4E : The Quantum Universe Lecture 29, May 24 Vivek Sharma - PowerPoint PPT Presentation

4E : The Quantum Universe Lecture 29, May 24 Vivek Sharma modphys@hepmail.ucsd.edu

2 Zeeman Effect Due to Presence of External B field Energy Degeneracy Is Broken

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 n gu l ar momentum) S is also Quantized � 3 + = � � |S| = ( 1) s s 4 1 = = ± � & S ; m m z s s 2 Spinning electron is an en titity defying any simple Spin angular momentum S classical description. . ... dd hi e n D.O. F also exhibits Space quantization 3

Stern-Gerlach Expt ⇒ An additional degree of freedom: “Spin” In an inhomogeneous field perpendicular to beam direction, magnetic moment µ experiences a force F z whose direction depends on Z component of the net magnetic moment & inhomogeneity dB/dz. The force deflects magnetic moment up or down. Space Quantization means expect (2 l + 1) deflections. For l =0, expect all electrons to arrive on the screen at the center (no deflection) � � � 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 ( ) moves particle up or down F m ∂ z B z � µ (in addition to torque causing magnetic m o me n t to precess about B field direction 4

An Additional degree of freedom: “Spin” for lack of a better name ! Expected Observed ! Hydrogen or l = 1 Silver ( l =0) This was a big surprise for Stern-Gerlach ! They had accidentally Discovered a new degree of freedom for electron : “spin” which Can take only two orientations for angular momentum S : up or down Leads to a new quantum number s=1/2. As a result: = � Z Component of Spin Angular Momentum S m z s = + � The magnitude | | ( 1) is FIXED, never changes ! S s s + = Allowed orientations are ( 1) 2 s s � � ⇒ µ S ; The corresponding Spin Magnetic Moment S 5

What Stern&Gerlach Saw in l=0 Silver Atoms B Field On ! B Field off Picture changes instantaneously as the external Field is switched off or on….discovery ! 6

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 current dipole moment s i � ⎛ ⎞ ⎛ ⎞ � � q q ∑ µ = µ = = ⎜ ⎟ ⎜ ⎟ ; 2 g S g s s ⎝ 2 ⎠ ⎝ 2 ⎠ m i m i e e � � � ⇒ = µ In a Magnetic Field B magnetic energy due to s pin 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 since g=2, net dipole moment vector is not to J (There are many such "ubiq t ui ous" quantum numbers for elementary particles!) 7

Magnetic Energy in an External B Field Contributions from Orbital and Spin motions. Defining Z axis to be the orientation of the B field: � � � e e { } { } µ = + = + U=- . B B L gS B m gm z z l s 2 2 m m Example: Zeeman spectrum in B=1T produced by Hyd rogen initially in n=2 state ⇒ = − = − 2 after taking spin into account: n=2 E 13.6 / 2 3.40 eV eV 2 = ± = m ω � Since m 0, 1, orbital contribution to Magnetic energy U l 0 l L ± ω = ± � This splits energy levels to E=E ; for m 1 sta tes 2 l L These states get further split in pairs due to spin magnetic moment 1 = ± ω � Since g=2 and m ; spin energy is again Zeeman energy= s L 2 As a result electrons in this shell have one of the following energi e s ± ω ± ω � � E E E 2 2 2 2 L L ∆ ± This leads to a variety of allowed ( (m +m )=0, 1) energy transitions with different l s intensities (Principal an d satellites) which a re vi sible when B field is large (ignore LS coupling See energy level diagram on next page 8

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 9

Spin-Orbit Interaction: L and S Momenta are Linked Magnetically B B B -e +Ze Equivalent to -e +Ze Electron revolving around Nucleus finds itself in a "internal" B field because in its frame of reference, the nucleus is orbiting around it � µ This B field, due to orbital motion , interacts with electron's spin dipole moment s � � � � = − µ ⇒ . Energy larger when S || B, smaller when anti-paralle l U B m � ⇒ ⇒ States with same ( , , ) but diff. spins e e n rg y level splitting/doubling due to S n l m l 10

Spin-Orbit Interaction: Angular Momenta are Linked Magnetically B B B -e +Ze Equivalent to -e +Ze � µ This B field, due to orbital motion , interacts with electron's spin dipole moment s � � � � = − µ ⇒ . Energy larger when S || B, smaller when anti-paralle l U B m � ⇒ ⇒ States with same ( , , ) but diff. spins e e n rg y level splitting/doubling due to S n l m l Under No External B Field There is Sodium Doublet Still a Splitting! & LS coupling 11

12 Vector Model For Total Angular Momentum J j 3/2 2P n

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 : = + = + + + � | | ( 1) w ith | |, -1, - 2......,... .,| - | J j j j l s l s l s l s = = � J with , -1, - 2.. ....., - m m j j j j z j j 1 = = Example: state with ( 1, ) l s 2 = ⇒ − 3/ 2 = -3/ 2, 1/ 2,1/ 2,3/ 2 j m j ⇒ ± j = 1/ 2 = 1/ 2 m j + In general takes (2 1) values m j j ⇒ Even # of or ientations 13

Addition of Orbital and Spin Angular Momenta When l =1, s=1/2; According to Uncertainty Principle, the vectors can lie anywhere on the cones, corresponding to definite values of their z component 14

Complete Description of Hydrogen Atom Full description of the Hydr oge n atom : { , , , } n l m m 2P l s n 3/2 ⇓ j LS Coupling ⇓ { , , , } n l j m How to describe multi-electrons atoms like He, Li etc? s corresponding How to order the Periodic table? 4 D .O F. . to • 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 15

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 Quantum Picture path, don’t know between which of the two scattering events actually happened 16