1. Harris Chapter 7 - Atomic Structure
7.1
Orbital Magnetic Moments, discovery of intrinsic spin
7.2 & 7.3
Identical Particles (warning: examples in book all inf-squ well)
Exclusion Principle
7.4 & 7.5
Multielectron Atoms, effective charges
Hartree Treatment
7.6
Spin-Orbit Effect
7.7
Adding QM Angular Momenta
7.9 & 7.8
Multielectron Spectroscopic Notation
Zeeman Effect

2.
Summary So Far

3. 7.1 Orbital Magnetic Moments and Discovery of Intrinsic Spin

4. Two kinds of Angular Momentum
Classical Angular Momentum
L = r x p
r vector, p vector  L vector
L obeys vector math
Any L possible, no contraints on Lx Ly Lz
Quantum
Quantum Mechanical Angular Momentum
r vector, p vector operator  L 3 component operator
L obeys …… got to be careful
L described by two labels l , m
L and Lz can be known, Lx and Ly cannot

5. Bohr Model of Ang Momentum
Classical or
Semi-classical
description
Note: s-states (l=0) have no Bohr model picture
Eisberg & Resnick: Fig 7-11

6. Vector Model of QM Ang. Momentum
quantum numbers
E&R Fig 7-12

7. Edmonds “A.M. in QM” pg 19: “We might imagine the vector moving in an
unobservable way about the z-axis...”
pg 29: “The QM probability density, not being time dependent,
gives us no information about the motion of the
particle in it’s orbit.”
Y*(r,t) Y(r,t)
Y(r,t)=Y(r) e-iwt

8. Morrison, Estle, Lane “Understanding More QM”, Prentice-Hall, 1991

9. Otto Stern & Walther Gerlach ~1922
Bohr’s Q hypothesis
Sommerfeld’s Q hypothesis 3 2 1
Assigned by advisor Max Born to demonstrate existence of the l, ml quantum numbers

10. Orbital Magnetic Moment
E&R Fig 7-11

11. Orbital Magnetic Moment
E&R Fig 7-11

12. E&R Fig 7-11
Bohr magneton

13. B m

14. B m
Different ml states experience different forces

15. Different ml states experience different forces
Use B as z-axis.
Different ml states experience different forces

16. Stern & Gerlach ~1922
Harris Fig 7.3, 7.4

17. Stern & Gerlach ~1922
Intended to demonstrate space quantization (l), & therefore expected odd number of spots,
but observed an even number.

18. Despite Stern's careful design and feasibility calculations, the experiment took more than a year to accomplish. In the final form of the apparatus, a beam of silver atoms (produced by effusion of metallic vapor from an oven heated to 1000°C) was collimated by two narrow slits (0.03 mm wide) and traversed a deflecting magnet 3.5 cm long with field strength about 0.1 tesla and gradient 10 tesla/cm. The splitting of the silver beam achieved was only 0.2 mm.
Accordingly, misalignments of collimating slits or the magnet by more than 0.01 mm were enough to spoil an experimental run. The attainable operating time was usually only a few hours between breakdowns of the apparatus. Thus, only a meager film of silver atoms, too thin to be visible to an unaided eye, was deposited on the collector plate.
Stern described an early episode:

19. Stern described an early episode:
After venting to release the vacuum, Gerlach removed the detector flange. But he could see no trace of the silver atom beam and handed the flange to me. With Gerlach looking over my shoulder as I peered closely at the plate, we were surprised to see gradually emerge the trace of the beam Finally we realized what [had happened]. I was then the equivalent of an assistant professor. My salary was too low to afford good cigars, so I smoked bad cigars. These had a lot of sulfur in them, so my breath on the plate turned the silver into silver sulfide, which is jet black, so easily visible. It was like developing a photographic film.7

20. Wolfgang Pauli ~ 1924 Pauli Exclusion Principle
No two electrons can have the same quantum number
Postulated an additional quantum number (i.e. label)
Believed it came from the interaction between electrons.

