Physics 2D Lecture Slides Lecture 29: Mar 10th Vivek Sharma UCSD - PDF document

This is the first course at UCSD for which “Lecture on Demand” have been made available. 2000 Minutes of Streaming video served to hundreds of “demands” without interruption (24/7) Pl. take 10 minutes to fill out the Streaming Video Survey sent to you last week by the Physics Department. Your input alone will decide whether to extend this educational service to other classes at UCSD and to the next generation of UC students. Physics 2D Lecture Slides Lecture 29: Mar 10th Vivek Sharma UCSD Physics

Φ 2 d + Φ = 2 m 0.. .................(1) φ 2 l d Typo Fixed ⎡ ⎤ ⎡ ⎤ ⎛ ⎞ + 2 1 m d d θ + − Θ θ = ⎢ l ⎥ ⎜ sin ⎟ ⎢ ( 1) ⎥ ( ) 0.....(2) l l θ θ θ θ ⎝ ⎠ 2 sin ⎣ sin ⎦ ⎣ d d ⎦ ⎡ ⎤ ⎡ ⎤ ∂ + ⎛ ⎞ + 2 2 1 2m ke ( 1) d r l l = 2 ⎢ ⎥ ⎜ ⎟ (E + )- ( ) 0....(3) r ⎢ ⎥ R r ∂ 2 ⎝ ⎠ � 2 2 ⎣ r ⎦ ⎣ r dr 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,.... = − 0,1,2,3 ,,4....( 1) l n = ± ± ± ± m 0 , 1, 2, 3,.. . l l The Spatial Wave Function of the Hydrogen Atom Ψ θ φ = Θ θ Φ φ = m ( , , ) ( ) . ( ) . ( ) Y (Spherical Harmonics) r R r R l m nl lm nl l l l 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 space ! Z Y

Magnetic Quantum Number m l � � � = × Angular Momentum of the electron (Right Hand Rule) L r p � Classically, direction & Magnitude of L always well define d � QM: Can/Does L have a definite direction ? Proof by Negatio n : � � ˆ Supp os e L was p rec isely known/defined ...say (L || z) � � � = × ⇒ Sinc e Electron MUST be PRECISELY in the x-y plane L r p 2 p ⇒ ∆ ∆ ∆ ⇒ ∆ ∞ = ∞ ∼ � ∼ ∼ z = 0 ; p p ; z !!! z KE z z z 2 m ⇒ breaks the Hydrogen bound state.... . � , in the bo und Hy d rogen atom, L can not have precise measurable valu e So ⇒ ∆ ∆ φ ∼ � Uncertainty Principle & Angular Moment um L z Magnetic Quantum Number m l = � Consider 2 = + = � � � | | ( 1) 6 L � 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 < : since | L | | | (always) Note L Z < + = = + � � � � sin c e ( 1) It can never be that |L | ( 1) m l l m l l 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 Where is it likely to be ? � Radial Probability Densities Ψ θ φ = Θ θ Φ φ = m ( , , ) ( ) . ( ) . ( ) Y r R r R l m nl l m nl l l l Probability Density Function in 3D: θ φ Ψ Ψ Ψ θ φ = * 2 2 m 2 P(r, , ) = =| ( , , ) | | | . | Y | r R l n l l θ θ φ 2 : 3D Volume element dV= r .sin . . . Note d r d d Prob. of finding parti cle in a ti n y volume dV is θ θ φ 2 m 2 2 P.dV = | | . | Y | .r .sin . . . R dr d d l n l l The Radial part of Prob. distribution: P(r)dr dv π π 2 ∫ ∫ Θ θ θ Φ φ φ 2 2 2 2 P(r)dr= | | . | ( ) | | ( ) | R r d r d d nl lm m l l 0 0 Θ θ Φ φ When ( ) & ( ) are auto-normalized then l m m l l 2 2 2 2 P(r)dr = | | . . ; in other words P(r)=r | | R r d 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 = ψ π 2 2 ( ) | ( ) | .4 P r dr 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 ∫ = 3 P 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 (such integrals called Error. Fn) P z e dz > r a 0 2 2 1 − ∞ + + = = ⇒ 2 z 2 =- [ z 2 z 2] e | 5 e 0.667 66. 7% !! 2 2 Most Probable & Average Distance of Electron from Nucleus Most Probable Distance: r − 4 2 = = = = 2 a In the ground state ( 1, 0, 0) ( ) n l m P r dr r e 0 l 3 a 0 ⇒ Most probable distance r from Nucleus What value of r is P(r) max? ⎡ ⎤ r ⎡ − ⎤ r 2 − − dP 4 d 2 2 r 2 ⇒ ⇒ = ⇒ + = ⎢ 2 ⎥ a a =0 . r e 0 ⎢ 2 r ⎥ e 0 0 0 3 dr a dr ⎢ ⎥ ⎣ a ⎦ ⎣ ⎦ 0 0 2 2 r ⇒ + = ⇒ = = 2 0 0 ... which solution is correct? r r or r a 0 a 0 (see past quiz) : Can the electron BE at the center of Nucleus (r=0)? 0 − 4 2 = = = ⇒ = 2 a ( 0) 0 0! Most Probable distance (Bohr guess ed rig ht) P r e 0 r a 0 3 a 0 What about the AVERAGE locati on <r> of the electron in Ground state? ∞ ∞ r − 4 2r 2 ∫ ∫ 2 a <r>= rP(r)dr= . ... cha nge of variable z= a r r e d r 0 3 a r=0 0 0 0 ∞ ∞ a ∫ ∫ − − ⇒< >= = = − − 0 3 z n z ....... Use general for m z ! ( 1)( 2)...(1) r z e dz e dz n n n n 4 = z 0 0 3 a a ⇒ < >= = ≠ 0 0 3! ! Average & most likely distance is not same. Why? r a 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 Normalized Spherical Harmonics & Structure in H Atom

