Spectra of Atoms and Molecules, 3 rd Ed., Peter F. Bernath, Oxford University Press, chapter 5. Engel/Reid, chapter 18.3 / 18.4

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1 Last class Today Atomic spectroscopy (part I) Absorption spectroscopy Bohr model QM of H atom (review) Atomic spectroscopy (part II)-skipped Visualization of wave functions Atomic spectroscopy (part III) Operator / Eigenvalue equations Angular momentum (some details) Orbital angular momentum Spin Spin-Orbit coupling Zeeman effect, 017 Uwe Burghaus, Fargo, ND, USA

2 Spectra of Atoms and Molecules, 3 rd Ed., Peter F. Bernath, Oxford University Press, chapter 5 Engel/Reid, chapter 18.3 / 18.4 Physics of Atoms and Molecules, B.H. Bransden, C.J. Joachain, Wiley Press chapter 5 Haken, Wolf chapter 1-14 not very good, too formal & not complete good/didactic, but does not go far enough, undergraduate book Similar to Bernath but more applied and less detailed Well written didactic outline, QM description in separate chapters

3 Sir William Rowan Hamilton Born: 1805 in Dublin, Ireland Died:

4 OPERATORS observable operator name symbol symbol operation position x multiply by x X position momentum momentum kinetic energy kinetic energy p x r p K x K p i x x p i K R K x V ( x) multiply by m x m potential energy V(x) Multiply with V(x) r angular momentum L x =yp z -zp y L x i( y z ) z y

5 Schrödinger equation with operators PChem Quantum mechanics Rewriting the Schrödinger equation with operators. H m +V i Ψ t = H Ψ Time dependent Schrödinger eq. H ψ = Eψ Time independent Schrödinger eq. Eigenvalues

6 Eigenvalue equations PChem Quantum mechanics ˆBf = cf B ˆ : Operator f: function c: constant f is eigenfunction of the operator B with eigenvalue c H ψ = Eψ Time independent Schrödinger equation as an eigenvalue equation. Eigenvalues

7 Postulate PChem Quantum mechanics The possible results in a measurement of a physical/chemical quantity are the Eigenvalues of the corresponding observable. (SCHAUM s outlines) If the system s state function Ψ happens to be an eigenfunction of eigenvalue c, i.e., if M Ψ = cψ with is obeyed, then a measurement of M is certain to give the value c as the result (Levine p. 64). M Eigenvalues

8 Why is the operator / eigenvalue formalism useful? PChem Quantum mechanics Math: An operator is a rule for a transformation of a function. QM: An operator defines a measurement/experiment. Apply an operator to a state of a system (the wave function). The answer/result of the system for this experiment is the eigenvalue. H ψ = Eψ Eigenvalues

9 Eigenvalue equations H ψ = Eψ ψ = ψ rule function the result function doing the experiment

10 Today Operator / Eigenvalue equation Atomic spectroscopy (part III) Angular momentum (details) Orbital angular momentum Spin Spin-Orbit coupling Zeeman effect, 017 Uwe Burghaus, Fargo, ND, USA

11 Why do we consider this? The angular momentum (orbital & spin) is closely related to the H atom description. Orbital angular momentum Spin angular momentum The quantum mechanics of atoms is the simplest meaningful application of quantum mechanics strategies example for spectroscopy

12 Angular momentum operators Linear momentum operators Angular momentum operators L = rxp L = rx i Cartesian coordinates Polar coordinates Spectra of Atoms and Molecules, 3 rd Ed., Peter F. Bernath, Oxford University Press, chapter 5

13 Eigenvalue equations PChem Quantum mechanics ˆBf = cf B ˆ : Operator f: function C: constant f is eigenfunction of the operator B with eigenvalue c Eigenvalues

14 Eigenvalue equations ψ = ψ rule function the result function doing the experiment

15 Eigenvalue equations H ψ = Eψ ψ = ψ rule function the result function doing the experiment Orbital angular momentum L ψ L ψ = l( l + 1) ψ n, l, m n, l, m = mψ z n, l, m n, l, m Spin angular momentum ˆ ss ( 1) S α = + α S z α = 1 α

16 Orbital angular momentum [ L, L ] x y [ L, L ] = z 0 0 L = rxp L = rx i L ψ n, l, m n, l, m L ψ = l( l + 1) ψ L = l( l +1) = mψ z n, l, m n, l, m L z = m PChem Quantum mechanics m: magnetic quantum number L z : z - component of angular momentum p: linear momentum L: angular momentum r: coordinates : gradient vector l: angular momentum quantum number ψ n, l, m : H - atom wave functions

