Background: We have discussed the PIB, HO, and the energy of the RR model. In this chapter, the H-atom, and atomic orbitals.

Transcription

1 Chaptr 7 Th Hydrogn Atom Background: W hav discussd th PIB HO and th nrgy of th RR modl. In this chaptr th H-atom and atomic orbitals. * A singl particl moving undr a cntral forc adoptd from Scott Kirby s nots - such a systm dpnds only on th distanc and th not th dirction and is sphrically symmtric - th potntial nrgy is dpndnt only on position V ˆ V() r V V -- in trms of sphrical coordinats: 0 r r - th Hamiltonian: ˆ ˆ ˆ H K V V() r m sin r r r r r sin r r sin r ˆ L sin sin r sin r ˆ ˆ H r L V() r mr r r mr - w hav shown that th angular momntum is consrvd - in ordr to b abl to obtain dfinit valus of both th nrgy and th angular momntum simultanously w nd to s if th Hamiltonian and our angular momntum oprator commut Hˆ Lˆ Kˆ Lˆ Vˆ Lˆ ˆ ˆ ˆ ˆ ˆ ˆ K L r L L L r L m r r r mr m r r r mr ˆ ˆ ˆ ˆ ˆ ˆ L K r L L L L r ˆ ˆ L L mr r r mr mr r r mr sinc th angular momntum oprator is dpndnt of r th kintic and angular momntum oprators commut Vˆ Lˆ 0 Th sam can b said about th angular momntum and potntial oprators thrfor th Hamiltonian and angular momntum oprators commut so w can masur both th angular momntum and th nrgy of any givn stat simultanously - Is our on particl dscription applicabl to th H-atom -- as w know thr ar two particls involvd in hydrogn: a proton & an

2 Elctron -- as it stands w would hav two sts of coordinats (on for particl) to dscrib this sytm -- lt s look at th rducd mass: 7 mm kg.670 kg p kg m 7 m mp kg.670 kg --- this indicats that w will loos littl accuracy by approximating our H-atom as hav th proton fixd at th origin --- in othr words w can trat our H-atom a singl particl systm * Schrödingr Eqn for th H-atom - our modl will b a proton fixd at th nuclus intracting with an lctron which has a mass m and sparatd from th proton by a distanc of r - rcall th Coulombic potntial: V() r whr is th charg of th 4 lctron/proton and 0 is th prmittivity of fr spac - our Hamiltonian bcoms: ˆ H m 4 - th Schrödingr qn Hˆ E r sin E m r r r r sin r sin 4 say w multiply thru by m r and mov all trms to on sid of th qn r sin m r E 0 r r sin sin 4 -- th wavfunction will hav dpndnts r and : (r ) -- w can onc again invok sparation variabls: (r ) = R(r)Y( ) --- thn w can sparat th r trms from thos possssing & r R() r Y( ) sin R() r Y( ) r r sin sin m r ER( r) Y( ) 0 4

3 r R() r Y( ) sin () ( ) R r Y RrY () ( ) r r RrY () ( ) sin sin mr ER() r Y( ) 0 RrY () ( ) 4 Sinc ths trms ar indpndnt of ach othr w can sparat our SE into r R() r sin ( ) Y Rr () r r Y( ) sin sin mr ER() r 0 Rr () 4 mr r E R() r Rr () r r 4 sin Y ( ) Y ( ) sin sin Y Y sin sin sin Y 0 -- th R(r) trm is calld th radial quation and will b addrssd latr -- th Y( )is th angular momntum dpndnc and w solvd this in th prvious chaptr * Angular Momntum & Quantum Numbrs ˆ m m LYl ll Yl l0 - I will lav it to you prov th abov rlationship to yourslf - this is th sam l that you larnd in gnchm and it is calld th angular momntum quantum numbr sinc it ariss whn w apply th angular momntum oprator twic ˆ - w hav sn ˆ L ˆ m l l m H HYl Yl l 0 I I ll and hnc th nrgy valus ar givn by: E I * Finally th Hydrogn Atomic Orbitals - Th radial componnt is: h d dr hll 0 r E R mr dr dr mr 4 -- th solution is: 4 4 m m E n n 8 hn n 0 0

4 -- rcall Bohr radius is a h m m -- so thn th nrgy can also b writtn as 4 En n 8 0an 0 - a consqunc of solving this radial componnt is n which must satisfy th condition nl or 0 l n l 0 n -- this should look familiar as th principl quantum numbr - th rsults for Rnl(r): l nl! r 0 () l na l r Rnl r r L n nn! na0 na0 l r -- whr Ln ar Lagurr (lә gr) polynomials hr ar th first fw: na0 n l 0 L Zr n l 0 L! a0 l L! -- it turns out that this wavfunction is normalizd you should prov this latr ( ) m nlm r Rnl r Yl -- ths wavfunctions ar orthonormal: - th total hydrogn atomic wavfunction * sin ' ' ' nlm ' ' ' rdr d r r d nlm nn ll mm th first fw wavfunctions whr Z = atomic numbr n l m wavfunction nlm r 0 0 Z Zr a 0 00 a0 0 0 Z Zr Zr 0 00 a a0 a0 0 Z Zr Zr a 0 0 cos a0 a0 Z Zr Zr a 0 sin i 64 a0 a0 * Th Atomic Orbitals - quantum numbrs a rhash -- n our principl quantum numbr rlatd to siz n

5 -- l angular momntum quantum numbr rlatd to shap 0l n --- dscribs th angular momntum of - about a p+: L ll Orbital Nam l s sharp 0 p principl d diffus f fundamntal --- ths nams com from spctroscopy --- highr orbitals (.g. g & h) do not hav nams -- m magntic quantum numbr rlatd to dirction th orbital points in ml l --- th z-componnt of th angular momntum is compltly spcifid by m: Lz m --- th nam coms from th nrgy of th hydrogn atom in a magntic fild ---- in th absnc of a fild l th splitting in th prsnc of a magntic fild is calld th Zman ffct this will b discussd in mor dtail whn w talk about magntic spctroscopy - Radial Distribution Plots -- dscrib th probabilistic location of an lctron using th distanc from th nuclus or radius r -- for th s: th maximum is locatd at a distanc of r = a0 -- in gnral th numbr of nods for ach orbital is givn by n l nods - p orbitals -- m = 0 is th pz orbital on th basis of our dscription whr th sphrical

6 harmonic has only ral componnts -- th m = corrspond to th px & py orbitals --- thr ar both ral and imaginary componnts for ths orbitals i i Y sin Y sin whr th probability dnsitis ar qual: Y Y sin ths ar oftn rprsntd by linar combos: px Y Y sin cos py Y Y sin sin 4 i 4 - thr ar similar combos for th d-orbitals * Sctions w will skip again for now. * Multi-lctron atoms - w can solv th SE xactly for our singl lctron systm but that is not th cas whn anothr lctron gts involvd - w will discuss how to handl such a systm in th nxt chaptr