Physics 2D Lecture Slides Mar 14 Vivek Sharma UCSD Physics 2 - PowerPoint PPT Presentation

Physics 2D Lecture Slides Mar 14 Vivek Sharma UCSD Physics

Φ 2 d + Φ = 2 m 0.. .................(1) l φ 2 d Typo Fixed     2   + m 1 d d θ + − Θ θ = l sin l l ( 1) ( ) 0.....(2)       θ θ θ θ 2 sin d  d  sin         ∂ + 2 2 1 d   + 2m r ke 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 Ψ θ φ = Θ θ Φ φ = m ( , r , ) R ( ) . r ( ) . ( ) R Y (Spherical Harmonics) l nl lm m nl l l l

Cross Sectional View of Hydrogen Atom prob. densities in r, θ , φ Birth of Chemistry (Can make Fancy Molecules; Bonds � Overlapping electron clouds) Z Y

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 i ; A rea of current lo op A = r π 2 r π T 2 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 

“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 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 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 L θ θ dt Lsin dt Lsin 2 m 2 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 d W d ( Bcos ) : This work is stored as orientational Pot. Energy U dW= - dU � � µ = − µ θ = − µ Define Magnetic Potential Ene rgy U=- . B cos . B B z e � = � ω Change in Potential Energy U = 2m m B m l L l e Zeeman Effect: Hydrogen Atom In External B Field 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 + � ω E=E m Lifting The Energy Level Degeneracy: 0 L l

Electron has “Spin”: An additional degree of freedom Electron possesses additional "hidden" degree of freedom : " Spinning around itself" ! � s s + |S| = ( 1) � 1 s = Spin Quantum # (either Up or Down) 2 ⇒ How do we know this ? Stern-Gerlach expt � Spin Vector (a form of a S n gu l ar momentum) is also Quantized � 3 + = |S| = s ( s 1) � � 2 1 = = ± & S m � ; m z s s 2 Spinning electron is an entitity defying any simple classical de scriptio n. D on t try to visualize it (e.g ee HW probl s em 7)... hidd en D.O.F

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 2 m   i   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

Spin-Orbit Interaction: Angular Momenta are Linked Magnetically Electron revolving around Nucleus finds itself in a "internal" B field because in its frame of reference, B B B the nucleus is orbiting around it. -e +Ze Equivalent to -e +Ze � µ This B field, due to orbital motion , interacts with electron's spin dipole moment s � � � � = − µ ⇒ U . B Energy larger when S || B, smaller when anti-paralle l m � ⇒ ⇒ States with same ( , , n l m ) but diff. spins e e n rg y level splitting/doubling due to S l Under No External B Field There is Sodium Doublet Still a Splitting! & LS coupling

Vector Model For Total Angular Momentum J ⇒ Coupling of Orbital & Spin magnetic moments Neither Orbital nor Spin angular Momentum are conserved seperately! � � � J = L + S is conserv e d so long as there are no ex ternal torque s pr esen t Rules for Tota l Angular Momentum Quanti zat ion : = + = + + + | J | j ( j 1) w � ith j | l s |, l s -1, l s - 2......,... .,| l - | s = = J m � with m , -1, - 2.. j j j ....., - j z j j 1 = = Example: state with ( l 1, s ) 2 = ⇒ − j 3/ 2 m = -3/ 2, 1/ 2,1/ 2,3/ 2 j ⇒ ± j = 1/2 m = 1/ 2 j + In general m takes (2 j 1) values j ⇒ Even # of orientations Spectrographic Notation: Final Label 1 S n 1/2 Complete Description of Hydrogen Atom 2 P j 3/2

Complete Description of Hydrogen Atom Full description of the Hydr oge n atom : 1 S n 1/2 { , , n l m m , } l s 2 P j ⇓ 3/2 LS Coupling ⇓ { , , , n l j m } How to describe multi-electrons atoms like He, Li etc? s corresponding How to order the Periodic table? to 4 D .O F. . • Four guiding principles: • Indistinguishable particle & Pauli Exclusion Principle •Independent particle model (ignore inter-electron repulsion) •Minimum Energy Principle for atom •Hund’s “rule” for order of filling vacant orbitals in an atom

Multi-Electron Atoms : >1 electron in orbit around Nucleus ψ θ φ Θ θ Φ φ ≡ In Hydrogen Atom (r, , )=R(r ). ( ). ( ) { , , , n l j m } j e - In n-electron atom, to simplify, ignore electron-electron inte rac tions complete wavefunction, in "independent"part icle ap prox" : ψ ψ ψ ψ ψ (1,2, 3,..n)= (1). (2). (3)... ( ) ??? n e - → Complication Electrons are identical particles, labeling meanin gless! Question: How many electrons can have same set of quan t um #s? Answer: No two elec trons in an atom can have SAME set of quantum#s (if not, all electrons would occupy 1s state (least energy). .. no struct ure!! Example of Indistinguishability: elec tron-ele ctron scatte ring Small angle scatter large angle scatter If we cant follow electron path, don’t know between which of the two scattering Event actually happened Quantum Picture