21. Ralph Kronig ~1925
Spinning Electron Idea

22. Goudsmit & Ulhenbeck ~ 1925
Studied high resolution spectra of alkali elements

23. Ocean Optics - Helium

24. Ocean Optics - Neon

25. Giancoli – fig 36.21

26. [5] S. Goudsmit and G.E. Uhlenbeck, Physica 6 (1926) 273.
The old and the new term scheme of hydrogen [5]. The scheme shows the multiplet splitting of the excited states of the hydrogen atom with principal quantum number n=3, presented by Goudsmit in the form in which it appeared in the original publications of1926. The assignment in the current notation has been added at the right. With the development of quantum mechanics the notation changed. The quantum numbers L and J now usedfor the orbital and total angular momentum, respectively, correspond to K-1/2 and J-1/2 in the figure. The "forbidden component" referred to by Goudsmit is of the type 3 2P1/2 --> 2 2S in which the total angular momentum is conserved and L changes by plus or minus 1.
[5] S. Goudsmit and G.E. Uhlenbeck, Physica 6 (1926) 273.

27. Uhlenbeck & Goudsmit ~ 1925
The discovery note in Naturwissenschaften is dated 17 October One day earlier Ehrenfest had written to Lorentz to make an appointment and discuss a "very witty idea" of two of his graduate students. When Lorentz pointed out that the idea of a spinning electron would be incompatible with classical electrodynamics, Uhlenbeck asked Ehrenfest not to submit the paper. Ehrenfest replied that he had already sent off their note, and he added: "You are both young enough to be able to afford a stupidity!"

28. Uhlenbeck & Goudsmit ~ 1925
Ehrenfest's encouraging response to his students ideas contrasted sharply with that of Wolfgang Pauli. As it turned out, Ralph Kronig, a young Columbia University PhD who had spent two years studying in Europe, had come up with the idea of electron spin several months before Uhlenbeck and Goudsmit. He had put it before Pauli for his reactions, who had ridiculed it, saying that "it is indeed very clever but of course has nothing to do with reality". Kronig did not publish his ideas on spin. No wonder that Uhlenbeck would later refer to the "luck and privilege to be students of Paul Ehrenfest".

29. “This isn't right. This isn't even wrong.”
There were some people thinking about electron spin in those days, but there was a lot of basic opposition to such an idea. One of the first was Ralph de Laer Kronig. He got the idea that the electron should have a spin in addition to its orbital motion. He was working with Wolfgang Pauli at the time, and he told his idea to Pauli. Pauli said, "No, it's quite impossible." Pauli completely crushed Kronig.
Then the idea occurred quite independently to two Young Dutch physicists, George Uhlenbeck and Samuel Goudsmit. They were working in Leiden with Professor Paul Ehrenfest, and they wrote up a little paper about it and took it to Ehrenfest. Ehrenfest liked the idea very much. He suggested to Uhlenbeck and Goudsmit that they should go and talk it over with Hendrik Lorentz, who lived close by in Haarlem.
His ability to make experiments self destruct simply by being in the same room was legendary, and has been dubbed the "Pauli effect"
(Frisch 1991, p. 48; Gamow 1985).
"The Birth of Particle Physics," edited by Laurie M. Brown and Lillian Hoddeson. The essay by Paul A.M. Dirac is entitled "Origin of Quantum Field Theory."

30. “This isn't right. This isn't even wrong.”
They did go and talk it over with Lorentz. Lorentz said, "No, it's quite impossible for the electron to have a spin. I have thought of that myself, and if the electron did have a spin, the speed of the surface of the electron would be greater than the velocity of light. So, it's quite impossible." Uhlenbeck and Goudsmit went back to Ehrenfest and said they would like to withdraw the paper that they had given to him. Ehrenfest said, "No, it's too late; I have already sent it in for publication "
His ability to make experiments self destruct simply by being in the same room was legendary, and has been dubbed the "Pauli effect"
(Frisch 1991, p. 48; Gamow 1985).
"The Birth of Particle Physics," edited by Laurie M. Brown and Lillian Hoddeson. The essay by Paul A.M. Dirac is entitled "Origin of Quantum Field Theory."

31. The calculation (using current values)
r < 2.8 E-19 m
b > 3 *
value from Bhabha scattering at CERN

32. “This isn't right. This isn't even wrong.”
That is how the idea of electron spin got publicized to the world. We really owe it to Ehrenfest's impetuosity and to his not allowing the younger people to be put off by the older ones. The idea of the electron having two states of spin provided a perfect answer to the duplexity.
His ability to make experiments self destruct simply by being in the same room was legendary, and has been dubbed the "Pauli effect"
(Frisch 1991, p. 48; Gamow 1985).
"The Birth of Particle Physics," edited by Laurie M. Brown and Lillian Hoddeson. The essay by Paul A.M. Dirac is entitled "Origin of Quantum Field Theory."

