Physics 2D Lecture Slides Lecture 30: Mar 12th

Transcription

1 Course revew s schedued for Sunday 4 th March at 0am n WLH 005 There w be no Streamng vdeo for ths sesson, so p. attend Physcs D Lecture Sdes Lecture 30: Mar th Vvek Sharma UCSD Physcs

2 d Φ + Φ = dφ m 0.. d d m sn θ + ( + ) ( θ) 0...() Θ = snθ dθ dθ sn θ r m ke ( ) r (E These 3 "smpe" dff. eqn descrbe the physcs of the Hydrogen atom. The hydrogen atom brought to you by the etters n =,,3,4,5,... = 0,,,3,,4...( n ) m = 0, ±, ±, ± 3,... ± The Spata Wave Functon of the Hydrogen Atom m Ψ(, r θφ, ) = R ( r). Θ ( θ). Φ ( φ) = R Y (Spherca Harmoncs) n m n...() d r )- R( r) = dr r r r 0...(3) m Normazed Spherca Harmoncs & Structure n H Atom

3 Excted States (n>) of Hydrogen Atom : Brth of Chemstry! Features of Wavefuncton n θ & φ : Consder n=, = 0 ψ = Sphercay Symmetrc (ast sde) 00 Excted States (3 & each wth same E ) : ψ, ψ, ψ 0 - are a p states 3/ Z Z r ψ =R Y = e π a0 8 a0 n Zr a0. snθ. e φ * π ψ = ψ ψ sn θ Max at θ=,mn at θ =0; Symm n φ What about (n=, =, m = 0) ψ = R (r) Y( θφ, ); 0 0 p z 0 3 Y( θφ, ) cos θ; π π Functon s max at θ =0, mn atθ = We ca ths p z state because of ts extent n z Excted States (n>) of Hydrogen Atom : Brth of Chemstry! Remember Prncpe of Lnear Superposton for the TISE whch s bascay a smpe dfferenta equaton: - m ψ + Uψ = Eψ p z Prncpe of Lnear Superposton If ψ and ψ are so. of TISE then a "desgner" wavefuncton made of near sum ' ψ = aψ+ bψ s aso a so. of the dff. equaton! ' To check ths, ust substtute ψ n pace of ψ & convnce yoursef that - m ' ' + = ' ψ Uψ Eψ The dversty n Chemstry and Boogy DEPENDS on ths superposton rue

4 Desgner Wave Functons: Soutons of S. Eq! Lnear Superposton Prncpe means aows me to "cook up" wavefunctons ψp = [ ψ ] x + ψ...has eectron "coud" orented aong x axs ψp = [ ψ ] y ψ...has eectron "coud" orented aong y axs So from 4 soutons ψ00, ψ0, ψ, ψ, s px, py,pz Smary for n=3 states...and so on...can get very compcated structure n θ & φ...whch I can then mx & match to make eectrons" most key" to be where I want them to be! Desgner Wave Functons: Soutons of S. Eq!

5 What s So Magnetc? Precessng eectron Current n oop Magnetc Dpoe moment µ The eectron s moton hydrogen atom s a dpoe magnet The Magnetsm of an Orbtng Eectron Precessng eectron Current n oop Magnetc Dpoe moment µ Eectron n moton around nuceus crcuatng charge curent e e ep = = = ; Area of current oop A= π r T π r π mr v -e -e -e Magnetc Moment µ =A= rp; µ = r p = L m m m Lke the L, magnetc moment µ aso precesses about "z" axs -e -e z component, µ z = L = m µ m quantzed! m z = = m B

6 Quantzed Magnetc Moment -e -e µ z = Lz = m m m = µ m µ B B = Bohr Magnetron e = m Why a ths? Need to fnd a way to break the Energy Degeneracy & get eectron n each ( nm,, ) state to dentfy tsef, so we can "tak" to t and make t do our bddng: " Wak ths way, tak ths way!" e Lftng Degeneracy : Magnetc Moment n Externa B Fed Appy an Externa B fed on a Hydrogen atom (vewed as a dpoe) Consder B Z axs (coud be any other drecton too) The dpoe moment of the Hydrogen atom (due to eectron orbt) experences a Torque τ = µ B whch does work to agn µ B but ths can not be (same Uncertanty prncpe argument) So, Instead, µ precesses (dances) around B... ke a spnnng top The Azmutha ange φ changes wth tme : cacuate frequency Look at eometry: proecton aong x-y pane : dl = Lsn θ.dφ dl q dφ = ; Change n Ang Mom. dl = τ dt = LBsnθ dt Lsnθ m dφ dl q qb ωl = = = LB snθ = dt Lsnθ dt Lsnθ m ω depends on B, the apped externa magnetc fed L m e Larmor Freq

