2. 1 Optically polarized atoms Marcis Auzinsh, University of Latvia Dmitry Budker, UC Berkeley and LBNL Simon M. Rochester, UC Berkeley

3. 2 A brief summary of atomic structure A brief summary of atomic structure Begin with hydrogen atom Begin with hydrogen atom The Schrödinger Eqn: The Schrödinger Eqn: In this approximation (ignoring spin and relativity): In this approximation (ignoring spin and relativity): Chapter 2: Atomic states Image from Wikipedia Principal quant. Number n=1,2,3,…

4. 3 Could have guessed me 4 /  2 from dimensions Could have guessed me 4 /  2 from dimensions me 4 /  2 = 1 Hartree me 4 /  2 = 1 Hartree me 4 /2  2 = 1 Rydberg me 4 /2  2 = 1 Rydberg E does not depend on l or m  degeneracy E does not depend on l or m  degeneracy i.e. different wavefunction have same E We will see that the degeneracy is n 2 We will see that the degeneracy is n 2

5. 4 Angular momentum of the electron in the hydrogen atom Orbital-angular-momentum quantum number l = 0,1,2,… Orbital-angular-momentum quantum number l = 0,1,2,… This can be obtained, e.g., from the Schrödinger Eqn., or straight from QM commutation relations This can be obtained, e.g., from the Schrödinger Eqn., or straight from QM commutation relations The Bohr model: classical orbits quantized by requiring angular momentum to be integer multiple of  The Bohr model: classical orbits quantized by requiring angular momentum to be integer multiple of  There is kinetic energy associated with orbital motion  an upper bound on l for a given value of E n There is kinetic energy associated with orbital motion  an upper bound on l for a given value of E n Turns out: l = 0,1,2, …, n-1 Turns out: l = 0,1,2, …, n-1

6. 5 Angular momentum of the electron in the hydrogen atom (cont’d) In classical physics, to fully specify orbital angular momentum, one needs two more parameters (e.g., to angles) in addition to the magnitude In classical physics, to fully specify orbital angular momentum, one needs two more parameters (e.g., to angles) in addition to the magnitude In QM, if we know projection on one axis (quantization axis), projections on other two axes are uncertain In QM, if we know projection on one axis (quantization axis), projections on other two axes are uncertain Choosing z as quantization axis: Choosing z as quantization axis: Note: this is reasonable as we expect projection magnitude not to exceed Note: this is reasonable as we expect projection magnitude not to exceed

7. 6 Angular momentum of the electron in the hydrogen atom (cont’d) m – magnetic quantum number because B-field can be used to define quantization axis m – magnetic quantum number because B-field can be used to define quantization axis Can also define the axis with E (static or oscillating), other fields (e.g., gravitational), or nothing Can also define the axis with E (static or oscillating), other fields (e.g., gravitational), or nothing Let’s count states: Let’s count states: m = -l,…,l i. e. 2l+1 states m = -l,…,l i. e. 2l+1 states l = 0,…,n-1  l = 0,…,n-1  As advertised !

8. 7 Angular momentum of the electron in the hydrogen atom (cont’d) Degeneracy w.r.t. m expected from isotropy of space Degeneracy w.r.t. m expected from isotropy of space Degeneracy w.r.t. l, in contrast, is a special feature of 1/r (Coulomb) potential Degeneracy w.r.t. l, in contrast, is a special feature of 1/r (Coulomb) potential

9. 8 Angular momentum of the electron in the hydrogen atom (cont’d) How can one understand restrictions that QM puts on measurements of angular-momentum components ? How can one understand restrictions that QM puts on measurements of angular-momentum components ? The basic QM uncertainty relation (*) leads to (and permutations) The basic QM uncertainty relation (*) leads to (and permutations) We can also write a generalized uncertainty relation We can also write a generalized uncertainty relation between l z and φ (azimuthal angle of the e): This is a bit more complex than (*) because φ is cyclic This is a bit more complex than (*) because φ is cyclic With definite l z, φ is completely uncertain… With definite l z, φ is completely uncertain…

10. 9 Wavefunctions of the H atom A specific wavefunction is labeled with n l m : A specific wavefunction is labeled with n l m : In polar coordinates : In polar coordinates : i.e. separation of radial and angular parts Further separation: Further separation: Spherical functions (Harmonics)

11. 10 Wavefunctions of the H atom (cont’d) Separation into radial and angular part is possible for any central potential ! Separation into radial and angular part is possible for any central potential ! Things get nontrivial for multielectron atoms Things get nontrivial for multielectron atoms Legendre Polynomials