33. Letter fm Thomas to Goudsmit
Part of a letter by L.H. Thomas to Goudsmit (25 March 1926). Reproduced from a transparency shown by Goudsmit during his 1971 lecture. The original is presumably in the Goudsmit archive kept by the AIP Center for History of Physics.

34. intrinsic spin Fundamental objects Composite objects electron spin – ½
neutrino spin – ½ , but LH only
photon spin – 1
Composite objects
proton spin – ½
neutron spin – ½
D delta spin – 3/2

35. How to Denote Wavefunctions (version 1)
the spinor has no ‘functional form’
because spin
is not a spatial feature

36. Two types of Magnetic MomentsL S

37. interesting fundamental constants
-2.002 319 304 3622 (15)
1.602 176  (40) x 10-19 C   

38. 7.2 & 7.3 Complications from having Identical Particles

39. Exchange Symmetry

40. 7.4 & 7.5 Multielectron Atoms

41. En = ( eV ) (Z2/n2)
r = n2 ao / Z
ao = Å

42. Prob = r2 R* R

43. orbitals get sucked down the most
Crossings occur for the upper orbitals 4p 4s 3d 3p 3s 2p 2s 1s
1s sucked off bottom of page

44. Note: This shows how the
orbitals shift as viewed
from the perspective
of an s-orbital.

45. Hartree-Fock Method

46. Hartree-Fock Methods Choose initial shape For Coulomb Potl V(r)
Solve Schro Eqn
for En Yn
Insert fine structure
corrections
Build atom according to
This set of orbital energies En
Loop until V(r) doesn’t change much
Use the collection of Yn*Yn to
Get new electron charge distrib
Use Gauss’ Law to
get new V(r) shape

48. Using effective charge is a very crude approximation.

49. Hartree-Fock Effective Charge Effects
r2 ~ n2 ao / Zeff
En ~ (Zeff2/n2) ( eV )
Hartree-Fock Effective Charge Effects

52. 7.6 Spin-Orbit Effect

53. Corrections to the Coulomb Potl for H-atom
Central Potential
Spin-Orbit (electron viewpoint)
Relativistic Spin (Thomas precession)
Relativistic Kinetic Energy
Spin-Orbit (nucleus viewpoint)
Spin-Spin
Impact of External Fields
Zeeman Effect (applied B-field)
Stark Effect (applied E-field)

54. Spin-Orbit InteractionL ms ms L
Note: L.H. Thomas showed that in the x-form between
non-inertial reference frames a factor of ½ appears.

55. L Goal: find expression for the orientational potential energy
of electron intrinsic mag moment (ms)
in terms of orbital motion (L) and forces (~ dV/dr). ms L
Note: L.H. Thomas showed that in the x-form between
non-inertial reference frames a factor of ½ appears.

56. ms L

57. E

58. -2 L S
NRG shift depends on
relative orientation
of L and S

59. How to evaluate DE and S·L
involved
in radial
integrations
depends
on A.M.
qu. no.s

61. J electronSpin-Orbit “locks” the angle between L & S
 J is now a well-defined direction. J S
NOTE Lz is
no longer
well-defined
ml not a
good q. no. L S L

62. Revised H-atom Level Scheme
add in
spin-orbit
correction
nlj
not required to specify NRG
mj ml s ms
not required to specify NRG
j mj l ml s ms
absolutely worthless

63. electron Spin-Orbit is more important in higher-Z atoms
fn’l expression only for H-atom,
for all others, must come from
Hartree procedure Li Na K Rb Cs
Splitting
(eV)
0.42E-4
21.E-4
72.E-4
295.E-4
687.E-4

64. Bigger atoms
larger Z (central charge)
~ same size
larger

65. 7.7 QM Angular Momentum

66. Bohr Model of Ang Momentum
Note: s-states (l=0) have no Bohr model picture
Eisberg & Resnick: Fig 7-11

67. Vector Model of Ang. Momentum
quantum numbers
E&R Fig 7-12

68. Edmonds “A.M. in QM” pg 19: “We might imagine the vector moving in an
unobservable way about the z-axis...”
pg 29: “The QM probability density, not being time dependent,
gives us no information about the motion of the
particle in it’s orbit.”