7 Lftng Degeneracy : Magnetc Moment n Externa B Fed WORK done to reorent µ aganst B fed: dw= τd θ =-µ Bsnθdθ dw = d( µ Bcos θ) : Ths work s stored as orentatona Pot. Energy U dw= -du Defne Magnetc Potenta Energy U=- µ. B = µ cos θ. B = µ B e Change n Potenta Energy U = m 0 e mb= ω m Zeeman Effect n Hydrogen Atom In presence of Externa B Fed, Tota energy of H atom changes to E=E + ω m L So the Ext. B fed can break the E degeneracy "organcay" nherent n the H atom. The Energy now depends not ust on n but aso m L z Zeeman Effect Due to Presence of Externa B fed Energy Degeneracy Is Broken

8 Eectron has Spn : An addtona degree of freedom Even as the eectron rotates around nuceus, t aso spns There are ony two possbe spn orentatons: Spn up : s = +/ ; Spn Down: s=-/ Spn s an addtona degree of freedom ust Lke r, θ and ϕ Quantum number correspondng to spn orentatons m = ± ½ Spnnng obect of charge Q can be thought of a coecton of eementa charges q and mass m rotatng n crcuar orbts So Spn Spn Magnetc Moment nteracts wth B fed Stern-erach Expt An addtona degree of freedom: Spn for ack of a better name µ n nhomogenous B fed, experences force F F= - U = ( µ.b) B B B B When gradent ony aong z, 0; = = 0 z x y B Fz = mµ B( ) moves partce up or down z (n addton to torque causng Mag. moment to precess about B fed drecton In an nhomogeneous fed, magnetc moment µ experences a force F z whose drecton depends on component of the net magnetc moment & nhomogenety db/dz. Quantzaton means expect (+) defectons. For =0, expect a eectrons to arrve on the screen at the center (no defecton) Sver Hydrogen (=0)!

9 Four (not 3) Numbers Descrbe Hydrogen Atom Æ n,,m,ms "Spnnng" charge gves rse to a dpoe moment µ s : q Imagne (sem-cascay,ncorrecty!) eectron as sphere: charge q, radus r Tota charge unformy dstrbuted: q= q ; as eectron spns, each "chargeet" rotates current dpoe moment µs q q µ s = g S me me µs = In a Magnetc Fed B magnetc energy due to spn US = µ s.b Net Anguar Momentum n H Atom J = L + S e Net Magnetc Moment of H atom: µ = µ0 + µ s = ( L + gs ) me Notce that the net dpoe moment vector µ s not & to J (There are many such "ubqutous" quantum numbers for eementary partce but we won't teach you about them n ths course!) Doubng of Energy Leves Due to Spn Quantum Number Under Intense B fed, each {n, m } energy eve spts nto two dependng on spn up or down In Presence of Externa B fed

10 Spn-Orbt Interacton: Anguar Momenta are Lnked Magnetcay Eectron revovng around Nuceus fnds tsef n a "nterna" B fed because n ts frame of reference, B B B the nuceus s orbtng around t. -e +Ze Equvaent to +Ze -e Ths B fed, due to orbta moton, nteracts wth eectron's spn dpoe moment µ s Um = µ. B Energy arger when S B, smaer when ant-parae States wth same ( nm,, ) but dff. spns enrg e y eve spttng/doubng due to S Under No Externa B Fed There s St a Spttng! Sodum Doubet & LS coupng z Vector Mode For Tota Anguar Momentum J Coupng of Orbta & Spn magnetc moments Nether Orbta nor Spn anguar Momentum are conserved seperatey! J = L + S s conserved so ong as there are no externa torques present Rues for Tota Anguar Momentum Quantzaton : J = ( + ) wth = + s, + s-, + s-...,...., - s J = m wth m Exampe: state wth ( =, s = ) = 3/ m = -3/, /,/,3/ = / m = ± / In genera m takes ( + ) vaues Even # of orentatons =, -, , - Spectrographc Notaton: Fna Labe n S P / 3/ Compete Descrpton of Hydrogen Atom

11 Compete Descrpton of Hydrogen Atom Fu descrpton of the Hydrogen atom: { nm,,, m} LS Coupng s n S P {,, n m, s} How to descrbe mut-eectrons atoms ke He, L etc? correspondng How to order the Perodc tabe? to 4 D.O. F. Four gudng prncpes: Indstngushabe partce & Pau Excuson Prncpe Independent partce mode (gnore nter-eectron repuson) / 3/ Mnmum Energy Prncpe for atom Hund s rue for order of fng vacant orbtas n an atom Mut-Eectron Atoms : > eectron n orbt around Nuceus In Hydrogen Atom ψ(r, θ, φ)=r(r). Θ( θ). Φ( φ) { n,, m, } In n-eectron atom, to smpfy, gnore eectron-eectron nteractons compete wavefuncton, n "ndependent"partce approx" : ψ (,, 3,..n)= ψ(). ψ(). ψ(3)... ψ( n)??? Compcaton Eectrons are dentca partces, abeng meanngess! Queston: How many eectrons can have same set of quantum #s? Answer: No two eectrons n an atom can have SAME set of quantum#s (f not, a eectrons woud occupy s state (east energy)... no structure!! e - e - Exampe of Indstngushabty: eectron-eectron scatterng Sma ange scatter arge ange scatter Quantum Pcture If we cant foow eectron path, don t know between whch of the two scatterng Event actuay happened

