Physics 2D Lecture Slides Lecture 30: March 9th 2005

Transcription

1 Physics D Lecture Sides Lecture 3: Mrch 9th 5 Vivek Shrm UCSD Physics Mgnetic Quntum Number m L = r p (Right Hnd Rue) Cssicy, direction & Mgnitude of L wys we defined QM: Cn/Does L hve definite direction? Proof by Negtion: Suppose L ws precisey known/defined (L z) ˆ Since L = r p # Eectron MUST be in x-y orbit pne p # $ z = ; $ pz$ z " # # $ p z " %; E = " % m So, in Hydrogen tom, L cn not hve precise mesurbe vue Uncertinty Principe & Angur Momentum : $ L $ " " # z

2 Mgnetic Quntum Number m Consider = L = ( + ) = 6" In Hydrogen tom, L cn not hve precise mesurbe vue Arbitrriy picking Z xis s reference direction: L vector spins round Z xis (precesses). The Z component of L : L = m "; m = ±, ±, ± 3... ± Z Note : since L < L (wys) since Z m " < ( + ) " It cn never be tht L = m " = ( + ) " (breks Uncertinty Principe) Z So...the Eectron's dnce hs begun L=, m =,±, ± : Pictoriy Eectron sweeps Conic pths of different ϑ: Cos ϑ = L Z /L On verge, the ngur momentum Component in x nd y cnce out <L X > = <L Y > =

3 Where is it ikey to be? Rdi Probbiity Densities #( r,,") = R ( r). $ ( ). % (" ) = R Y n m P(r,,") = # # = #( r,,") = R. Y m Probbiity Density Function in 3D: m * m n n Note : 3D Voume eement dv= r.sin. dr. d. Prob. of finding prtice in tiny voume dv is m P.dV = Rn. Y.r.sin. dr. d. The Rdi prt of Prob. distribution: P(r)dr m d" & & P(r)dr= Rn. r dr' $ m ( ) d m (" ) d" ' % When $ m d" ( ) & % (" ) re uto-normized then P(r)dr= Rn. r. dr; in other words P(r)=r R n dv ( ' n Normiztion Condition: = r R dr Expecttion Vues ( ' <f( r)>= f(r).p(r)dr P( r) dr = ( r).4" r dr 4 % P( r) dr = r e r> r> 3 3 r # * + Ground Stte: Rdi Probbiity Density r # Probbiity of finding Eectron for r> P 4 r e To sove, empoy chnge of vribe & r ' Define z= ( ); chnge imits of integrtion P = $, = $, # z z e dz dr (such integrs ced Error. Fn) # z $ =- [ z + z + ] e = 5e =.667 % 66. 7% 3

4 Most Probbe & Averge Distnce of Eectron from Nuceus Most Probbe Distnce: 3 r 4 In the ground stte ( n =, =, m = ) P( r) dr = r e Most probbe distnce r from Nuceus " Wht vue of r is P(r) mx? r r dp 4 d # $ # r $ " = ". r e r e 3 % & = " % + & = dr dr %' &( ' ( r " + r = " r = or r =... which soution is correct? (see pst quiz) : Cn the eectron BE t the center of Nuceus (r=)? ( 4 = ) = = " Most Probbe distnce = 3 (Bohr guess P r e r Wht bout the AVERAGE oction <r> of the eectron in Ground stte? <r>= ) ) r 4 r * rp(r)dr= r. r e dr... chnge of vribe z= 3 * r= "< >= = = 4 * * ) ) 3 z n z r z e dz... Use gener form z e dz n n( n z= " < r >= = + ed right) )( n )...() 3 3 Averge & most ikey distnce is not sme. Why? 4 Asnwer is in the form of the rdi Prob. Density: Not symmetric Rdi Probbiity Distribution P(r)= r R(r) Becuse P(r)=r R(r) No mtter wht R(r) is for some n The prob. Of finding eectron inside nuceus = 4

5 Normized Spheric Hrmonics & Structure in H Atom Excited Sttes (n>) of Hydrogen Atom : Birth of Chemistry Fetures of Wvefunction in " & # : Consider n =, = % = Sphericy Symmetric (st side) Excited Sttes (3 & ech with sme E ) :,, - re p sttes 3/ & ' & Z ' & Z '& r ' =R Y = ( ) ( ) ( )( ) e * $ + * + * 8 + * + n, Zr. sin". e i # * $ = - sin " Mx t " =, min t " =; Symm in # Wht bout (n=, =, m = ) = R (r) Y (",#); p z 3 Y (",#) - cos "; $ $ Function is mx t " =, min t" = We c this p z stte becuse of its extent in z 5

