Physics 2D Lecture Slides Lecture 30: March 9th 2005 Vivek Sharma - PDF document

Physics 2D Lecture Slides Lecture 30: March 9th 2005 Vivek Sharma UCSD Physics Magnetic Quantum Number m l � � � = � L r p (Right Hand Rule) � Classically, direction & Magnitud e of L always well defi n ed � QM: Can/Does L have a definite direction ? Proof by Negat io n : � � ˆ Suppose L was precisely known/defined (L || z) � � � = � � S ince L r p Electron MUST be in x-y orbit plane 2 p � � � � � � � = � z = 0 ; p z � � p � ; E � !!! z z 2 m � So , in Hydrogen atom, L can not have precise measurable value � � � Uncertainty Principle & An gular Momentum : L � � z 1

Magnetic Quantum Number m l Consider � = 2 | L | = � ( � + 1) = 6 � � In Hydrogen atom, L can not have precise measurable value Arbitararily picking Z axis as a reference direction : � L vector spins around Z axis (precesses). � = = ± ± ± ± The Z component of L : | L | m � ; m 1 , 2, 3... l Z l l < Note : since | L | | L | (always) Z < + = = + sin c e m � l l ( 1) It can never be that |L | � m � l ( l 1) � l Z l (break s Uncertainty Pri ncip le) So......the Electron's dance has be gun ! L=2, m l =0, ± 1, ± 2 : Pictorially Electron “sweeps” Conical paths of different ϑ : Cos ϑ = L Z /L On average, the angular momentum Component in x and y cancel out <L X > = 0 <L Y > = 0 2

Where is it likely to be ?  Radial Probability Densities � � � = � � � � = m ( , , ) r R ( ) . r ( ) . ( ) R Y l nl l m m nl l l l Probability Density Function in 3D: � � � � � � � = m P(r, , ) = * =| ( , , ) | r 2 | R | . 2 | Y | 2 l n l l 2 � � � Note : 3D Volume element dV= r .sin . d r . d . d Prob. of finding parti cle in a ti n y volume dV is m � � � P.dV = | R | . 2 | Y | .r .sin . 2 2 dr . d . d l n l l The Radial part of Prob. distribution: P(r)dr dv � 2 � � � � � � � � � P(r)dr= | R | 2 . r d 2 r | ( ) | 2 d | ( ) | 2 d nl lm m l l 0 0 � � � � When ( ) & ( ) are auto-normalized then l m m l l 2 2 2 2 P(r)dr = | R | . . r d r ; in other words P(r)=r | R | n l nl � � 2 2 Normalization Condition: 1 = r |R | dr nl 0 � � Expectation Values <f( r)>= f(r).P(r)dr 0 Ground State: Radial Probability Density = � 2 � 2 P r dr ( ) | ( ) | .4 r r dr r 4 � 2 � = 2 a P r dr ( ) r e 0 3 a 0 Probability of finding Electron for r>a 0 r 4 � 2 � 2 a r e dr 0 � = P 3 a r a > 0 0 a To solve, employ change of variable � � 2r Define z= ; change limits of integra tion � � a � � 0 � 1 � = � 2 z P z e dz (such integrals called Error. Fn) > r a 2 0 2 1 + + � � = = � 2 z 2 =- [ z 2 z 2] e | 5 e 0.667 66. 7% !! 2 2 3

Most Probable & Average Distance of Electron from Nucleus Most Probable Distance: r 4 � 2 = = = = In the ground state ( n 1, l 0, m 0) P r dr ( ) r e 2 a 0 l 3 a 0 Most probable distance r from Nucleus � What value of r is P(r) max? � r � r � � � dP 4 d � 2 r 2 � 2 2 � � 2 a = � + a = =0 . � r e � 0 2 r e 0 0 � � 0 dr a 3 dr a � � � � � � 0 0 2 r 2 � + = � = = 2 r 0 r 0 or r a ... which solution is correct? 0 a 0 (see past quiz) : Can the electron BE at the center of Nucleus (r=0)? 0 4 � 2 = = a = � = P r ( 0) 0 2 e 0! Most Probable distance r a (Bohr guess ed rig ht) 0 3 0 a 0 What about the AVERAGE locati on <r> of the electron in Ground state? � � r 4 � 2r 2 � � <r>= rP(r)dr= r r e . 2 a d r ... cha nge of variable z= a 0 a 3 r=0 0 0 0 � � a � � � < >= 3 � z n � z = = � � r 0 z e dz ....... Use general for m z e dz n ! n ( n 1)( n 2)...(1) 4 z = 0 0 a 3 a � < >= = � r 0 3! 0 a ! Average & most likely distance is not same. Why? 0 4 2 Asnwer is in the form of the radial Prob. Density: Not symmetric Radial Probability Distribution P(r)= r 2 R(r) Because P(r)=r 2 R(r) No matter what R(r) is for some n The prob. Of finding electron inside nucleus =0 4