17 Angular momentum vector model x z y z [ L, L ] [ L, L ] = 0 0 m =1 m = 0 m = 1 L = rxp L = rx i L ψ n, l, m n, l, m L ψ = l( l + 1) ψ L = l( l +1) = mψ z n, l, m n, l, m L z = m PChem Quantum mechanics m: magnetic quantum number L z : z - component of angular momentum p: linear momentum L: angular momentum r: coordinates : gradient vector l: angular momentum quantum number ψ n, l, m : H - atom wave functions

18 Analogy: H atom & rigid rotor Rigid rotor J, M J ψ rot H atom l, m Y l, m l l Y l m, = ψ l rot The eigenfunctions are the spherical harmonics.

19 Compare Bohr s model and quantum mechanics Angular momentum of the H atom s ground state Bohr theory L = mvr nh h = = π π Quantum mechanics 1s state l = 0; n = 1 L = l( l + 1) = 0

20 Matrix representation of angular momentum

21 North electron South electron South North Electron spin, classical analog

22 Paul Adrien Maurice Dirac ( ) was a British theoretical physicist. Dirac made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics. He held the Lucasian Chair of Mathematics at the University of Cambridge and spent the last fourteen years of his life at Florida State University. Among other discoveries, he formulated the Dirac equation, which describes the behaviour of fermions which led to the prediction of the existence of antimatter. Dirac shared the Nobel Prize in physics 1933 with Erwin Schrödinger, "for the discovery of new productive forms of atomic theory."

23 Stern-Gerlach experiment Ag: closed-shell + 5s electron No angular momentum Effect must be related to 5s electron l=0 for s-electrons It cannot be the angular momentum Electron spin

24 Electron spin vs. orbital angular momentum Orbital momentum L = l( l +1) l = 0,1,, (n-1) S = s( s + 1) = Electron Spin 3 4 s = 1/ L z = m l m l = +l,,0, -l S z = m s m s = +1/, -1/

25 Electron spin vs. orbital angular momentum Orbital momentum L = l( l +1) l = 0,1,, (n-1) S = s( s + 1) = Electron Spin 3 4 s = 1/ L z = m l m l = +l,,0, -l S z = m s m s = +1/, -1/ z vector model z m =1 m s = m = m = 1 ms = 1 example for l = 1

26 Quantum numbers for one electron system (H atom) n = 1,, 3, l = 0, 1,, 3,, n-1 principal quantum number m l = 0, ±1, ±, ± 3,, ± l angular momentum quantum number magnetic quantum number m s = +1/, -1/ spin orientation quantum number

27 Wave functions including the electron spin ψ spatial wave n, l, m function α spin up = + β spin ψ ψ down n, l, m n, l, m α β m s m s = spin orbitals 1 1 L ψ = l( l + 1) ψ n, l, m n, l, m ˆ ss ( 1) S α = + α S 3 4 α = α S 3 4 β = β L ψ = mψ z n, l, m n, l, m S z α = 1 α S β = β ˆz 1

28 Matrices & vectors nomenclature for the spin Spin eigenfunctions 1 0 α = and β = 0 1 Spin operators i 1 0 sˆ, ˆ, ˆ x = sy sz 1 0 = = i 0 0 1

29 P1.7) In this problem we represent the spin eigenfunctions and operators as vectors and matrices. a) The spin eigenfunctions are often represented as the column vectors Show that α and β are orthogonal using this representation. 1 0 α = and β = 0 1 Using the rules of matrix multiplication, 1 αβ = ( 0 1 ) = = 0. 0 Therefore α and β are orthogonal.

30 b) If the spin angular momentum operators are represented by the matrices i 1 0 sˆx =, sˆy, sˆz 1 0 = = i 0 0 1, show that the commutation rule sˆ, ˆ ˆ x sy = i sz holds. ss ˆˆ ˆˆ i i ss = 1 0 i 0 i x y y x i 0 i = i i i sˆ 0 i = = = 0 i z

31 α and β d) Show that are eigenfunctions of What are the eigenvalues? sˆ z and sˆ. sˆ α and sˆzβ = = = = z S z α = 1 α S β = β ˆz 1

32 Need a break?

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34 Is that all? Any other effects? What about Electric fields Magnetic fields Etc.

35 image from Haken, Wolf, p. 18 Experimental results related to angular & spin momentum Einstein-de Haas effect Stern-Gerlach experiment Schematics for alkali atoms doublets (class #4) E = E(n)???