12 Heum Atom: Two eectrons around a Nuceus In Heum, each eectron has : knetc energy + eectrostatc potenta energy If eectron "" s ocated at r & eectron ""s ocated at r then TISE has terms ke: (e) e H = - + k m r ( ) ( e)( e) ; H = - + k m r Hψ + Hψ = Eψ ; H & H are same except for "abe" Independent Partce Approx e gnore repusve U=k term r r * Heum WaveFuncton: ψ= ψ(r, r ); Probabty P= ψ (r, r ) ψ (r, r ) such that e- a But f we exchange ocaton of (dentca, ndstngushabe) eectrons ψ (r, r ) = ψ (r, r) In genera, when ψ(r, r ) = ψ(r, r)......bosonc System (made of photons, e.g) when ψ(r, r ) = ψ(r, r)...fermonc System (made of eectron, proton e.g) Heum wavefuncton must be ODD; f eectron "" s n state a & eectron "" s n state b Then the net wavefuncton ψ (r,r )= ψ ( r). ψ ( r ) satsfes ab a b H ψ ( r). ψ ( r ) = Eψ ( r). ψ ( r ) a b a a b H ψ ( r). ψ ( r ) = Eψ ( r). ψ ( r ) a b b a b and the sum [H +H ] ψ ( r). ψ ( r ) = ( E + E ) ψ ( r). ψ ( r ) a b a b a b Tota Heum Energy E E +E =sum of Hydrogen atom ke E a b e - b Heum Atom: Two eectrons around a Nuceus e- a Heum wavefuncton must be ODD ant-symmetrc: ψ (r,r )=- ψ (r,r ) So t must be that ψ (r, r) = ψ ( r). ψ ( r ) ψ ( r ). ψ ( r) ab a b a b ab It s mpossbe to te, by ookng at probabty or energy whch partcuar eectron s n whch state If both are n the same quantum state a=b & ψ aa ab (r,r )= ψ (r,r )=0... Pau Excuson prncpe bb enera Prncpes for Atomc Structure for n-eectron system:. n-eectron system s stabe when ts tota energy s mnmum.ony one eectron can exst n a partcuar quantum state n an atom...not or more! 3. Shes & SubShes In Atomc Structure : (a) gnore nter-eectron repuson (crude approx.) (b) thnk of each eectron n a constant "effectve" mean Eectrc fed (Effectve fed: "Seen" Nucear charge (+Ze) reduced by parta screenng due to other eectrons "buzzng" coser (n r) to Nuceus) Eectrons n a SHELL: have same n, are at smar <r> from nuceus, have smar energes Eectons n a SubShe: have same prncpa quantum number n - Energy depends on, those wth ower coser to nuceus, more tghty bound - a eectrons n sub-she have same energy, wth mnor dependence on m, m s e - b

13 Energy She & Sub-She Energes & Capacty. She & subshe capacty mted due to Pau Excuson prncpe. She s made of sub-shes ( of same prncpa quantum # n) 3. Subshe ( n, ), gven n = 0,,,3,..( n-), for any m = 0, ±, ±,.. ( + ), ms =± Max. # of eectrons n a she = subshe capacty n NMAX =.(+ ) = [ ( n ) + ] = ( n) ( + (n )) = n = 0 4. The "K" She (n=) hods eectron s, "L" She (n=) hods 8 eectrons, M she (n=3) hods 8 eectrons She s cosed when fuy occuped 6. Sub-She cosed when (a) L = 0, S = 0, Effectve charge dstrbuton= symmetrc (b) Eectrons are tghty bound snce they "see" arge nucear charge (c) Because L = 0 No dpoe moment No abty to attract eectrons Inert! Nobe gas! 6.Aka Atoms: have a snge "s" eectron n outer orbt; nucear charge heavy sheded by nner she eectrons very sma bndng energy of "vaence"eectron arge orbta radus of vaence eectron Eectronc Confguratons of n successve eements from Lthum to Neon Hund s Rue: Whenever possbe -eectron n a sub-she reman unpared - States wth spns parae occuped frst - Because eectrons repe when cose together - eectrons n same sub-she () and same spn -Must have dff. m -(very dff. anguar dstrbuton) - Eectrons wth parae spn are further apart -Than when ant-parae esser E state -et fed frst Perodc tabe s formed That s a I can teach you ths quarter; Rest s a Chemstry!