# SCORES 50 40 30 0 10 MTE 3 Rsults P08 Exam 3 0 30 40 50 60 70 80 90 100 SCORE Avrag 79.75/100 std 1.30/100 A 19.9% AB 0.8% B 6.3% BC 17.4% C 13.1% D.1% F 0.4% Final Mon. Ma 1, 1:5-:5, Ingraam B10 Gt prpard for t Final Rmmbr Final counts 5% of final grad It will contain nw matrial and MTE1-3 matrial (no altrnat xams but notif SOON an potntial and VERY srious problm ou av wit tis tim) Atomic Psics Prvious Lctur: Particl in a Box, wav functions and nrg lvls Quantum-mcanical tunnling and t scanning tunnling microscop Start Particl in D,3D boxs Tis Lctur: Mor on Particl in D,3D boxs Otr quantum numbrs tan n: angular momntum H-atom wav functions Pauli xclusion principl HONOR LECTURE PROF. R. Wakai (Mdical Psics) Biomagntism Biomagntism dals wit t rgistration and analsis of magntic filds wic ar producd b organ sstms in t bod. 3 From last wk: particl in a box Classical: particl bouncs back and fort. Somtims vlocit is to lft, somtims to rigt Quantum mcanics: Particl is a wav: p = mv = / standing wav: suprposition of wavs travling lft and rigt => intgr numbr of wavlngts in t tub Summar of quantum information Enrg is quantizd E n = p m = n " % $ ' # 8mL & t largr t box t lowr t nrg of t particl in t box A quantum particl in a box cannot b at rst Fundamntal stat nrg is not zro: En=1 = 0.38 V for an lctron in a quantum wll of L = 1 nm Consqunc of uncrtaint principl: "x = L # "p x $1/L % 0 Enrg n=5 n=4 n=3 n= n=1
Classical/Quantum Probabilit n=3 Probabilit (D) x Ground stat: sam wavlngt (longst) in bot x and Nd two quantum # s, on for x-motion on for -motion Us a pair (nx, n) Ground stat: (1,1) n= Tunnling: nonzro probabilit of scaping t box. Tunnling Microscop: tunnling lctron currnt from sampl to prob snsitiv to surfac variations n=1 Particl in a D box Sam nrg but diffrnt probabilit in spac Similar wn n $% (nx, n) = (,1) Particl in 3D box (nx, n) = (1,) Wit incrasing nrg... () (1) Ground stat surfac of constant probabilit (nx, n, nz)=(1,1,1) px = = nx "n x L sam for,z (11) (11) (11) E= p p px + + z = E o (n x + n + n z ) quantum stats wit m m m sam nx, n, nz av sam E ( mano 3Dx Eg: owe = E n + n + n All ts stats av t sam nrg, but diffrnt probabilitis ) particl zstats av 18E0? (n,n,n ) = (4,1,1), (1,4,1), (1,1,4) x z Otr quantum numbrs? H-atom Quantization of angular momntum Bor modl fails dscribing atoms avir tan H Dos it violat t Hisnbrg uncrtaint principl? A) YES B) No 13.6 V rn = n ao Bor n radius and nrg of lctron cannot b xactl known at t sam tim Hdrogn atom is 3D structur. Scrödingr: L = l(l + 1) En = " " is t orbital quantum numbr Stats wit sam n, av sam nrg and can av " = 0,1,,...,n-1 orbital quantum numbr " =0 orbits ar most lliptical " =n-1 most circular T z componnt of t angular momntum must also b quantizd Sould av 3 quantum numbrs. Coulomb potntial (lctron-proton intraction) is spricall smmtric. x,, z not as usful as r,, # Modifid H-atom sould av 3 quantum numbrs Lz = m l m ℓ rangs from - ℓ, to ℓ intgr valus=> (ℓ+1) diffrnt valus
