PH Modrn Physics SP11 Final Essay Thr will b an ssay portion on th xam, but you don t nd to answr thos qustions if you submit a final ssay by th day of th final: Sat. 5/7 It dosnʼt mattr how smart you ar, or how bautiful your thory is, if it dosnʼt agr with xprimnt thn itʼs wrong. " - Richard Fynman" Thos who turn in a papr will consquntly hav mor tim to answr th MC probs. Day 5,4/1: Qustions? H-atom and Quantum Chmistry Up Nxt: Priodic Tabl Molcular onding I will rad rough draft paprs submittd by class on Tusday, 5/ Rcntly: 1. Quantum tunnling. Schrödingr quation in -D. Hydrogn atom Today: 1. Hydrogn atom (cont.). Multi-lctron atoms. Priodic tabl 4. onding (?) Coming Up: Finish bonding/banding Exam Nxt Thursday, 4/8 Lctur for Tusday, 5/? ψ nlm (r,θ,φ) = R nl (r)y lm (θ,φ) Shaps of hydrogn wav functions: ψ nlm (r,θ,φ) = R nl (r)y lm (θ,φ) l=1, calld p-orbitals: angular dpndnc (n=) l=1, m=: p z = dumbbll shapd. l=1, m=-1: bagl shapd around z-axis (travling wav) l=1, m=+1 In th stat, th most likly singl plac to find th lctron is: A) r = ) r = a C) Why ar you confusing us so much? n =, l = 1, m = ψ n =, l = 1, m = 1 ψ 11 11 1 = 6a 1 = 6a cosθ 4π Suprposition applis: p x =suprposition (addition of m=-1 and m=+1) p y =suprposition (subtraction of m=-1 and m=+1) r a r a r r / / a a iφ sinθ 8π Dumbblls (chmistry) 1
Physics vs Chmistry viw of orbits: wav functions (Physics viw) (n=, l=1) m=1 m=-1 m= Dumbbll Orbits (chmistry) p x p z p y p x =suprposition (addition of m=-1 and m=+1) p y =suprposition (subtraction of m=-1 and m=+1) Chmistry: Shlls st of orbitals with similar nrgy, 6 (p x, p y, p z ), 6, 1 n l Ths ar th wav functions (orbitals) w just found: n=1,, = Principl Quantum Numbr l=s, p, d, f = Angular Momntum Quantum Numbr =, 1,, (rstrictd to, 1, n-1) m =... -1,, 1.. = z-componnt of Angular Momntum (rstrictd to l to l) (for Hydrogn, sam as ohr) n=1,, = Principl Quantum Numbr E n = E 1 / n (for Hydrogn, sam as ohr) l=s, p, d, f = Angular Momntum Quantum Numbr =, 1,, (rstrictd to, 1, n-1) l + 1) m =... -1,, 1.. = z-componnt of Angular Momntum (rstrictd to -l to l) L z = m What is th magnitud of th angular momntum of th ground stat of Hydrogn? a. b. ħ c. sqrt()ħ d. not nough information Answr is a. n=1 so l= and m=... Angular momntum is Enrgy of a Currnt Loop in a Magntic Fild: τ = µ τ = µsin φ du = dw = τ dφ ΔU = µcos φ ( ) = µ ( ) For an lctron moving in a circular orbit: (old HW problm) µ = m L According to Schrödingr: l + 1) L(n=1, l =) = ( + 1) = Strn-Grlach Exprimnt with Silvr Atoms (19) Ag = 4d 1 5s 1 µ = m L = (S-stat) According to ohr: L = n L (n=1) = 1 = µ = m L = m (ground stat) ohr magnton!! µ z = ± µ!! What givs?!?