12. 11 Electron spin and fine structure Experiment: electron has intrinsic angular momentum -- spin (quantum number s) Experiment: electron has intrinsic angular momentum -- spin (quantum number s) It is tempting to think of the spin classically as a spinning object. This might be useful, but to a point. It is tempting to think of the spin classically as a spinning object. This might be useful, but to a point. Experiment: electron is pointlike down to ~ 10 -18 cm

13. 12 Electron spin and fine structure (cont’d) Another issue for classical picture: it takes a 4π rotation to bring a half-integer spin to its original state. Amazingly, this does happen in classical world: Another issue for classical picture: it takes a 4π rotation to bring a half-integer spin to its original state. Amazingly, this does happen in classical world: from Feynman's 1986 Dirac Memorial Lecture (Elementary Particles and the Laws of Physics, CUP 1987)

14. 13 Electron spin and fine structure (cont’d) Another amusing classical picture: spin angular momentum comes from the electromagnetic field of the electron: Another amusing classical picture: spin angular momentum comes from the electromagnetic field of the electron: This leads to electron size This leads to electron size Experiment: electron is pointlike down to ~ 10 -18 cm

15. 14 Electron spin and fine structure (cont’d) s=1/2  s=1/2  “Spin up” and “down” should be used with understanding that the length (modulus) of the spin vector is >  /2 ! “Spin up” and “down” should be used with understanding that the length (modulus) of the spin vector is >  /2 !

16. 15 Electron spin and fine structure (cont’d) Both orbital angular momentum and spin have associated magnetic moments μ l and μ s Both orbital angular momentum and spin have associated magnetic moments μ l and μ s Classical estimate of μ l : current loop Classical estimate of μ l : current loop For orbit of radius r, speed p/m, revolution rate is For orbit of radius r, speed p/m, revolution rate is Gyromagnetic ratio

17. 16 Electron spin and fine structure (cont’d) In analogy, there is also spin magnetic moment : In analogy, there is also spin magnetic moment : Bohr magneton

18. 17 Electron spin and fine structure (cont’d) The factor  2 is important ! The factor  2 is important ! Dirac equation for spin-1/2 predicts exactly 2 Dirac equation for spin-1/2 predicts exactly 2 QED predicts deviations from 2 due to vacuum fluctuations of the E/M field QED predicts deviations from 2 due to vacuum fluctuations of the E/M field One of the most precisely measured physical constants:  2=2  1.00115965218085(76) One of the most precisely measured physical constants:  2=2  1.00115965218085(76) Prof. G. Gabrielse, Harvard (0.8 parts per trillion) New Measurement of the Electron Magnetic Moment Using a One-Electron Quantum CyclotronUsing a One-Electron Quantum Cyclotron, B. Odom, D. Hanneke, B. D'Urso, and G. Gabrielse, Phys. Rev. Lett. 97, 030801 (2006)

19. 18 Electron spin and fine structure (cont’d)

20. 19 Electron spin and fine structure (cont’d) When both l and s are present, these are not conserved separately When both l and s are present, these are not conserved separately This is like planetary spin and orbital motion This is like planetary spin and orbital motion On a short time scale, conservation of individual angular momenta can be a good approximation On a short time scale, conservation of individual angular momenta can be a good approximation l and s are coupled via spin-orbit interaction: interaction of the motional magnetic field in the electron’s frame with μ s l and s are coupled via spin-orbit interaction: interaction of the motional magnetic field in the electron’s frame with μ s l and s, i.e., on Energy shift depends on relative orientation of l and s, i.e., on

21. 20 Electron spin and fine structure (cont’d) QM parlance: states with fixed m l and m s are no longer eigenstates States with fixed j, m j are eigenstates Total angular momentum is a constant of motion of an isolated system |m j |  j If we add l and s, j ≥ |l-s| ; j  l+s s=1/2  j = l  ½ for l > 0 or j = ½ for l = 0

22. 21 Electron spin and fine structure (cont’d) Spin-orbit interaction is a relativistic effect Including rel. effects : Correction to the Bohr formula   2 The energy now depends on n and j

23. 22 Electron spin and fine structure (cont’d)   1/137  relativistic corrections are small ~ 10 -5 Ry  E  0.366 cm -1 or 10.9 GHz for 2P 3/2, 2P 1/2  E  0.108 cm -1 or 3.24 GHz for 3P 3/2, 3P 1/2

24. 23 Electron spin and fine structure (cont’d) The Dirac formula : predicts that states of same n and j, but different l remain degenerate In reality, this degeneracy is also lifted by QED effects (Lamb shift) For 2S 1/2, 2P 1/2 :  E  0.035 cm -1 or 1057 MHz

25. 24 Vector model of the atom Some people really need pictures… Recall: for a state with given j, j z We can draw all of this as (j=3/2) m j = 3/2m j = 1/2

