Chemistry 342 Spring, The Hydrogen Atom.

Transcription

1 Th Hyrogn Ato. Th quation. Th first quation w want to sov is φ This quation is of faiiar for; rca that for th fr partic, w ha ψ x for which th soution is Sinc k ψ ψ(x) a cos kx a / k sin kx ± ix cos x ± i sin x a or gnra soution to quations of this typ is A iφ B iφ In orr that (φ) (φ ) Th vau of at so vau of φ ust b th sa at φ, sinc is prioic. it is ncssary that A iφ B iφ A A i(φ ) iφ i B B i(φ ) iφ i Sinc ±i cos ( ) ± i sin ( ) ony whn, ±, ±, th quation has soutions A i φ,, ±, ±,

2 W can trin A by rquiring that th wavfunctions b noraiz, * φ iφ iφ A φ A φ so A ( ) A, A iφ, ±, ±, ar th fina soutions to th quation. A postscript. Ths wavfunctions ar copx. Sotis it is or usfu to hav ra wavfunctions. Ths can b construct by first fining i φ (cos φ i sin φ) i φ (cos φ i sin φ) an thn aing an subtracting an w say, foring inar cobinations : sy antisy ( ( ) ) cos φ sin φ Ths functions ar aso soutions to th quation. Try it! ach of which is a ra function. W cannot associat with ths functions a particuar vau, but ony with. Th first thr of ths functions ar ± cos φ sin φ tc.

3 Th Θ quation. Th Θ quation is sin θ θ sin θ Θ θ Θ sin θ βθ. arranging, Θ θ cosθ sinθ Θ θ β sin Θ θ. Now, ak th substitutions x cosθ, sin θ x θ x x θ sinθ x, θ sin θ x cosθ x Aftr so agbra, w gt Θ Θ ( x ) x β Θ x x ( x ). This quation is intica to th associat quation of Lgnr P P ( x ) x ( ) P x x ( x ) if w intify P with Θ an β with ( ). Th soutions P of th associat Lgnr quation ar ca th associat Lgnr functions; ths ay b xprss in cos for as (sinc x cos θ) P (cosθ) ( cos θ) k k ( ) ( k)!(cosθ) ( k)!k!( k)! k

4 Hr, P is a poynoia of gr an orr, whr an ar intgrs. k is an (intgr) inx, an th su (Σ) runs fro k to an uppr iit of k ( ) / if ( ) is vn k ( ) / if ( ) is o Sinc is an intgr, an sinc th soutions to th associat Lgnr quation ar accptab ony if ( ) is an intgr, it is ncssary that int gr, with. Th soutions P ( Θ) ust of cours b noraiz; th rquirnt that Θ *, Θ, θ A P * (cosθ) P (cosθ) (cosθ) as to A ( (!! which givs Θ, (θ) P ( )! ( )! (cosθ). Ths wavfunctions, though thy appar to b copicat, ar not, at ast for sa. For xap,,. Θ, (θ) (s) 6,. Θ, (θ) cosθ (p), ±. Θ, ± (θ) sin θ (p)

5 ,. Θ,(θ) (cos θ ) () 5, ±. Θ, ± (θ) sin θ cosθ () 5, ±. Θ, ± (θ) sin θ () You hav aray t ths functions bfor, though possiby not in this for. Ths ar th anguar functions scribing th probabiity apitus in s, p, orbitas! So postscripts. Th associat Lgnr functions ar rivativs of th Lgnr poynoias P (cos θ) P (x) Th L. poynoias x ( x ) P (x) P (x) k k k ( ) ( k)!x ( k)!k!( k)! uppr iit on Σ: / if vn. (-)/ if o. ar, in turn, soutions of th Lgnr quation z z ( x ) x ( ) z (z x x P). Th functions P (cosθ) an (cosθ) for an orthonora st in th intrva - cos θ. Th L. functions ar sytric or antisytric as is vn or o P P ( cosθ) ( ) P (cosθ) P ( cosθ) ( ) P (cosθ) Th functions o not xc in absout vau

