Physics 2D Lecture Slides Lecture 30: Mar 12th Vivek Sharma UCSD - PDF document

Course review is scheduled for Sunday 14 th March at 10am in WLH 2005 There will be no Streaming video for this session, so pl. attend Physics 2D Lecture Slides Lecture 30: Mar 12th Vivek Sharma UCSD Physics

Φ 2 d + Φ = 2 m 0.. .................(1) φ 2 l d ⎡ ⎤ ⎡ ⎤ ⎛ ⎞ + 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 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

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 ( , , ) state to identify its elf , so n l m l we can "talk" to it and make it do our bidding: " Walk this wa y , ta lk th s i way!" “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) � � � � � τ = µ × µ experi e nces a Torque which does work to align || B B but this can not be (same Uncertainty principle argument) � � ⇒ µ So, Instead, precesses (dances) around ... like a spinning B top φ T he Azimuthal angle changes with time : calculate frequency θ φ Look at Geometry: |projection along x-y plane : |dL| = Lsin .d |dL| ; Change in Ang Mom. q ⇒ φ = = τ = θ d | dL | | | dt LB sin dt θ Ls in 2 m φ d 1 |dL 1 q LB qB ⇒ ω θ = = = = sin Larmor Freq θ θ L dt Lsin dt Lsin 2 2 m m e ω L depends on B, the applied externa l magnetic f l ie d

“Lifting” Degeneracy : Magnetic Moment in External B Field � � µ τ θ µ θ θ WORK done to reorient against field: dW= d =- Bsin d B = µ θ ( Bcos ): This work is stored as orientational Pot. Energy U d W d dW= - dU � � µ = − µ θ = − µ Define Magnetic Potential Ene rgy U=- . cos . B B B z � e = � ω 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 + � ω 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 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 ∆ q and mass ∆ m rotating in circular orbits So Spin � Spin Magnetic Moment � interacts with B field Stern-Gerlach Expt ⇒ An additional degree of freedom: “Spin” for lack of a better name � � � µ in inhomogenous B field, experiences force F � � � ∇ = −∇ − µ F= - U ( .B) B ∂ ∂ ∂ B B B ≠ = = When gradient only along z, 0; 0 ∂ ∂ ∂ z x y ∂ B = µ F m ( ) moves particle up or down z B ∂ z (in addition to torque causing Mag. momen t t o precess about B field direction In an inhomogeneous field, magnetic moment µ experiences a force F z whose direction depends on component of the net magnetic moment & inhomogeneity dB/dz. Quantization means expect (2l+1) deflections. For l=0, expect all electrons to arrive on the screen at the center (no deflection) ! Silver Hydrogen (l=0)

Four (not 3) Numbers Describe Hydrogen Atom � n,l,m l ,m s � µ "Spinning" charge gives rise to a dipole moment : s ∆ q Imagine (semi-clasically , in correctl y ! ) electron as s phere : charge q, radius r ∑ ∆ Total charg e uniformly dist ribut ed : q= q ; i i � ⇒ ⇒ µ a s electron spins, each "chargelet" rotates c urrent dipole mo ment s i ⎛ ⎞ ⎛ ⎞ � � � q ∑ q µ = µ = ⎜ ⎟ ⎜ ⎟ g S s s ⎝ 2 m ⎠ i ⎝ 2 m ⎠ i e e � � � ⇒ = µ In a Magnetic Field B magnetic energy due to spin U . B S s � � � Net Angular Momentum in H Atom J = L + S � ⎛ ⎞ � − � � � e µ = µ + µ = + Net Magnetic Moment of H atom: ⎜ ⎟ ( ) L gS 0 s ⎝ 2 ⎠ m e � � µ � Notice that the net dipole moment vector is not to J (There are many such "ubiqui tous" qu antum numbers for elementary particle but we won't teach you about them in this course !) Doubling of Energy Levels Due to Spin Quantum Number Under Intense B field, each {n , m l } energy level splits into two depending on spin up or down In Presence of External B field