36 single electron system - coupling of l and s j = l + s single electron case! j = l + s p-electron, example Orbital angular momentum: l=1 Spin angular moment: s=1/ j=3/ and j=1/ s=1/ j=3/ l=1 one electron system 3 1 z 3 4 m j = 3 m j = 1 j z m j = m j = j, j 1,..., j total angular momentum m j = 1 m j = 3

37 Term symbols with j n s+1 l j n S+1 L j Single electron system More than one electron p-electron, example p 1/ n= principal quantum number l=1 orbital angular momentum s=1/ j=1/ =multiplicity, m j = ½, -½

38 Spectroscopy nomenclature: term symbols for atoms PChem Quantum mechanics multiplicity (number of possible different wave functions) n S+1 L J principal quantum number (defines the energy) angular momentum L=0 s state L=1 p state L= d state L+S L-S coupling J-J coupling

39 coupling of l and s multi electron systems J = L + S Usually small symbols: single electron system large symbols: multi electron systems L: orbital angular momentum S: spin angular momentum total angular momentum J-J coupling j 1 = l 1 + s 1 j = l + s J = j 1 + j + L-S coupling L = l 1 + l + S = s 1 + s + J = L + S m j = j, j 1,..., j

40 SUMMARY Spin orbit coupling / fine structure Rather complex topic Mathematically involved when done in detail Experimental result Spectra of alkali atoms do show peak splitting, doublets Systems with one valence electron Exception s-states, these do not show doublets Plausible explanation (quasi classical) The electron (with orbital angular moment l) loping the nucleus generate a current That current generates a magnetic field, B l l The magnetic moment of the spin, µ s, interacts with that field B l Different orientations of µ s generate different interaction energies, V ls = - µ s B l LS For a one electron system: duplet peaks One can also say that the magnetic moments interact; µ l and µ s interact p-electron, example Orbital angular momentum: l=1 Spin angular momentum: s=1/ j=3/ and j=1/ 3 1 z 3 4 m j = 3 m j = 1 All states with the same j have the same energy m j = 1 m j = 3

41 image from Haken, Wolf, p. 18 Spin orbit coupling / fine structure In-class homework 10) Why do s-states not show a doublet? Schematics for alkali atoms (treaded at oneelectron states including spin-orbit coupling) Dashed lines: H atom

42 image from Haken, Wolf, p. 18 Spin orbit coupling / fine structure In-class homework 10) Why do s-states not show a doublet? Schematics for alkali atoms (treaded at oneelectron states including spin-orbit coupling) Dashed lines: H atom RESULT B l l=0 No magnetic field (magnetic moment) generated by looping electron RESULT

43 image from Haken, Wolf, p. 18 Selection rules doublets triplets Schematics for alkali atoms (treaded at oneelectron states including spin-orbit coupling) Dashed lines: H atom Selection rules l = +-1 j = +-1, 0 J=0 j=0 (not allowed)

44 A few more equations Using basically classical physics one can write e Magnetic dipole moment of the orbital angular moment µ l = l = glµ B m e l Magnetic dipole moment of the electron spin Magnetic field of the orbital angular moment Potential energy of electron spin in B l field: Selection rules l = +-1 j = +-1, 0 J=0 j=0 (not allowed) µ = g s s e m e s Magnetic moment to angular moment ratio Zeµ 0 Bl = l 1Tesla 8 π r 3 m e Ze µ 0 Vl, s = µ sbl= sl πr m0 4 Z V, s 3 Very large effect for e.g. Cs n Bohr magnetron ground state H atom magnetic moment 4 ev l Small effect for H-atom (smaller than Doppler broadening)

45 image from Haken, Wolf, p. 184 (or p. 190) A few more equations Bohr model, no fine structure effect E j = Enα n ( 1 j ) 4n 1 j for H-atom (Dirac equation), including fine structure, independent of l All states with the same j have the same energy

46 A complete QM description of Spin orbit coupling Fine structure splitting Etc. is rather involved. Spectra of Atoms and Molecules, 3 rd Ed., Peter F. Bernath, Oxford University Press, chapter 5 Physics of Atoms and Molecules, B.H. Bransden, C.J. Joachain, Wiley Press chapter 5 Haken, Wolf Chapter 14

47 Spin orbit coupling / fine structure In-class homework 11) Why do we see a doublet for Na-atoms?

48 Spin orbit coupling / fine structure In-class homework 11) Why do we see a doublet for Na-atoms? RESULT RESULT One electron system. [Ne] 3s 1 The spin of the single electron has two possible orientations with respect to the orbital angular momentum.