6 P (cosθ) ;.g. P (), P ( ) ( ). Sinc th P (x) ar poynoias, thr xist roots, or vaus of cos θ, for which P (x) changs sign. Th sign of P (x) is oftn inicat by a circuar iagra, - θ θ θ At th north po in this iagra, θ an x cos θ ; at th quator, x cos ; at th south po, x cos -. W thn us ins on th circ to inicat th vaus of θ at which th poynoia is zro: - - nos P P cosθ P ½ ( cos θ - ) currnc rations xist for both th P an P,.g. ( ) (cosθ) P P P Th prouct functions Y (θ,φ) Y (θ, φ) Θ, (θ) (φ) ar ca sphrica haronis. Ths ar givn by th forua Y (θ, φ) ( ) ( )! ( )! P (cosθ) iφ.

7 Th quation. Th raia quation for th ctron in orbit about th nucus of th hyrogn ato is r r r Z ( ) E r r " If w consir boun stats (E < ) ony, an introuc th nw variabs n an ρ, whr E Z n h Z n a a! r n! Z ρ n a Z ρ th raia quation bcos ρ ρ ρ n ( ) ρ ρ. (A) W sk soutions of th for ρ/ c u(ρ) ρ (B) If (B) is substitut into (A), w fin that u (ρ) ust satisfy th iffrntia quation u ρ ρ u ( ρ) (n ) u (C) ρ Eq. (C) is of th sa for as th associat quation of Lagurr, x L x L (β x) (α β) L. (D) x (D) has soutions, known as th associat Lagurr poynoias, which ar of th for L β α (x) α β k ( ) k (α!) (α β k)!(β k)!k! x k whr α an β ar intgrs, k is an inx running fro to (α - β), an (α - β) is an intgr gratr than zro. Thus, th soutions u(ρ) of Eq. (C) ar of th for L β α (x), proviing on aks th intifications

8 x ρ, (β ) ( ), (α β) (n ) Cobining ths rations, on fins β, α n. an n k ( ) k [( n )!] k Ln (p) (n k)!( k)!k! ρ Eignvaus. Sinc th conition for soution is ( α β) (n ) > an sinc,,,, n ay tak th vaus n,,, with th rstriction that n This givs th aow (ngativ) vaus of th nrgy E n Z n ", n,,, (inpnnt of, ). This rsut is intica with th vaus obtain by ans of th Bohr thory. Th rsuting nrgy v iagra is shown on th right. - nl! (Z ) 8!

9 Eignfunctions. Th raia wavfunctions for th hyrogn ato ar of th for ρ / (ρ) c ρ L (ρ) n To trin th noraizing constant c, w rquir that ρ (r) r r c ρ L (ρ) r r n Substituting r (na /Z)ρ, this bcos na ρ c ρ Ln (ρ) Z na c Z [( n )!] n (n )! ρ (EWK, p. 66). so that c ± Z na (n )! [( n ) ] n! W choos c < to ak th (tota) wavfunction positiv, so (n )! Zr na Zr / nl(r) L Z na Th first fw nl (r) ar, xprss in trs of ρ Zr/na, n Zr [( )] n n! na na Z a ρ / 9 Z a (6 6ρ 6ρ ) ρ / Z a ( ρ) ρ / 9 6 Z a ( ρ) ρ ρ / 6 Z a ρ ρ / 9 Z a ρ ρ /

10 Not th vry iportant structur of ths wavfunctions. Each function consists of a constant, tis a poynoia in ρ, tis an xponntia factor in -ρ/. Th ast factor ooks, of cours, ik -ρ/ ρ ρ ρ so is a sip xponntia. But, which contains, in aition, th factor (-ρ), has a no at ρ, as shown abov. An, which contains th factor ρ, gos to zro at th origin, aso as shown abov. Aso not that, as n incrass, th nubr of nos incrass as (n - ) this structur bing ictat by th highst powr of ρ apparing in th poynoia!

11

12