6 Excited Sttes (n>) of Hydrogen Atom : Birth of Chemistry Remember Principe of Liner Superposition for the TISE which is bsicy simpe differenti eqution: - m " + U = E p z Principe of Liner Superposition # If nd re so. of TISE then "designer" wvefunction mde of iner sum ' = + b To check this, just substitute & convince yoursef tht is so so. of the diff. eqution ' in pce of - m ' ' " + U = ' E The diversity in Chemistry nd Bioogy DEPENDS on this superposition rue Designer Wve Functions: Soutions of S. Eq Liner Superposition Principe mens ows me to "cook up" wvefunctions p = [ ] x + "...hs eectron "coud" oriented ong x xis p = [ ] y " "...hs eectron "coud" oriented ong y xis So from 4 soutions,,, " % s, px, py, pz Simiry for n=3 sttes...nd so on...cn get very compicted structure in # & $...which I cn then mix & mtch to mke eectrons " most ikey" to be where I wnt them to be 6

7 Designer Wve Functions: Soutions of S. Eq Typo Fixed d + d" m =.. These 3 "simpe" diff. eqn describe the physics of the Hydrogen tom. The hydrogen tom brought to you by the etters n =,,3,4,5,... # =,,,3,,4...( n $ ) m =, ±, ±, ± 3,... ±...() # d % d & # m $ $ ( * sin + + ( ( + ) / ( )...() )) =. sin d, d -. sin # d % ' & # mr ke ( + ) $ $ ( r (E+ )- R( r) * + + ) = r dr ' r ( r r )...(3)., -. The Spti Wve Function of the Hydrogen Atom %( r,," ) ( ). ( ). ( ) Y m = R r & ' " = R (Spheric Hrmonics) n m m n 7

8 Cross Section View of Hydrogen Atom prob. densities in r,θ,φ Birth of Chemistry (Cn mke Fncy Bonds Overpping eectron couds ) Wht s the eectron coud : Its the Probbiity Density in r, θ,φ spce Z Y Wht s So Mgnetic? Precessing eectron Current in oop Mgnetic Dipoe moment µ The eectron s motion hydrogen tom is dipoe mgnet 8

9 The Mgnetism of n Orbiting Eectron Precessing eectron Current in oop Mgnetic Dipoe moment µ Eectron in motion round nuceus " circuting chrge " curent i # e # e # ep i = = = ; Are of current oop A= r T r mr v $ -e % $ -e % $ -e % Mgnetic Moment µ =ia= & ' rp; µ = & ' r ( p = & ' L ) m * ) m * ) m * Like the L, mgnetic moment µ so precesses bout "z" xis $ -e % $ -e" % z component, µ z = & ' L = m µ m quntized m z & ' = # = ) * ) m * B Quntized Mgnetic Moment -e " -e " µ z = # $ Lz = # $ m % m & % m & = ' µ m µ B B = Bohr Mgnetron e " = # $ m % e & Why this? Need to find wy to brek the Energy Degenercy & get eectron in ech ( n,, m ) stte to identify itsef, so we cn "tk" to it nd mke it do our bidding: " Wk this wy, tk this wy" 9

10 Lifting Degenercy : Mgnetic Moment in Extern B Fied Appy n Extern B fied on Hydrogen tom (viewed s dipoe) Consider B Z xis (coud be ny other direction too) The dipoe moment of the Hydrogen tom (due to eectron orbit) experiences Torque = µ " B which does work to ign µ B but this cn not be (sme Uncertinty principe rgument) # So, Insted, µ precesses (dnces) round B... ike spinning top The Azimuth nge % chnges with time : ccute frequency Look t Geometry: projection ong x-y pne : dl = Lsin $.d% dl q # d% = ; Chnge in Ang Mom. dl = dt = LBsin$ dt Lsin$ m d% dl q qb # & L = = = LB sin$ = dt Lsin$ dt Lsin$ m & depends on B, the ppied extern mgnetic fied L m e Lrmor Freq Lifting Degenercy : Mgnetic Moment in Extern B Fied WORK done to reorient µ ginst B fied: dw= d " =-µ Bsin" d" dw = d( µ Bcos " ) : This work is stored s orienttion Pot. Energy U dw= -du Define Mgnetic Potenti Energy U=- µ. B = $ µ cos ". B = $ µ B e" Chnge in Potenti Energy U = m B = "# Lm m Zeemn Effect in Hydrogen Atom In presence of Extern B Fied, Tot energy of H tom chnges to E=E + m L So the Ext. B fied cn brek the E degenercy "orgnicy" inherent in the H tom. The Energy now depends not just on n but so m e z

