Physics 2D Lecture Slides Lecture 31: March 11th 2005 Vivek Sharma UCSD Physics 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 1
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 2
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 ! 3
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 4
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 5
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!" 6
“Lifting” Degeneracy : Magnetic Moment in External B Field � Apply an External B field on a Hydrogen atom (viewed as a dipole) � � Consider B || Z axis (could be any other direction too) The dipole moment of the Hydrogen atom (due to electron orbit) � � � � � � = µ � µ experi e nces a Torque B which does work to align || B but this can not be (same Uncertainty principle argument) � � � µ So, Instead, precesses (dances) around ... like a spinning B top � T he Azimuthal angle changes with time : calculate frequency � � Look at Geometry: |projection along x-y plane : |dL| = Lsin .d |dL| ; Change in Ang Mom. q � � = = � = � d | dL | | | dt LB sin dt � Ls in 2 m � d 1 |dL 1 q LB qB � � � = = = = sin Larmor Freq L � � dt Lsin dt Lsin 2 m 2 m e � L depends on B, the applied externa l magnetic f l ie d “Lifting” Degeneracy : Magnetic Moment in External B Field � � µ � � µ � � WORK done to reorient against field: dW= d =- Bsin d B = µ � d W d ( Bcos ) : This work is stored as orientational Pot. Energy U dW= - dU � � µ = � µ � = � µ Define Magnetic Potential Ene rgy U=- . B cos . B B z e � = � � Change in Potential Energy U = 2m m B m l L l e Zeeman Effect in Hydrogen Atom In presence of External B Field, Total energy of H atom changes to + � � E=E m 0 L l So the Ext. B field can break the E degeneracy "organically" inherent in the H atom. The E nergy now depends not just on but also n m l 7
Zeeman Effect Due to Presence of External B field Energy Degeneracy Is Broken Electron has “Spin”: An additional degree of freedom Even as the electron rotates around nucleus, it also “spins” There are only two possible spin orientations: Spin up : s = +1/2 ; Spin Down: s=-1/2 “Spin” is an additional degree of freedom just Like r, θ and ϕ Quantum number corresponding to spin orientations m l = ± ½ Spinning object of charge Q can be thought of a collection of elemental charges Δ q and mass Δ m rotating in circular orbits So Spin Spin Magnetic Moment interacts with B field 8
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