Excited States (n>1) of Hydrogen Atom : Birth of Chemistry ! θ φ Features of Wavefunction in & : = = ⇒ ψ = Consider 2, 0 Spherically Symmetric (last slide) n l 200 Excited States (3 & each with same E ) : n ψ ψ ψ , , are all 2 p states 211 210 21- 1 3/ 2 − ⎛ ⎞ ⎛ ⎞ Zr ⎛ ⎞ ⎜ ⎛ ⎞ 1 Z Z r ψ θ φ 1 a =R Y = a ⎟ ⎜ ⎟ . sin . i ⎜ ⎟ ⎜ ⎟ e e 0 π 211 21 1 ⎝ ⎠ ⎝ ⎝ ⎠ ⎠ 8 ⎝ a ⎠ 0 0 π ψ = ψ ψ ∝ θ θ θ φ 2 * 2 | | | | sin Max at = ,min at =0; Symm in 211 211 21 1 2 = � W hat about (n=2, =1, m 0 ) l ψ = θ φ 0 2p z (r) Y ( , ); R 210 21 1 1 3 θ φ ∝ θ 0 Y ( , ) cos ; 1 π 2 π θ θ Function is max at =0, min a t = 2 We call this 2p state because of its extent in z z 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 ∇ ψ + ψ = ψ 2 - 2m U E ⇒ ψ ψ Principle of Linear Superposition If are sol. of TISE a nd 1 2 then a "des igne r" wavefunction made of linear sum ψ = ψ + ψ ' a b i s also a sol. of the diff. equ ation ! 1 2 ψ ψ ' To check this, just substitute in pla ce of & convince yourself that � 2 ∇ ψ + ψ = ψ 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 [ ] ψ = ψ + ψ ......has electron "cloud" oriented along x axis − 2p 211 21 1 x 2 1 [ ] ψ = ψ − ψ ......has electron "cloud" oriented along y axis − 2p 211 21 1 y 2 ψ ψ ψ ψ − → So from 4 solutio ns , , , 2 ,2 ,2 ,2 s p p p 200 210 211 21 1 x y z Similarly for n=3 states ...and so on ...can get very complicated structure θ φ in & .......whic h I can then mix & match to make electron s " most likely" to be where I want them to be ! Designer Wave Functions: Solutions of S. Eq !

What’s So “Magnetic” ? Precessing electron � Current in loop � Magnetic Dipole moment µ The electron’s motion � hydrogen atom is a dipole magnet The “Magnetism”of an Orbiting Electron Precessing electron � Current in loop � Magnetic Dipole moment µ ⇒ ⇒ E lectron in m otion around nucleus circulating charge curent i − − − e e ep = = = π 2 ; A rea of current lo op A = r i π π 2 r 2 T m r v � ⎛ ⎞ � ⎛ ⎞ � � ⎛ ⎞ -e -e -e µ µ = × = M agnetic M om ent | |=i A = ⎜ ⎟ ; ⎜ ⎟ ⎜ ⎟ r p r p L ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 2m 2m 2m � � µ Like the L, m agneti c m om ent also prece sses about "z" axi s ⎛ ⎞ ⎛ � ⎞ -e -e µ = = = − µ = z com ponent, ⎜ ⎟ L ⎜ ⎟ m m quantized ! z z l B l ⎝ ⎠ ⎝ ⎠ 2 m 2 m