11 Zeemn Effect Due to Presence of Extern B fied Energy Degenercy Is Broken Eectron hs Spin : An ddition degree of freedom Even s the eectron rottes round nuceus, it so spins There re ony two possibe spin orienttions: Spin up : s = +/ ; Spin Down: s=-/ Spin is n ddition degree of freedom just Like r, θ nd ϕ Quntum number corresponding to spin orienttions m = ± ½ Spinning object of chrge Q cn be thought of coection of eement chrges Δq nd mss Δm rotting in circur orbits So Spin Spin Mgnetic Moment intercts with B fied

12 Stern-Gerch Expt An ddition degree of freedom: Spin for ck of better nme µ in inhomogenous B fied, experiences force F F= - U = "("µ.b) B # B # B # B When grdient ony ong z, $ ; = = # z # x # y # B Fz = mµ B( ) moves prtice up or down # z (in ddition to torque cusing Mg. moment to precess bout B fied direction µ Siver Hydrogen (=) Four (not 3) Numbers Describe Hydrogen Atom n,,m,m s "Spinning" chrge gives rise to dipoe moment : µ s Imgine (semi-csicy, incorrecty ) eectron s sphere: chrge q, rdius r Tot chrge uniformy distributed: q= " q i; i s eectron spins, ech "chrgeet" rottes ) current ) dipoe moment µ # q $ # q $ µ s = % &" µ s = g i % & S ' me ( i ' me ( In Mgnetic Fied B ) mgnetic energy due to spin U S = µ s. B Net Angur Momentum in H Atom J = L + S # * e $ Net Mgnetic Moment of H tom: µ = µ + µ s = % &( L + gs) ' me ( Notice tht the net dipoe moment vector µ is not " to J si Δq (There re mny such "ubiquitous" quntum numbers for eementry prtice but we won't tech you bout them in this course )

13 Doubing of Energy Leves Due to Spin Quntum Number Under Intense B fied, ech {n, m } energy eve spits into two depending on spin up or down In Presence of Extern B fied Spin-Orbit Interction: Angur Moment re Linked Mgneticy Eectron revoving round Nuceus finds itsef in "intern" B fied becuse in its frme of reference, B B B the nuceus is orbiting round it. -e +Ze +Ze -e This B fied, due to orbit motion, intercts with eectron's spin dipoe moment µ s Um = µ. B " Energy rger when S B, smer when nti-pre " Sttes with sme ( n,, m ) but diff. spins " energy eve spitting/doubing due to S Under No Extern B Fied There is Sti Spitting Sodium Doubet & LS couping 3

14 Vector Mode For Tot Angur Momentum J Couping of Orbit & Spin mgnetic moments Neither Orbit nor Spin ngur Momentum re conserved sepertey J = L + S is conserved so ong s there re no extern torques present Rues for Tot Angur Momentum Quntiztion : J = j( j + ) " with j = + s, + s -, + s -...,...., - s J = m " with m z j j Exmpe: stte with ( =, s = ) j = 3/ " m = -3/, /,/,3/ j = / " m = ± / j In gener m tkes ( j + ) vues " Even # of orienttions j j = j, j -, j , - j Spectrogrphic Nottion: Fin Lbe n S P / 3/ Compete Description of Hydrogen Atom j Fu description { n,, m, m } LS Couping Compete Description of Hydrogen Atom of the Hydrogen tom: { n,, j, m } to 4 D.O. F. corresponding s s n S P / 3/ How to describe muti-eectrons toms ike He, Li etc? How to order the Periodic tbe? Four guiding principes: Indistinguishbe prtice & Pui Excusion Principe Independent prtice mode (ignore inter-eectron repusion) Minimum Energy Principe for tom Hund s rue for order of fiing vcnt orbits in n tom j 4