26. 25 Vector model of the atom (cont’d) These pictures are nice, but NOT problem-free Consider maximum-projection state m j = j Q: What is the maximal value of j x or j y that can be measured ? A: that might be inferred from the picture is wrong… m j = 3/2

27. 26 Vector model of the atom (cont’d) So how do we draw angular momenta and coupling ? Maybe as a vector of expectation values, e.g., ? Simple Has well defined QM meaning BUT Boring Non-illuminating Or stick with the cones ? Complicated Still wrong…

28. 27 Vector model of the atom (cont’d) A compromise : j is stationary l, s precess around j What is the precession frequency? Stationary state – quantum numbers do not change Say we prepare a state with fixed quantum numbers |l,m l,s,m s  This is NOT an eigenstate but a coherent superposition of eigenstates, each evolving as Precession  Quantum Beats  l, s precess around j with freq. = fine-structure splitting

29. 28 Multielectron atoms Multiparticle Schrödinger Eqn. – no analytical soltn. Many approximate methods We will be interested in classification of states and various angular momenta needed to describe them SE: This is NOT the simple 1/r Coulomb potential  Energies depend on orbital ang. momenta

30. 29 Gross structure, LS coupling Individual electron “sees” nucleus and other e’s Approximate total potential as central: φ(r) Can consider a Schrödinger Eqn for each e Central potential  separation of angular and radial parts; l i (and s i ) are well defined ! Radial SE with a given l i  set of bound states Label these with principal quantum number n i = l i +1, l i +2,… (in analogy with Hydrogen) n i - l i -1 Oscillation Theorem: # of zeros of the radial wavefunction is n i - l i -1

31. 30 Gross structure, LS coupling (cont’d) Set of, for all electrons  electron configuration Set of n i, l i for all electrons  electron configuration Different configuration generally have different energies Different configuration generally have different energies In this approximation, energy of a configuration is just sum of In this approximation, energy of a configuration is just sum of E i No reference to projections of l i or to spins  degeneracy If we go beyond the central-field approximation some of the degeneracies will be lifted Also spin-orbit (l  s) interaction lifts some degeneracies In general, both effects need to be considered, but the former is more important in light atoms

32. 31 Gross structure, LS coupling (cont’d) Beyond central-field approximation (cfa) Non-centrosymmetric part of electron repulsion (  1/r ij ) = residual Coulomb interaction (RCI) Non-centrosymmetric part of electron repulsion (  1/r ij ) = residual Coulomb interaction (RCI) The energy now depends on how l i and s i combine The energy now depends on how l i and s i combine Neglecting Neglecting (l  s) interaction  LS coupling or Russell-Saunders coupling This terminology is potentially confusing….. This terminology is potentially confusing….. ….. but well motivated ! ….. but well motivated ! Within cfa, individual orbital angular momenta are conserved; RCI mixes states with different projections of l i Within cfa, individual orbital angular momenta are conserved; RCI mixes states with different projections of l i Classically, RCI causes precession of the orbital planes, so the direction of the orbital angular momentum changes Classically, RCI causes precession of the orbital planes, so the direction of the orbital angular momentum changes

33. 32 Gross structure, LS coupling (cont’d) Beyond central-field approximation (cfa) Projections of l i are not conserved, but the total orbital momentum L is, along with its projection ! Projections of l i are not conserved, but the total orbital momentum L is, along with its projection ! This is because l i form sort of an isolated system This is because l i form sort of an isolated system So far, we have been ignoring spins So far, we have been ignoring spins One might think that since we have neglected One might think that since we have neglected (l  s) interaction, energies of states do not depend on spins WRONG !

34. 33 Gross structure, LS coupling (cont’d) The role of the spins Not all configurations are possible. For example, U has 92 electrons. The simplest configuration would be 1s 92 Not all configurations are possible. For example, U has 92 electrons. The simplest configuration would be 1s 92 Luckily, such boring configuration is impossible. Why ? Luckily, such boring configuration is impossible. Why ? e’s are fermions  Pauli exclusion principle: no two e’s can have the same set of quantum numbers e’s are fermions  Pauli exclusion principle: no two e’s can have the same set of quantum numbers This determines the richness of the periodic system This determines the richness of the periodic system Note: some people are looking for rare violations of Pauli principle and Bose-Einstein statistics…  new physics Note: some people are looking for rare violations of Pauli principle and Bose-Einstein statistics…  new physics So how does spin affect energies (of allowed configs) ? So how does spin affect energies (of allowed configs) ?   Exchange Interaction

