4E : The Quantum Universe. Lecture 25, May 14 Vivek Sharma

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1 4E : The Quntum Universe Lecture 5, My 14 Vivek Shrm modphys@hepmil.ucsd.edu

2 Interpreting Orbitl Quntum Number (l) 1 d dr m ke l( l+ 1) Rdi l prt of S.Eqn: r + ( E+ )- R( r) r dr dr r r = ke For H Atom: E = K + U = KRADIAL + K ORBITAL ; substitute this in E r 1 d dr m l( l+ 1) r K + RADIAL + KORBI TAL - R r dr dr m r () r = Exmine the eqution, if we set K ORBITAL ll ( + 1) = then m r wht remins is differentil eqution in r 1 d dr m r + [ KRAD IAL ] R( r) = which depends only on rdius r of orbit r dr dr 1 L Further, we lso know t ht K ORBITAL = mvorbit; L= r p ; L =mv orbr KORBIT = mr ll L Putting it ll togt her: K mgnitude of Ang m r mr L p r AL ( + 1) ORBITAL =. Mom L l l = = + ( 1) Since l = positive integer =,1,,3...(n-1) ngulr momentum L = l( l + 1) = discrete vlues L = l( l + 1) : QUANTIZATION OF Elect ron's Angulr Mom entu m

3 Mgnetic Quntum Number : m l L = r p (Right Hnd Rule) Clssiclly, direction & Mgnitude of L lwys well defined QM: Cn/Does L hve definite direction? Proof by Negtion: Suppose L ws precisely known/defined (L z) ˆ Since L = r p Electron MUST be in x-y orbit plne p z = ; pz z p z ; E =!!! m So, in Hydrogen tom, L cn not hve precise mesurble vlue Uncertinty Principle & Angulr Momentum : L z φ

The Z component of L : L = m; m =± 1, ±, ± 3.

4 Mgnetic Quntum Number m l Consider = L = ( + 1) = 6 In Hydrogen tom, L cn not hve precise mesurble vlue Arbitrrily picking Z xis s reference direction: L vector spins round Z xis (precesses). The Z component of L : L = m; m =± 1, ±, ± 3... ± l Z l l Note : since L < L (lwys) since Z m < l( l+ 1) It cn never be tht L = m= l( l + 1) l (breks Uncertinty Principle) Z l So...the Electron's dnce hs begun!

5 L=, m l =,±1, ± : Pictorilly Electron sweeps conicl pths of different ϑ: Cos ϑ= L Z /L On verge, the ngulr momentum component in x nd y cncel out <L X > = <L Y > =

6 Where is it likely to be? Æ Rdil Probbility Densities Ψ ( r,θ, φ ) = Rnl ( r ). Θlml (θ ). Φ ml (φ ) = Rnl Ylml Probbility Density Function in 3D: P(r,θ,φ ) = Ψ*Ψ = Ψ ( r,θ, φ ) = Rnl. Ylml Note : 3D Volume element dv= r.sin θ.dr.dθ.dφ Prob. of finding prticle in tiny volume dv is P.dV = Rnl. Ylml.r.sin θ.dr.dθ.dφ The Rdil prt of Prob. distribution: P(r)dr π π dv P(r)dr= Rnl.r dr Θlml (θ ) dθ Φ ml (φ ) dφ When Θlml (θ ) & Φ ml (φ ) re uto-normlized then P(r)dr= Rnl.r. dr; in other words P(r)=r Rnl Normliztion Condition: 1 = r R nl dr Expecttion Vlues <f(r)>= f(r).p(r)dr 6

7 Ground Stte: Rdil Probbility Density () = ψ().4π Prdr r rdr Prdr () = 4 r re 3 r re 3 Probbility of finding Electron for r> P r> = 4 dr To solve, employ chnge of vrible r Define z= ; chnge limits of integrtion P r> = 1 z z e dz (such integrls clled Error. Fn) 1 z =- [ z + z+ ] e = 5e = %!!

Most Probble & Averge Distnce of Electron from Nucleus Most Probble Distnce: r 4 n = l = ml = P r dr = r e 3 In the ground stte ( 1,, ) ( ) Most probble distnce r from Nucleus Wht vlue of r is P(r)

8 Most Probble & Averge Distnce of Electron from Nucleus Most Probble Distnce: r 4 n = l = ml = P r dr = r e 3 In the ground stte ( 1,, ) ( ) Most probble distnce r from Nucleus Wht vlue of r is P(r) mx? r dp 4 d r =. re r 3 = + e dr dr ( )! Most Probble distnce (Bohr guess 3 r = r + r = r = or r =... which solution is correct? (see pst quiz) : Cn the electron BE t the center of Nucleus (r=)? 4 Pr= = e = r = Wht bout the AVERAGE loction <r> of the electron in Ground stte? <r>= r 4 rp(r)dr= r= 3 z n z r z e dz e dz n n n z = rr e dr r chnge of vrible z= < >=... Use generl form z =! = ( 4 < r >= = ed right) 1)( n )...(1) 3 3!! Averge & most likely distnce is not sme. Why? 4 Asnwer is in the form of the rdil Prob. Density: Not symmetric