Th Zman Effct: ΔU = µcos( φ) = µ Th Anomalous Zman Effct: ΔU = µcos( φ) = µ Spctrum: With no xtrnal -fild Extrnal -fild ON Spctrum: With no xtrnal -fild Extrnal -fild ON Enrgy m = +1,, -1 1. V +µ µ m = +1 m = m = -1 Enrgy m = +µ µ m = Hlium ( - ) in th xcitd stat 1 1 m = stats unaffctd m = +/- 1 stats split into ΔE = ±µ Hydrogn (1 - ) in th ground stat: 1 m = stat splits into: ΔE = ±µ For th orbital angular momntum of an lctron: µ z,orb = L m z = m m What if thr wr an additional componnt of angular momntum? µ z,spin = m S z S z = ± ( ) µ z,tot = L m z + S z µ z,tot = ( + ) = m m µ z,tot = + m = m For th total angular momntum of an lctron: J = L + S For th total magntic momnt du to th lctron: µ tot = m p ohr solvd: + V (x) = E m ( ) Dirac solvd: ( ˆpc) + m c L + S ( ) ˆp Schrödingr solvd: + V ( x) m Ψ x,t Ψ x,t Why th factor of? It is a rlativistic corrction! ( ) = Ê Ψ ( x,t ) ( ) = Ê Ψ( x,t) ˆp x = i x Ê = i t ( ) Dirac solvd: ( ˆpc) + m c Ψ x,t ( ) = Ê Ψ( x,t) Solutions to th Dirac quation rquir: Elctrons hav an intrinsic angular momntum - SPIN S = s(s + 1) s = 1 S z = ± Positiv and ngativ nrgy solutions, ±E n=1,, = Principl Quantum Numbr E n = E 1 / n (for Hydrogn, sam as ohr) l=s, p, d, f = Angular Momntum Quantum Numbr =, 1,, (rstrictd to, 1, n-1) l + 1) m =... -1,, 1.. = z-componnt of Angular Momntum (rstrictd to -l to l) L z = m è ngativ E solutions corrspond to th lctron s antiparticl POSITRON Dirac s rlativistic quation prdictd th xistnc of antimattr!!!
Enrgy Diagram for Hydrogn n= n= n=1 l= (s) l=1 (p) l=,m= l= (d) In HYDROGEN, nrgy only dpnds on n, not l and m. (NOT tru for multi-lctron atoms!) An lctron in hydrogn is xcitd to Enrgy = -1.6/9 V. How many diffrnt wav functions in H hav this nrgy? a. 1 b. c. 6 d. 9. 1 n= Principl Quantum Numbr: E E / n n = 1 n= l=(rstrictd to, 1, n-1) m=(rstrictd to -l to l) n l m 1-1 1 1 1 - -1 1 stats stats (l=1) stats (l=) Answr is d: l=,1, 9 stats all with th sam nrgy With th addition of spin, w now hav 18 possibl quantum stats for th lctron with n= Schrodingr finds quantization of nrgy and angular momntum: n=1,, l=, 1,, (rstrictd to, 1, n-1) E n = E 1 / n l + 1) How dos Schrodingr compar to what ohr thought? I. Th nrgy of th ground stat solution is sam II. Th orbital angular momntum of th ground stat solution is diffrnt III. Th location of th lctron is diffrnt a. sam, sam, sam b. sam, sam, diffrnt c. sam, diffrnt, diffrnt d. diffrnt, sam, diffrnt. diffrnt, diffrnt, diffrnt ohr got nrgy right, but h said orbital angular momntum L=nħ, and thought th lctron was a point particl orbiting around nuclus. ohr modl: + Postulats fixd nrgy lvls Givs corrct nrgis. Dosn t xplain WHY nrgy lvls fixd. Dscribs lctron as point particl moving in circl. drogli modl: Also givs corrct nrgis. + Explains fixd nrgy lvls by postulating lctron is standing wav, not orbiting particl. Only looks at wav around a ring: basically 1D, not D oth modls: Gts angular momntum wrong. Can t gnraliz to multi-lctron atoms. How dos Schrodingr modl of atom compar with othr modls? Why is it bttr? Schrodingr modl: Givs corrct nrgis. Givs corrct orbital angular momntum. Dscribs lctron as D wav. Quantizd nrgy lvls rsult from boundary conditions. Schrodingr