Hydrogen atom. Energy levels and wave functions Orbital momentum, electron spin and nuclear spin Fine and hyperfine interaction Hydrogen orbitals

Transcription

1 Hydogn atom Engy lvls and wav functions Obital momntum, lcton spin and nucla spin Fin and hypfin intaction Hydogn obitals

2 Hydogn atom A finmnt of th Rydbg constant: R ~ cm -1 A hydogn mas micowa las) and svs as an atomic clock. Th Lyman α lin is compisd of two componnts P 3/ -1 S 1/ and P 1/ -1 S 1/ - tst fo th lativistic Diac Hamiltonian solution Splitting of P 1/ and S 1/ tst fo quantum lctodynamics dvlopd by Fynman and oths. Diac s lativistic modl: ths lvls hav th sam ngy) Masumnts of Lamb and Rthfod showd that thy a split by about 1058 MHz. In th gound stat th F1-F0 splitting is 140 MHz 1 cm) hypfin stuctu du to contibution of poton). Th micowa adiation of 1 cm was fist dtctd in intstlla spac, lading to th dvlopmnt of adio astonomy. Th hydogn mas oscillats on this 140 MHz.

3 Hydogn atom: pocdu fo solving th Schöding quation ih t H Φ cm x, y, z) Φ, ϑ, ϕ ) H x1,... z) T V -Tansfomation to th cnt of mass H R cm, l ) T V p m pot p M pot cm p m p μ l l l d l pot -Tansfomation to th sphical pola coodinats Φ cm x, y, z) Φ, ϑ, ϕ ) - Spaation of th adial and angula pats θ z Φ, ϑ, ϕ ) R ) Y ϑ, ϕ ) - Spaation of angula pats Y ϑ, ϕ ) Θ ϑ) Φ ϕ ) y φ x

4 Radial quation It dscibs motion of a point paticl moving in on spatial dimnsion in an ffcti potntial consisting of V) plus th cntifugal potntial Rducd mass Is only littl small than th st mass m of th lcton 0.05:%)

5 Intaction potntial: adial Coulomb potntial including cntifugal potntial

6 { } na a m P m l m l l Y L l n n l n na R m m im m l lm l l n l nl 0 0 1/ 1 ) 1/ 1/ ) ) cos ] )! )! 4 1 [ ), ) )!] [ 1)! ) ) 0, > ρ μ ε ε θ π ε ϕ ϑ ρ ρ ϕ ρ h

7 Tm schm fo n-quantum numb

8 Angula obital momntum L, spin S and nucla spin I quantum numbs

9 Th obital angula momntum L: Angula momntum L

10 Th th opatos Lx, Ly, Lz a gnatos of th otations about th th coodinat axs though th angls φx, φy, φz. L do not commut with ach oth but thy commut with th hamiltonian. L is thfo a constant of th motion. [ H, L ] [ H, L z ] 0 H L nlm nlm E n nlm l l 1) h nlm Lz nlm mh nlm Elcton obital angula momntum in th hydogn atom dpnds on th θ, φ pola coodinats of th lcton. L is stictd to intg valus.

11

12 Elcton and nucla spin Thotical intptation of lcton spin is givn in th solution of Diac Hamiltonian. This is consistnt with th quimnts of th spcial thoy of lativity. Th solutions of Diac quations a intptd as a spin stats of th lcton and th calculations pdicts fin splitting of th ngy lvls fo hydogn atom. Similaly, th nuclus poton) posssss spin I. In cas of poton, th spin Quantum I is also ½. Th ignvalu of I is II1)h /..