9 Rdil Probbility 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 electron inside the nucleus =!!

10 Normlized Sphericl Hrmonics & Structure in H Atom

11 Excited Sttes (n>1) of Hydrogen Atom : Birth of Chemistry! Fetures of Wvefunction in θ & φ : Consider n=, l = ψ = Sphericlly Symmetric (lst slide) Excited Sttes (3 & ech with sme E ) : ψ, ψ, ψ ψ re ll p sttes 3/ 1 Z Z r =R Y = e π 8 n Zr. sinθ e. i φ * π ψ11 = ψ ψ sin θ Mx t θ=,min t θ =; Symm in φ Wht bout (n=, =1, m = ) ψ = R (r) Y( θφ, ); l p z 1 3 Y( 1 θφ, ) cos θ; π π Function is mx t θ =, min tθ = We cll this p z stte becuse of its extent in z

12 Excited Sttes (n>1) of Hydrogen Atom : Birth of Chemistry! Remember Principle of Liner Superposition for the TISE which is bsiclly simple differentil eqution: - m ψ + Uψ = Eψ Principle of Liner Superposition If ψ nd ψ re sol. 1 then "designer" wvefunction mde of liner sum ' ψ = ψ1 + bψ is lso sol. of the diff. eqution! ' To check this, just substitute in pl & convince yourself tht ψ ce of ψ of TISE - m ' ' + = ' ψ Uψ Eψ The diversity in Chemistry nd Biology DEPENDS on this superposition rule

13 Designer Wve Functions: Solutions of S. Eq! Liner Superposition Principle mens llows me to "cook up" wvefunctions 1 ψp = [ ψ ] x 11 + ψ1 1...hs electron "cloud" oriented long x xis 1 ψ p = [ ψ ] y 11 ψ1 1...hs electron "cloud" oriented long y xis So from 4 solutions ψ, ψ1, ψ11, ψ1 1, s px, py,pz Similrly for n=3 sttes...nd so on...cn get very complicted structure in θ & φ...which I cn then mix & mtch to mke electrons" most likely" to be where I wnt them to be!

14 Designer Wve Functions: Solutions of S. Eq!

15 Cross Sectionl View of Hydrogen Atom prob. densities in r,θ,φ Birth of Chemistry (Cn mke Fncy Bonds Overlpping electron clouds ) Wht s the electron cloud : Its the Probbility Density in r, θ,φ spce! Z Y

16 Wht s So Mgnetic? Precessing electron Current in loop Mgnetic Dipole moment µ The electron s motion hydrogen tom is dipole mgnet

The Mgnetism of n Orbiting Electron Precessing electron Current in loop Mgnetic Dipole moment µ Electron in motion round nucleus circulting chrge curent e e ep i = = = ; Are of current loop

17 The Mgnetism of n Orbiting Electron Precessing electron Current in loop Mgnetic Dipole moment µ Electron in motion round nucleus circulting chrge curent e e ep i = = = ; Are of current loop A= π r T π r π mr v -e -e -e Mgnetic Moment µ =ia= rp ; µ = r p = L m m m Like the L, mgnetic moment µ lso precesses bout "z" xis -e -e z component, µ z = L = m µ m quntized! m z = = m l B l i

18 Quntized Mgnetic Moment µ µ z B -e -e = Lz = m m = µ m B = Bohr Mgnetron e = m e l m l Why ll this? Need to find wy to brek the Energy Degenercy & get electron in ech ( nlm,, l ) stte to identify itself, so we cn "tlk" to it nd mke it do our bidding: " Wlk this wy, tlk this wy!"

19 Lifting Degenercy : Mgnetic Moment in Externl B Field Apply n Externl B field on Hydrogen tom (viewed s dipole) Consider B Z xis (could be ny other direction too) The dipole moment of the Hydrogen tom (due to electron orbit) experiences Torque τ = µ B which does work to lign µ B but this cn not be (sme Uncertinty principle rgument) So, Insted, µ precesses (dnces) round B... like spinning top The Azimuthl ngle φ chnges with time : clculte frequency Look t Geometry: projection long x-y plne : dl = Lsin θ.dφ dl q dφ = ; Chnge in Ang Mom. dl = τ dt = LBsinθ dt Lsinθ m dφ 1 dl 1 q ωl = = = LB sinθ = dt Lsinθ dt Lsinθ m L qb ω depends on B, the pplied externl mgnetic field m e Lrmor Freq

20 Lifting Degenercy : Mgnetic Moment in Externl B Field WORK done to reorient µ ginst B field: dw= τd θ =-µ Bsinθdθ dw = d( µ Bcos θ) : This work is stored s orienttionl Pot. Energy U dw= -du Define Mgnetic Potentil Energy U=- µ. B = µ cos θ. B = µ B e Chnge in Potentil Energy U = m e mb l = ω m Zeemn Effect in Hydrogen Atom In presence of Externl B Field, Totl energy of H tom chnges to E=E + ω m L So the Ext. B field cn brek the E degenercy "orgniclly" inherent in the H tom. The Energy now depends not just on n but lso ml l L l z

21 Zeemn Effect Due to Presence of Externl B field Energy Degenercy Is Broken

University of Washington Department of Chemistry Chemistry 453 Winter Quarter 2010 Homework Assignment 4; Due at 5p.m. on 2/01/10

University of Washington Department of Chemistry Chemistry 453 Winter Quarter 2010 Homework Assignment 4; Due at 5p.m. on 2/01/10 University of Wshington Deprtment of Chemistry Chemistry 45 Winter Qurter Homework Assignment 4; Due t 5p.m. on // We lerned tht the Hmiltonin for the quntized hrmonic oscilltor is ˆ d κ H. You cn obtin