4E : The Quantum Universe. Lecture 28, May 19 Vivek Sharma

Transcription

1 4E : The Quantum Univere Lecture 28, May 19 Vivek Sharma modphy@hepmail.ucd.edu

2 The Magnetim of an Orbiting Electron Preceing electron Current in loop Magnetic Dipole moment µ Electron in motion around nucleu circulating charge curent e e ep 2 i = = = ; Area of current loop A= π r T 2π r 2π mr v -e -e -e Magnetic Moment µ =ia= rp ; µ = r p = L 2m 2m 2m Like the L, magnetic moment µ alo precee about "z" axi -e -e z component, µ z = L = m µ m quantized! 2 m z = = 2 m l B l i

3 Bar Magnet Model of Magnetic Moment In external B field, magnet experience a torque which tend to align it with the field direction If the magnet i pinning, torque caue magnet to prece around the ext. B field with a contant frequency: Larmor frequency

4 Lifting Degeneracy : Magnetic Moment in External B Field Apply an External B field on a Hydrogen atom (viewed a a dipole) Conider B Z axi (could be any other direction too) The dipole moment of the Hydrogen atom (due to electron orbit) experience a Torque τ = µ B which doe work to align µ B but thi can not be (ame Uncertainty principle argument) So, Intead, µ precee (dance) around B... like a pinning top The Azimuthal angle φ change with time : calculate frequency Look at Geometry: proection along x-y plane : dl = Lin θ.dφ dl q dφ = ; Change in Ang Mom. dl = τ dt = LBinθ dt Linθ 2m dφ 1 dl 1 q ωl = = = LB inθ = dt Linθ dt Linθ 2m L qb 2 ω depend on B, the applied external magnetic field m e Larmor Freq

5 Lifting Degeneracy : Magnetic Moment in External B Field WORK done to reorient µ againt B field: dw= τd θ =-µ Binθdθ dw = d( µ Bco θ) : Thi work i tored a orientational Pot. Energy U dw= -du Define Magnetic Potential Energy U=- µ. B = µ co θ. B = µ B e Change in Potential Energy U = 2m e mb l = ω m Zeeman Effect in Hydrogen Atom In preence of External B Field, Total energy of H atom change to E=E 0 + ω m L So the Ext. B field can break the E degeneracy "organically" inherent in the H atom. The Energy now depend not ut on n but alo ml l L l z

6 Zeeman Effect Due to Preence of External B field Energy Degeneracy I Broken

7 Electron ha Spin : An additional degree of freedom Electron poee additional "hidden" degree of freedom : " Spinning around itelf"! Spin Quantum # 1 = 2 (either Up or Down) How do we know thi? Stern-Gerlach expt Spin Vector S (a form of angular momentum) i alo Quantized S = ( + 1) = & S = m ; m =± z 1 2 Spinning electron i an entitity defying any imple claical decription. Don t try to viualize it (e.g ee HW problem 7)...hidden D.O.F 3 2 S = + ( 1)

8 B Stern-Gerlach Expt An additional degree of freedom: Spin for lack of a better name µ in inhomogenou B field, experience force F F= - U = ( µ.b) B B B When gradient only along z, 0; = = 0 z x y B Fz = mµ B( ) move particle up or down z (in addition to torque cauing Mag. moment to prece about B field direction In an inhomogeneou field, magnetic moment µ experience a force F z whoe direction depend on component of the net magnetic moment & inhomogeneity db/dz. Quantization mean expect (2l+1) deflection. For l=0, expect all electron to arrive on the creen at the center (no deflection) Silver Hydrogen (l=0)!

9 Four (not 3) Number Decribe Hydrogen Atom n,l,m l,m "Spinning" charge give rie to a dipole moment : µ Imagine (emi-claically, incorrectly! ) electron a phere: charge q, radiu r Total charge uniformly ditributed: q= q i; i a electron pin, each "chargelet" rotate current dipole moment µ q q µ = µ = g i S 2me i 2me In a Magnetic Field B magnetic energy due to pin Net Angular Momentum in H Atom J = L + S U = µ. B e Net Magnetic Moment of H atom: µ = µ 0 + µ = ( L+ gs) 2me Notice that the net dipole moment vector µ i not to J S i q (There are many uch "ubiquitou" quantum number for elementary particle!)

10 Doubling of Energy Level Due to Spin Quantum Number Under Intene B field, each {n, m l } energy level plit into two depending on pin up or down In Preence of External B field

11 Spin-Orbit Interaction: Angular Momenta are Linked Magnetically Electron revolving around Nucleu find itelf in a "internal" B field becaue in it ref. frame the -enucleu i orbiting around it B B B +Ze Equivalent to -e +Ze Thi B field, due to orbital motion, interact with electron' pin dipole moment µ Um = µ. B Energy larger when S B, maller when anti-parallel State with ame ( nlm,, ) but diff. pin enrg e y level plitting/doubling due to S l Under No External B Field There i Still a Splitting! Sodium Doublet & LS coupling

12 Vector Model For Total Angular Momentum J Coupling of Orbital & Spin magnetic moment Neither Orbital nor Spin angular Momentum are conerved eperately! J = L + S i conerved o long a there are no external torque preent Rule for Total Angular Momentum Quantization : J = ( + 1) with = l+, l+ -1, l ,...., l J z = m with m 1 Example: tate with ( l = 1, = ) 2 = 3/2 m = -3/2, 1/2,1/2,3/2 = 1/2 m = ± 1/2 In general m take (2 + 1) value Even # of orientation =, -1, , - - Spectrographic Notation: Final Label n 1 S 2 P 1/2 3/2 Complete Decription of Hydrogen Atom

13 Complete Decription of Hydrogen Atom Full decription of the Hydrogen atom: { nlm,,, m} LS Coupling {,, nl m, } l correponding to 4 D.O. F. n 1 S 2 P How to decribe multi-electron atom like He, Li etc? How to order the Periodic table? Four guiding principle: Inditinguihable particle & Pauli Excluion Principle Independent particle model (ignore inter-electron repulion) Minimum Energy Principle for atom Hund rule for order of filling vacant orbital in an atom 1/2 3/2