15 Muti-Eectron Atoms : > eectron in orbit round Nuceus In Hydrogen Atom (r,",#)=r(r). $ (" ).%(#) & { n,, j, m j } In n-eectron tom, to simpify, ignore eectron-eectron interctions compete wvefunction, in "independent"prtice pprox" : (,, 3,..n)= (). (). (3)... ( n)??? Compiction ' Eectrons re identic prtices, being meningess Question: How mny eectrons cn hve sme set of quntum #s? Answer: No two eectrons in n tom cn hve SAME set of quntum#s (if not, eectrons woud occupy s stte (est energy)... no structure e - e - Exmpe of Indistinguishbiity: eectron-eectron scttering Sm nge sctter rge nge sctter Quntum Picture If we cnt foow eectron pth, don t know between which of the two scttering Event ctuy hppened Heium Atom: Two eectrons round Nuceus In Heium, ech eectron hs : kinetic energy + eectrosttic potenti energy If eectron "" is octed t r & eectron ""is octed t r then TISE hs terms ike: (e) e H = - " + k m r ( ) ( e)( e) ; H = - " + k such tht m r H # + H # = E# ; H & H re sme except for "be" e Independent Prtice Approx $ ignore repusive U=k term r r * Heium WveFunction: # = # (r, r ); Probbiity P = # (r, r )# (r, r ) But if we exchnge oction of (identic, indistinguishbe) eectrons $ # (r, r ) = # (r, r ) In gener, when # (r, r ) = # (r, r )...Bosonic System (mde of photons, e.g) when # (r, r ) = # (r, r )...fermionic System (mde of eectron, proton e.g) $ Heium wvefunction must be ODD; if eectron "" is in stte & eectron "" is in stte b Then the net wvefunction # (r,r )=# ( r ).# ( r ) stisfies b b H # ( r ).# ( r ) = E # ( r ).# ( r ) b b H # ( r ).# ( r ) = E # ( r ).# ( r ) b b b nd the sum [H +H ]# ( r ).# ( r ) = ( E + E )# ( r ).# ( r ) b b b Tot Heium Energy E " E +E =sum of Hydrogen tom ike E b e- e - b 5

16 Heium Atom: Two eectrons round Nuceus e- Heium wvefunction must be ODD " nti-symmetric: (r,r )=- (r,r ) So it must be tht (r, r ) = ( r ). ( r ) # ( r ). ( r ) b b b b It is impossibe to te, by ooking t probbiity or energy which prticur eectron is in which stte If both re in the sme quntum stte " =b & (r,r )= (r,r )=... Pui Excusion principe b bb Gener Principes for Atomic Structure for n-eectron system:. n-eectron system is stbe when its tot energy is minimum.ony one eectron cn exist in prticur quntum stte in n tom...not or more 3. Shes & SubShes In Atomic Structure : () ignore inter-eectron repusion (crude pprox.) (b) think of ech eectron in constnt "effective" men Eectric fied (Effective fied: "Seen" Nucer chrge (+Ze) reduced by prti screening due to other eectrons "buzzing" coser (in r) to Nuceus) Eectrons in SHELL: hve sme n, re t simir <r> from nuceus, hve simir energies Eectons in SubShe: hve sme princip quntum number n - Energy depends on, those with ower coser to nuceus, more tighty bound - eectrons in sub-she hve sme energy, with minor dependence on m, m s e - b She & Sub-She Energies & Cpcity. She & subshe cpcity imited due to Pui Excusion principe Energy. She is mde of sub-shes ( of sme princip quntum # n) 3. Subshe ( n, ), given n " =,,,3,..( n -), for ny " m =, ±, ±,.. " ( + ), ms = ± " Mx. # of eectrons in she = subshe cpcity * n# $ % NMAX = *.( + ) = [ ( n # ) + ] = ( n) & ( + (n # )) = n = ( ' ) 4. The "K" She (n=) hods eectron s, "L" She (n=) hods 8 eectrons, M she (n=3) hods 8 eectrons She is cosed when fuy occupied 6. Sub-She cosed when * i * S i i i () L =, =, " Effective chrge distribution= symmetric (b) Eectrons re tighty bound since they "see" rge nucer chrge (c) Becuse * L i = " No dipoe moment " No biity to ttrct eectrons i " Inert Nobe gs 6.Aki Atoms: hve singe "s" eectron in outer orbit; nucer chrge heviy shieded by inner she eectrons " very sm binding energy of "vence"eectron " rge orbit rdius of vence eectron 6

17 Eectronic Configurtions of n successive eements from Lithium to Neon Hund s Rue: Whenever possibe -eectron in sub-she remin unpired - Sttes with spins pre occupied first - Becuse eectrons repe when cose together - eectrons in sme sub-she () nd sme spin -Must hve diff. m -(very diff. ngur distribution) - Eectrons with pre spin re further prt -Thn when nti-pre esser E stte -Get fied first Periodic tbe is formed Tht s I cn tech you this qurter; Rest is Chemistry 7