T xprimnt: Strn and Grlac It is possibl to masur t numbr of possibl valus of Lz rspct to t axis of t B-fild producd b t lctron r currnt Orbital magntic dipol For a quantum stat wit " =, ow man diffrnt orintations of t orbital angular momntum rspct to t z-axis ar tr? A. 1 B. C. 3 D. 4 E. 5 µ lctron Currnt - s: ℓ=0 p: ℓ=1 d: ℓ= f: ℓ=3 g: ℓ=4 T lctron moving on t orbit is lik a currnt tat producs a magntic momntum µ=ia r v µ = "r = "r = (mvr) = L µ = µb l(l + 1) T "r m atomic slls m Summar of quantum numbrs 3D Surfacs of constant prob. for H-atom Elctron cloud: probabilit dnsit in 3D of lctron around t nuclus P(r,",# )dv = $(r,",# ) dv For drogn atom: En = " 13.6 V n : dscribs nrg of orbit n "dscribs t magnitud of orbital angular momntum m dscribs t angl of t orbital angular momntum L = l(l + 1) Spricall smmtric. Probabilit dcrass xponntiall wit radius. Sown r is a surfac of constant probabilit Lz = m l n = 1, l = 0, ml = 0 Nxt igst nrg: n = s-stat n =, l = 0, ml = 0 p-stat 3p-stat 3s-stat p-stat n =, l = 1, ml = 0 n = 3: s-stats, 6 p-stats and... 3p-stat n = 3, l = 0, ml = 0 n =, l = 1, ml = ±1 n = 3, l = 1, m l = 0 Sam nrg, but diffrnt probabilitis n = 3, l = 1, ml = ±1
...10 d-stats Radial probabilit " n,l,m (r,#,$) = R n,l (r)y l,m (#,$) Radial Angular For 1s, p, 3d, rpak = a0, 4a0, 9a0 Ts ar t Bor orbit radii n = 3, l =, m = 0 l n = 3, l =, m = ±1 l n = 3, l =, m = ± l r n = n a o most probabl distanc of lctron from nuclus T bav lik Bor orbits bcaus for stats wit sam E, largr angular momntum corrsponds to mor sprical orbits, orbits ar lliptical for small " Elctron spin Nw lctron proprt: Elctron acts lik a bar magnt wit N and S pol. Magntic momnt fixd but possibl orintations of magnt: up and down Dscribd b spin quantum numbr m s z-componnt of spin angular momntum S z = m s All quantum numbrs of lctrons in atoms Quantum stat spcifid b four quantum numbrs: ( n, l, m l, m s ) Tr spatial quantum numbrs (3-dimnsional) On spin quantum numbr How man diffrnt quantum stats xist wit n=? A. 1 B. C. 4 D. 8 = 0 : s m l = 0 : m s = 1/, -1/ = 1 : p 6 m l = +1: m s = 1/, -1/ m l = 0: m s = 1/, -1/ m l = -1: m s = 1/, -1/ stats stats stats stats Pauli xclusion principl Elctrons ob Pauli xclusion principl Onl on lctron pr quantum stat (n,, m, m s ) occupid Hdrogn: 1 lctron on quantum stat occupid = +1/) Hlium: lctrons two quantum stats occupid = +1/) = "1/) unoccupid n=1 stats n=1 stats Atom Building Atoms H 1s 1 H 1s Configuration Li 1s s 1 B 1s s B 1s s p 1 tc N 1s s p 6 1s sll filld s sll filld p sll filld (n=1 sll filld - nobl gas) (n= sll filld - nobl gas)
T priodic tabl Atoms in sam column av similar cmical proprtis. Quantum mcanical xplanation: similar outr lctron configurations. H 1s 1 Li s 1 Na 3s 1 K 4s 1 B s Mg 3s Ca 4s Sc 3d 1 Y 3d 8 mor transition mtals B p 1 Al 3p 1 Ga 4p 1 C p Si 3p G 4p N p 3 P 3p 3 As 4p 3 O p 4 S 3p 4 S 4p 4 F p 5 Cl 3p 5 Br 4p 5 H 1s N p 6 Ar 3p 6 Kr 4p 6