13 Elcton spin, nucla spin It is dfin by commutation uls [S,S]ih / S, S1/ H z S S E ± n s s 1) h 1 h Simila fo th nucla

14 spinos two basis spinos th gnal on-lcton wav function

15

16 Engy dgnacy Engy valus dpnds on n and so a dgnat with spct to both l and m. Thus fo ach valu of n, l can vay fom 0 to n-1, and fo ach of ths l valus, m can vay fom l to l. Total dgnacy of th ngy lvl E n is thn n 1 l 0 l 1) n Th dgnacy with spct to m is chaactistic of any cntal fild V)). Th l dgnacy is chaactistic of th Coulomb fild. Baking this symmtis would mov dgnacy. Extnal fild.g.,magntic fild) can dstoy th sphical symmty imposd, and ngy lvls will b split up into n distinct lvls

17 Magntic momnts Magntic momnts a associatd with th vaious angula momnta.g., th obit of ngatily chagd lcton aound th nuclus is quialnt to a small cunt loop that cats a magntic momnt /.003 ) 1) 1) ) / p B N p I N I I I B S B L L m m m I g I g S g S g l l l l m m L μ μ γ μ γ μ μ γ μ μ μ γ γ μ h h h Boh magnton Gyomagntic constant

18 Spin-obit coupling Magntic fild gnatd in th obital motion of an lcton can affcts th ointation of th magntic momnt of th lcton du to th psnc of lcton spin. Intaction of spin magntic momntum and obital magntic momntum is dscibd by lativistic Diac quation. This phnomnon is known as spin-obit coupling and is sponsibl fo th fin stuctu in th spcta.g., hydogn) H SO 1 ξ ) l s ξ ) μ c 1 V V is th potntial ngy du to th Coulombic attaction btwn lcton and nuclus A dtaild divation quis th us of lativistic quantum lctodynamics

19 Inclusion of spin obit quation to hydogn Hamiltonian 0) [ H H ] E SO Hydognic hamiltonian Small spin-obit coupling tm E E R n H 0) 0)* 0)* H' SO 0) 0) ξ ) L S dτ Dgnat ptubation thoy can b usd to find th coct n 0) sinc spin-obit coupling movs som of th L and S dgnacy in th hydogn atom

20 Th L and S quantum numbs a no long good whn spin-obit coupling is takn into account sinc th opatos L z and S z do not commut with th hamiltonian bcaus of th poduct tm. Th total angula momntum: JLS Is constant of motion J and J z commut with th hamiltonian; us of th coupld basis functions ILSJM J > is appopiat Ip, J3/, mj3/> Ip, J3/, mj1/> Ip, J3/, mj-1/> Ip, J3/, mj-3/> Ip, J1/, mj1/> Ip, J1/, mj-1/> J LS) L S LS ξls ξ/ J -L -S )

21 τ ξ ς ς τ ξ τ ξ d R R S S L L J J E d S L J E d S L E E nl nl p p p ) ) ) 1)] 1) 1) [ 1/ ) ) 1/ ) * 0) 0) 0)* 0) 0) 0)* 0) p nl p nl L E E L E E 0) 0) 1) 1/ 1/ ς ς Fo JL1/ Fo JL-1/ p 3/ ) p 1/ ) Spin doubls obital momntum lvls!

22 Hypfin stuctu Th psnc of nucla spin poducs futh splittings in th lins of many lmnts.th splittings du to th nucla spin a calld hypfin stuctu. Th nucla spin I coupls with J to fom th total angula momntum F via vcto coupling: FJI Whn nucla spin is psnt, th only stictly good quantum numb is F. Splittings du to th hypfin stuctu a latily small, typically lss than 1 cm -1 so J mains a naly good quantum numb In spctum of th hydogn atom, th poton has a nucla spin of ½. Hypfin stuctu doubls all th ngy lvls. In th gound stat th F0-F1 splitting is 140 MHz, which cosponds to a wavlngth of 1 cm.

23 Hypfin stuctu Th nucla spin I coupls with J to fom th total angula momntum F via vcto coupling: FJI In cas of hydogn: Engy lvls dfind by J will b doubl bcaus I p 1/

24 Tm dfinition n L J1 L0 S L1 P L D n- pincipial quantum numb JLS

25

26 Wav functions and obitals

27 Th most pobabl distanc of an lcton fom th nuclus in cas of a 1s obital Pobability of finding th lcton on a shll of adius is popotional to: 0 / 0 / 0 ) 0 0 a a d dp x P a a

28 s- and p-obitals

29 d-obitals