quation can gnraliz to multi-lctron atoms. How? Schrodingr s solution for multi-lctron atoms What s diffrnt for ths cass? Potntial nrgy (V) changs! (Now mor protons AND othr lctrons) V (for q 1 ) = kq nuclus q 1 /r n-1 + kq q 1 /r -1 + kq q 1 /r -1 +. Nd to account for all th intractions among th lctrons Must solv for all lctrons at onc! (us matrics) Gts vry difficult to solv hug computr programs! Solutions chang: - wav functions chang highr Z à mor protonsà lctrons in mor strongly bound à radial distribution quit diffrnt gnral shap (p-orbital, s-orbital) similar but not sam - nrgy of wav functions affctd by Z (# of protons) highr Z à mor protonsà lctrons in mor strongly bound (mor ngativ total nrgy) 4
A brif rviw of chmistry Elctron configuration in atoms: How do th lctrons fit into th availabl orbitals? What ar nrgis of orbitals? Total Enrgy A brif rviw of chmistry Elctron configuration in atoms: How do th lctrons fit into th availabl orbitals? What ar nrgis of orbitals? Filling orbitals lowst to highst nrgy, s pr orbital H H Li C N O Total Enrgy Oxygn = 4 Shll not full ractiv Shll full stabl Will th orbital b at th sam nrgy lvl for ach atom? Why or why not? What would chang in Schrodingr s quation? No. Chang numbr of protons Chang potntial nrgy in Schrodingr s quation hld tightr if mor protons. H H Li C N O Th nrgy of th orbitals dpnds on th atom. Total Enrgy Shll not full ractiv Shll full stabl A brif rviw of chmistry Elctron configuration in atoms: How do th lctrons fit into th availabl orbitals? What ar nrgis of orbitals? 1,, principl quantum numbr, tlls you som about nrgy s, p, d tlls you som about gomtric configuration of orbital Shll Shll 1 Can Schrodingr mak sns of th priodic tabl? For a givn atom, Schrodingr prdicts allowd wav functions and nrgis of ths wav functions. l= l=1 l= 4p 4s m=-,-1,,1, Enrgy Li ( s) Na (11 s) m=-1,,1 1869: Priodic tabl (basd on chmical bhavior only) 1897: Thompson discovrs lctron 199: Ruthrford modl of atom 191: ohr modl Why would bhavior of Li b similar to Na? a. bcaus shap of outr most lctron is similar. b. bcaus nrgy of outr most lctron is similar. c. both a and b d. som othr rason 5
Wav functions for sodium Li ( s) Na (11 s) In cas of Na, what will nrgy of outrmost lctron b and WHY? a. much mor ngativ than for th ground stat of H b. somwhat similar to th nrgy of th ground stat of H c. much lss ngativ than for th ground stat of H Wav functions for sodium Sodium has 11 protons. lctrons in lctrons in 6 lctrons in Lft ovr: 1 lctron in Elctrons in,, gnrally closr to nuclus that lctron, what ffctiv charg dos lctron fl pulling it towards th nuclus? Clos to 1 proton 1 lctrons closr in shild (cancl) a lot of th nuclar charg. In cas of Na, what will nrgy of outrmost lctron b and WHY? a. much mor ngativ than for th ground stat of H b. somwhat similar to th nrgy of th ground stat of H c. much lss ngativ than for th ground stat of H Schrodingr prdicts wav functions and nrgis of ths wav functions. l= l=1 l= 4p 4s m=-,-1,,1, Enrgy Li Na m=-1,,1 Why would bhavior of Li b similar to Na? a. bcaus shap of outr most lctron is similar. b. bcaus nrgy of outr most lctron is similar. c. both a and b d. som othr rason 6