49 Fine structure splitting of hydrogen The splitting is caused by an interaction between the electron spin S and the orbital angular momentum L. Spin-orbit interaction, fine structure.

50 image from Haken, Wolf, p Lamb shift The effect: H atom, states with the same j show different energies. Explanation : quantum electrodynamics effect vacuum energy fluctuations related to black holes theory Willis Eugene Lamb Jr. ( ) Nobel Prize in Physics (1955) Can one lift the l-degeneracy of states for H atoms?

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52 Need a break?

53 Last topic today Pieter Zeeman ( ) Nobel Prize for Physics (190) Dutch physicist Normal Zeeman effect (l only) Anomalous Zeeman effect (j only) Paschen-Back effect (B large)

54 Angular momentum vector model z m l =1 m l = 0 ˆ ψ = ψ L z n, l, m ml n, l, m L z = m l m l = 1 Orientation quantization Space quantization

55 Zeeman effect The effect: Splitting of spectral lines due to an external magnetic field Line splitting: Splitting is described by orientation quantization L z = m l V m j = µ B j = m g 0 j jb 0 (m j = j, j-1,, -j) Selection rules m j = 0, +-1 Normal Zeeman effect (l only) Anomalous Zeeman effect (j only) Paschen-Back effect (B large)

56 image from Haken, Wolf, p. 198 Normal Zeeman effect - example Example Cd-atom; [Kr] 4d 10 5s Two electron system, but S=0 Equidistant energy levels In-class homework 1) Where is the fine structure splitting in this figure? Polarization of emitted radiation Pi: perpendicular σ: circular

57 image from Haken, Wolf, p. 198 Normal Zeeman effect - example Example Cd-atom; [Kr] 4d 10 5s Two electron system, but S=0 Equidistant energy levels Polarization of emitted radiation Pi: perpendicular σ: circular In-class homework 1) Where is the fine structure splitting in this figure? RESULT RESULT S=0 no fine structure splitting

58 Anomalous Zeeman effect - example Actually the general case Example Na-atom; [Ne] 3s 1 One electron system, but S not 0, consider J Not equidistant energy levels More spectral lines In-class homework 13) Where is the fine structure splitting in this figure?

59 Anomalous Zeeman effect - example Actually the general case Example Na-atom; [Ne] 3s 1 One electron system, but S not 0, consider J Not equidistant energy levels More spectral lines P1/ and P3/ are the fine structure splitting, I guess. In-class homework 13) Where is the fine structure splitting in this figure? RESULT RESULT

60 Paschen Back effect - example Line splitting due to external magnetic field is large compared with fine structure splitting Magnetic field is so large that spin-orbit coupling is absent. l and j become again independent quantum numbers By the way, is this Mr. Paschen Back or Mrs. Paschen and Back? Who cares well German physicists Friedrich Paschen and Ernst E. A. Back Paschen Back

61 Spin-Orbit coupling No external fields: All states with the same J value have the same energy With an external (magnetic) field: In an external magnetic field states with the same J but different M J have different energies. (Zeeman effect)

62 What happens here? l degeneracy lifted j degeneracy lifted same term same energy same j same energy Zeeman effect

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64 Spin-orbit coupling, one electron system Fine structure z m j = 3 n S+1 L j m j = m j = 1 m j = 3 j z m j = m j = j, j 1,..., j Schematics for alkali atoms (treaded at oneelectron states including spin-orbit coupling) Pieter Zeeman

65 Read at least Haken, Wolf Chapter 1-13 or equivalent Explain qualitatively what spin-orbit coupling and the Zeeman effect are. What is wrong/correct on the historic Bohr model? What are the main conclusions of the Einstein-de Haas effect and Stern- Gerlach experiment?

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67 Spectra of Atoms and Molecules, 3 rd Ed., Peter F. Bernath, Oxford University Press, chapter 5.4 Engel/Reid chapter??? Physics of Atoms and Molecules, B.H. Bransden, C.J. Joachain, Wiley Press chapter 6-7 Next class Many-electron atoms Pauli Principle Singlet vs. triplet Term symbols LS vs. jj coupling Haken, Wolf Chapter 17 How to measure these small peak splitting? See Next Next class

68 Figure acknowledgement All images shown in this power point presentation were made by the author except the following with are excluded for the copyright of the author: xxx No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means except as permitted by the United States Copyright Act, without prior written permission of the author. Trademarks and copyrights are property of their respective owners., 017 Publisher and author: Uwe Burghaus, Fargo, ND, USA