y e z e πi 0 6 ( 4) πi 0 6 ( 4) (,ω, ω ) (,ω, ω ) (9a) (9b) where (, ω, ω ) are the cube roots of uity (). For large the required solutios are πi πi α ( 4) [ e. + e ] = ( 4) (0a) 66

8 πi πi α ( 4) [ e. ω + e. ω ] = ( 4) (0b) α πi πi ( 4) e ω + e ω = 0 (0c) The sum of the largest ad the smallest values of viz. ( + ) is approximately equal to ( ) or 4, whe is large. The legth of the major axis, by our assumptio, is the η ad we are agai led to the coclusio that the eergy E depeds o this particular quatity. l = 4 I this last algebraically solvable case + L l + l [( 4)!] [ + ]!( 7)!!( 6)! ( 5)!! ( 4)!! () The miima of R r are placed at = 4) + α (, where α is a solutio of the cubic equatio α ( 4) α ( 4) = 0 () The required values of α ca be obtaied by Carda s method. For large the solutios are πi πi 6 6 α ( 4) [ e. + e ] = ( 4) (a) πi πi 6 6 α ( 4) [ e. ω + e. ω ] = ( 4) (b) α πi πi 6 4) e ω + e ω 0 (c) 6 ( = 67

9 To fid the values of where R r has its maxima, let us put = (-) + β. Some algebraic maipulatios lead to the followig equatio forβ : 4 β 6( 5) β 8(5 9) β + ( ) = 0 (4) Collectig the leadig terms (remember that we are iterested i the large situatio) we have 4 β 8β 40β + 4 = 0 (5) This equatio ca be solved by the techique iveted by Descartes [Hall ad Kight, loc cit]. For pedagogical reasos we give a brief outlie of the method. Cosider the equatio Assume x 4 + qx + rx + s = 0 (6) 4 x + qx + rx + s = ( x + kx + l)( x kx + m) (7) The by equatig coefficiets we get k qk + ( q 4s) k r = 0 (8) This is a cubic equatio i k. After solvig this equatio we ca fid the values of m ad l. Fially x (four possible values) is obtaied from the pair of equatios x + kx + l = 0 (9) ad x kx + m = 0. (40) Applyig this techique to our equatio for β, we get ± 8 4(5 ) 8 ± 8 4(5 + ) β 5, 5 (4) 68

10 I the large situatio the highest ad the lowest values of β are respectively ( 8 + ad ( 8 + 8). Clearly the sum of the correspodig values of is approximately equal to 4, as aticipated. Our stad is therefore vidicated. 8) 69

11 Fig.4. Plots (schematic) of the variatio of R r with for differet values of l. Coclusios The cases for other l values may be hadled by umerical methods. However, the tred is evidet, for it has to be cosistet with Bohr s correspodece priciple. We hope that through this article we have bee able to illustrate how quatum effects disappear ad classical results emerge i the limit of large values. It may also help the studets to accept quatum mechaics as a deeper ad refied descriptio of ature. Refereces Ghatak, A. K. ad Lokaatha, S. (984). Quatum Mechaics Theory ad Applicatios, rd eds. McMilla Idia Ltd. Hall, H. S. ad Kight, S. R. (969). Higher Algebra. McMilla ad Co. Limited: Lodo. Paulig, L. ad Wilso, E. B. (95). Itroductio to Quatum Mechaics, Itl. Studet ed. McGraw-Hill- Kogakusha Book Co., Ic./ Kogakusha Compay, Ltd.: Tokyo. Raa, N. C. ad Joag, P. S. (99). Classical Mechaics. Tata-McGraw Hill Publishig Compay Limited. Roy, A. (00), Itroductio to Quatum Mechaics (Miley-Mishey) Syge, J. L. ad Griffith, B. A. (959). Priciples of Mechaics, rd ed. /Itl.Studet ed. McGraw Hill Book Compay. 70

12 7