69. Morrison, Estle, Lane “Understanding More QM”, Prentice-Hall, 1991

70. ADDITION OF ANGULAR MOMENTUM
Ltot = L1 + L2 L1

71. Ltot = L1 + L2

72. Ltot = L1 + L2

73. Addition of Angular Momentum
aligned configuration
“aligned” does not mean straight
jack-knife configuration
“jack-knife” does not mean antiparallel

75. Detailed Example L2 L1
Problem: Two objects each travel in a p-orbit ( l=1 ).
The total energy of each object is degenerate wrt ml, so we have no detailed knowledge of ml.
What are the allowed values of ltot, mtot ?

76. l1=1, l2=1, m’s degenerate +2 +1 -1 -2 mtot Possibilities (m1,m2) m1“ -1 -2
mtot
Possibilities (m1,m2) +2 +1 -1 -2

77. Allowed Values of ltot mtot

78. Basic A.M. Math
J = L + S J S L

79. Vector Representation of J

80. Annoying Pictures #1
Jeff’s Qs: i) what am I supposed to think about the S & L cones as drawn?
ii) I thought I was told earlier that L & S were about z ??

81. Annoying Pictures #2
Jeff: Pictures such as this confuse the vector symbols L and S
with the quantum numbers ℓ and s .
For instance, how could L and S ever point in the same direction?

82. TOTAL ANGULAR MOMEMTUM
J = L + S

83. More Detailed H-atom Level Scheme
Energies & Spectra not sensitive to
j mj l ml s ms
till next page
Energies & Spectra not sensitive to
l ml

84. Ocean Optics - Helium
Because of the doublets, the states cannot be completely degenerate
 “spin-orbit effect” + …

85. Ocean Optics - Neon
Because of the doublets, the states cannot be completely degenerate
 “spin-orbit effect” + …

86. 7.9 Multi-electron Spectra

87. Multi-e Spectroscopic Notation
QUANTUM NUMBERS
principal: n
ltot , stot
jtot .
stot = 1, ltot=0, jtot=1
2S1
Stot = S1 + S2 + …
Ltot = L1 + L2 + …
Jtot = Ltot + Stot

88. Two Kinds of Notation Where an individ electron is at n l j
2p1/2
2p3/2
A.M. for whole atom
2Stot+1 ltot jtot
1S0
3S1
3P0 , 3P1, 3P2

90. Curious Things That Happen: Ground State of Helium
Ltot = L1 + L2
asym 1s
Stot = S1 + S2
sym
Ysystem = (spatial wfn) (spin wfn) 1s
1S0
3S1

92. 7.8 Atoms in External Magnetic Fields -- the Zeeman Effect

93. Corrections to the Coulomb Potl for H-atom
Central Potential
Spin-Orbit (electron viewpoint)
Relativistic Spin (Thomas precession)
Relativistic Kinetic Energy
Spin-Orbit (nucleus viewpoint)
Spin-Spin
Impact of External Fields
Zeeman Effect (applied B-field)
Stark Effect (applied E-field)

94. Weak-Field Zeeman Hartree-Fock Coulomb & related Procedures
Fine Structure
spin-orbit ( jtot becomes important )
relativistic
Zeeman
H’Zeeman = - mtot * Bext
Bext < few 0.1’s Tesla

95. Weak Field Zeeman
mtot

96. J electronSpin-Orbit “locks” the angle between L & S
 J is now a well-defined direction. J S
NOTE Lz is
no longer
well-defined
ml not a
good q. no. L S L

97. Weak-Field Zeeman eSO makes jtot good quantum number,
mltot & mstot become ‘confused’ (near worthless).
Jtot
mtot
Jtot is ‘well-defined’ direction; jtot mjtot g
project average mtot onto Jtot

98. Weak Field Zeeman DEZeeman = - mtot * Bext a Jtot
projection of mtot onto J onto B
monto J a
Bext
DEZeeman = - mtot * Bext

100. stot=0

101. Strong-Field Zeeman Hartree-Fock Coulomb & related procedures Zeeman
Fine Structure
spin-orbit
relativistic
H’Zeeman = - mtot * Bext

102. Strong Field Zeeman
Ltot
Stot
Bext
H’Zeeman = - mtot * Bext