Normalized Spherical Harmonics & Structure in H Atom Excited States (n>1) of Hydrogen Atom : Birth of Chemistry ! � � Features of Wavefunction in & : = = � � = Consider n 2, l 0 Spherically Symmetric (last slide) 200 Excited States (3 & each with same E ) : n � � � , , are all 2 p states 211 210 21- 1 3/2 � Zr � � � � � 1 � � Z � Z � r � a � � =R Y 1 = a e . sin . i e � � � � � � � 0 211 21 1 � � � � � 8 � a � � � 0 0 � � 2 = � � * � 2 � � � � | | | | sin Max at = ,min at =0; Symm in 211 211 2 21 1 = W hat about (n=2, =1, � m 0 ) l 2p z � = � � R (r) Y ( , ); 0 210 21 1 1 3 0 � � � � Y ( , ) cos ; 1 � 2 � � � Function is max at =0, min a t = 2 We call this 2p state because of its extent in z z 5

Excited States (n>1) of Hydrogen Atom : Birth of Chemistry ! Remember Principle of Linear Superposition 2p z for the TISE which is basically a simple differential equat ion: � 2 � � + � = � - 2m 2 U E � � � Principle of Linear Superposition If a nd are sol. of TISE 1 2 then a "des igne r" wavefunction made of linear sum � ' = � + � a b i s also a sol. of the diff. equ ation ! 1 2 � ' � To check this, just substitute in pla ce of & convince yourself that � 2 � 2 � ' + � ' = � ' - U E 2m The diversity in Chemistry and Biology DEPENDS on this superposition rule Designer Wave Functions: Solutions of S. Eq ! Linear Superposition Principle means allows me to "cook up" wavefunctions 1 [ ] � = � + � ......has electron "cloud" oriented along x axis � 2p 211 21 1 x 2 1 [ ] � = � � � ......has electron "cloud" oriented along y axis � 2p 211 21 1 2 y � � � � � � So from 4 solutio ns , , , 2 ,2 s p ,2 p ,2 p 200 210 211 21 1 x y z Similarly for n=3 states ...and so on ...can get very complicated structure � � in & .......whic h I can then mix & match to make electron s " most likely" to be where I want them to be ! 6

Designer Wave Functions: Solutions of S. Eq ! � 2 d + 2 � = m 0.. .................(1) � l 2 d Typo Fixed � � � � 2 1 d � d � + m � + � � � = sin l l ( 1) l ( ) 0.....(2) � � � � � � � � � 2 � sin d � d � sin � � � � � � � � � + 1 d � � + 2m r 2 ke 2 l l ( 1) 2 = r (E + )- R r ( ) 0....(3) � � � � � � 2 � 2 2 r dr � r � � r r � � � � These 3 "simple" diff. eqn describe the physics of the Hydrogen atom. The hydrogen atom brought to you by the letters � n = 1,2,3,4,5,.... = � l 0,1,2,3 ,,4....( n 1) = ± ± ± ± m 0 , 1, 2, 3,.. . l l The Spatial Wave Function of the Hydrogen Atom � � � = � � � � = m ( , r , ) R ( ) . r ( ) . ( ) R Y (Spherical Harmonics) l nl lm m nl l l l 7

Cross Sectional View of Hydrogen Atom prob. densities in r, θ , φ Birth of Chemistry (Can make Fancy Bonds  Overlapping electron “clouds”) What’s the electron “cloud” : Its the Probability Density in r, θ , φ space ! Z Y What’s So “Magnetic” ? Precessing electron  Current in loop  Magnetic Dipole moment µ The electron’s motion  hydrogen atom is a dipole magnet 8

The “Magnetism”of an Orbiting Electron Precessing electron  Current in loop  Magnetic Dipole moment µ � � Electron in motion around nucleus circulating charge curent i � � � e e ep = = = � i ; Area of current lo op A= r 2 � 2 r � T 2 mr v � � -e � � � -e � � � � -e � µ µ = � = Magnetic Moment | |=i A= r p ; r p L � � � � � � � 2m � � 2m � � 2m � � � µ Like the L, magneti c moment also prece sses about "z" axi s � � � � -e -e � µ = = = � µ = z component, L m m quantized ! � � � � z z l B l � 2 m � � 2 m � Quantized Magnetic Moment � � � � -e -e � µ = = L m � � � � z z l � 2m � � 2m � = � µ m B l µ = Bohr Magnetron B � � e � = 2m � � � � e Why all this ? Need to find a way to break the Energy Degeneracy & get electron in each ( , , n l m ) state to identify its elf , so l we can "talk" to it and make it do our bidding: " Walk this wa y , ta lk th s i way!" 9