35. 34 Gross structure, LS coupling (cont’d) Exchange Interaction The value of the total spin S affects the symmetry of the spin wavefunction The value of the total spin S affects the symmetry of the spin wavefunction Since overall spatial wavefunction is affected Since overall ψ has to be antisymmetric  symmetry of spatial wavefunction is affected  this affects Coulomb repulsion between electrons  effect on energies Thus, energies depend on L and S. Term: 2S+1 L 2S+1 is called multiplicity Example: He(g.s.): 1s 2 1 S

36. 35 Gross structure, LS coupling (cont’d) Within present approximation, energies do not depend on (individually conserved) projections of L and S Within present approximation, energies do not depend on (individually conserved) projections of L and S This degeneracy is lifted by spin-orbit interaction (also spin- spin and spin-other orbit) This degeneracy is lifted by spin-orbit interaction (also spin- spin and spin-other orbit) This leads to energy splitting within a term according to the value of total angular momentum This leads to energy splitting within a term according to the value of total angular momentum J (fine structure) If this splitting is larger than the residual Coulomb interaction (heavy atoms)  breakdown of LS coupling

37. 36 Example: a two-electron atom (He) Example: a two-electron atom (He) Quantum numbers: Quantum numbers: J, m J “good” no restrictions for isolated atoms l 1, l 2, L, S “good” in LS coupling m l, m s, m L, m S “not good”=superpositions “Precession” rate hierarchy: “Precession” rate hierarchy: l 1, l 2 around L and s 1, s 2 around S: residual Coulomb interaction (term splitting -- fast) L and S around J (fine-structure splitting -- slow) Vector Model

38. 37 jj and intermediate coupling schemes Sometimes (for example, in heavy atoms), spin-orbit interaction > residual Coulomb  LS coupling Sometimes (for example, in heavy atoms), spin-orbit interaction > residual Coulomb  LS coupling To find alternative, step back to central-field approximation To find alternative, step back to central-field approximation Once again, we have energies that only depend on electronic configuration; lift approximations one at a time Since spin-orbit is larger, include it first 

39. 38 jj and intermediate coupling schemes (cont’d) In practice, atomic states often do not fully conform to LS or jj scheme; sometimes there are different schemes for different states in the same atom  intermediate coupling In practice, atomic states often do not fully conform to LS or jj scheme; sometimes there are different schemes for different states in the same atom  intermediate coupling Coupling scheme has important consequences for selection rules for atomic transitions, for example Coupling scheme has important consequences for selection rules for atomic transitions, for example L and S rules: approximate; only hold within LS coupling L and S rules: approximate; only hold within LS coupling J, m J rules: strict; hold for any coupling scheme J, m J rules: strict; hold for any coupling scheme

40. 39 Notation of states in multi-electron atoms Spectroscopic notation Configuration (list of n i and l i ) n i – integers l i – code letters 1s 2 2s 2 2p 6 3s = [Ne]3s Numbers of electrons with same n and l – superscript, for example: Na (g.s.): 1s 2 2s 2 2p 6 3s = [Ne]3s Term 2S+1 L  State 2S+1 L J Term 2S+1 L  State 2S+1 L J 2S+1 = multiplicity (another inaccurate historism) 2S+1 = multiplicity (another inaccurate historism) Complete designation of a state [e.g., Ba (g.s.)]: [Xe]6s 2 1 S 0 Complete designation of a state [e.g., Ba (g.s.)]: [Xe]6s 2 1 S 0

41. 40

42. 41 Fine structure in multi-electron atoms LS states with different J are split by spin-orbit interaction LS states with different J are split by spin-orbit interaction Example: 2 P 1/2 - 2 P 3/2 splitting in the alkalis Example: 2 P 1/2 - 2 P 3/2 splitting in the alkalis Splitting  Z 2 (approx.) Splitting  Z 2 (approx.) Splitting  with n Splitting  with n

43. 42 Hyperfine structure of atomic states Nuclear spin I  magnetic moment Nuclear spin I  magnetic moment Nuclear magneton Nuclear magneton Total angular momentum: Total angular momentum:

44. 43 Hyperfine structure of atomic states (cont’d) Hyperfine-structure splitting results from interaction of the nuclear moments with fields and Hyperfine-structure splitting results from interaction of the nuclear moments with fields and gradients produced by e’s  Lowest terms: Lowest terms: M1 E2 E2 term: B  0 only for I,J>1/2 E2 term: B  0 only for I,J>1/2

45. 44 Hyperfine structure of atomic states A nucleus can only support multipoles of rank κ  2I A nucleus can only support multipoles of rank κ  2I E1, M2, …. moments are forbidden by P and T E1, M2, …. moments are forbidden by P and T B  0 only for I,J>1/2 Example of hfs splitting (not to scale) Example of hfs splitting (not to scale) 85 Rb (I=5/2) 87 Rb (I=3/2)