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Optically polarized atoms
Marcis Auzinsh, University of Latvia
Dmitry Budker, UC Berkeley and LBNL
Simon M. Rochester, UC Berkeley
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Chapter 2: Atomic states
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A brief summary of atomic structure
Begin with hydrogen atom
The Schrödinger Eqn:
Image from Wikipedia
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In this approximation (ignoring spin and
relativity):
Principal quant. Number
n=1,2,3,…
2
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Could have guessed me 4/2 from dimensions
me 4/2 = 1 Hartree
me 4/22 = 1 Rydberg
E does not depend on l or m  degeneracy
i.e. different wavefunction have same E
We will see that the degeneracy is n2
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Angular momentum of the electron
in the hydrogen atom
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Orbital-angular-momentum quantum number l = 0,1,2,…
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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 
There is kinetic energy associated with orbital motion 
an upper bound on l for a given value of En
Turns out: l = 0,1,2, …, n-1
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Angular momentum of the electron
in the hydrogen atom (cont’d)
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In classical physics, to fully specify orbital angular
momentum, one needs two more parameters (e.g., two
angles) in addition to the magnitude
In QM, if we know projection on one axis (quantization
axis), projections on other two axes are uncertain
Choosing z as quantization axis:
Note: this is reasonable as we expect projection
magnitude not to exceed
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Angular momentum of the electron
in the hydrogen atom (cont’d)
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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
Let’s count states:
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m = -l,…,l i. e. 2l+1 states
l = 0,…,n-1  n 1
1  2( n  1)  1
2
(2
l

1)


n

n

2
l 0
As advertised !
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Angular momentum of the electron
in the hydrogen atom (cont’d)
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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
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Angular momentum of the electron in
the hydrogen atom (cont’d)
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How can one understand restrictions that QM puts on
measurements of angular-momentum components ?
The basic QM uncertainty relation
leads to
(and permutations)
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We can also write a generalized uncertainty relation
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between lz and φ (azimuthal angle of the e):
This is a bit more complex than (*) because φ is cyclic
With definite lz , cos  0
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(*)
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Wavefunctions of the H atom
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A specific wavefunction is labeled with n l m :
In polar coordinates :
i.e. separation of radial and angular parts
Spherical
functions
(Harmonics)
Further separation:
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Wavefunctions of the H atom (cont’d)
Legendre Polynomials
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Separation into radial and angular part is possible for any
central potential !
Things get nontrivial for multielectron atoms
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Electron spin and fine structure
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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
L  I ~ mr 2
(1)
Presumably , we want  finite
The surface of the object has linear vel ocity ~ ωr  c (2)

If we have L ~ , Eqs. (1,2)  r 
  c  3.9 10 11 cm
mc
Experiment: electron is pointlike down to ~ 10-18 cm
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Electron spin and fine structure (cont’d)
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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)
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Electron spin and fine structure (cont’d)
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Another amusing classical picture: spin angular
momentum comes from the electromagnetic field of the
electron:
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This leads to electron size
Experiment: electron is pointlike down to ~ 10-18 cm
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Electron spin and fine structure (cont’d)
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s=1/2 
“Spin up” and “down” should be used with understanding
that the length (modulus) of the spin vector is >/2 !
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Electron spin and fine structure (cont’d)
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Both orbital angular momentum and spin have
associated magnetic moments μl and μs
Classical estimate of μl : current loop
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For orbit of radius r, speed p/m, revolution rate is
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Gyromagnetic ratio
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Electron spin and fine structure (cont’d)
Bohr magneton
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In analogy, there is also spin magnetic moment :
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Electron spin and fine structure (cont’d)
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The factor 2 is important !
Dirac equation for spin-1/2 predicts exactly 2
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)(0.8 parts per trillion)
New Measurement of the Electron Magnetic Moment
Using a One-Electron Quantum Cyclotron,
B. Odom, D. Hanneke, B. D'Urso, and G. Gabrielse,
Phys. Rev. Lett. 97, 030801 (2006)
Prof. G. Gabrielse,
Harvard
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Electron spin and fine structure (cont’d)
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Electron spin and fine structure (cont’d)
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When both l and s are present, these are not conserved
separately
This is like planetary spin and orbital motion
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 μs
Energy shift depends on relative orientation of l and s, i.e., on
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Electron spin and fine structure (cont’d)
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QM parlance: states with fixed ml and ms are no
longer eigenstates
States with fixed j, mj are eigenstates
Total angular momentum is a constant of motion of
an isolated system
|mj|  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
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Electron spin and fine structure (cont’d)
Spin-orbit interaction is a relativistic effect
 Including rel. effects :
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Correction to the Bohr formula 2
 The energy now depends on n and j
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Electron spin and fine structure (cont’d)
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1/137  relativistic corrections are small
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~ 10-5 Ry
En , j l 1/ 2  En , j l 1/ 2
E  0.366 cm-1 or 10.9 GHz for 2P3/2 , 2P1/2
 E  0.108 cm-1 or 3.24 GHz for 3P3/2 , 3P1/2
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Electron spin and fine structure (cont’d)
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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 2S1/2 , 2P1/2: E  0.035 cm-1 or 1057
MHz
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Electron spin and fine structure (cont’d)
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Example: n=2 and n=3 states in H (from C. J. Foot)
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Vector model of the atom
Some people really need pictures…
 Recall: for a state with given j, jz
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jx  j y  0;
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j2 = j ( j  1)
We can draw all of this as (j=3/2)
mj = 3/2
mj = 1/2
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Vector model of the atom (cont’d)
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These pictures are nice, but NOT problem-free
Consider maximum-projection state mj = j
mj = 3/2
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Q: What is the maximal value of jx or jy that can be
measured ?
A:
that might be inferred from the picture is wrong…
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Vector model of the atom (cont’d)
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So how do we draw angular momenta and coupling ?
Maybe as a vector of expectation values, e.g.,
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Simple
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Has well defined QM meaning
?
BUT
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Boring
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Non-illuminating
Or stick with the cones ?
Complicated
 Still wrong…
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Vector model of the atom (cont’d)
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A compromise :
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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,ml,s,ms
 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
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Multielectron atoms
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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
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Gross structure, LS coupling
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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; li (and si) are well defined !
Radial SE with a given li  set of bound states
Label these with principal quantum number
ni = li +1, li +2,… (in analogy with
Hydrogen)
Oscillation Theorem: # of zeros of the radial
wavefunction is ni - li -1
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Gross structure, LS coupling (cont’d)
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Set of ni , li for all electrons  electron configuration
Different configuration generally have different
energies
In this approximation, energy of a configuration is
just sum of Ei
No reference to projections of li 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
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Gross structure, LS coupling (cont’d)
Beyond central-field approximation (cfa)
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Non-centrosymmetric part of electron repulsion (1/rij )
= residual Coulomb interaction (RCI)
The energy now depends on how li and si combine
Neglecting (ls) interaction  LS coupling or
Russell-Saunders coupling
This terminology is potentially confusing…..
….. but well motivated !
Within cfa, individual orbital angular momenta are
conserved; RCI mixes states with different projections of li
Classically, RCI causes precession of the orbital planes, so
the direction of the orbital angular momentum changes
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Gross structure, LS coupling (cont’d)
Beyond central-field approximation (cfa)
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Projections of li are not conserved, but the total orbital
momentum L is, along with its projection !
This is because li form sort of an isolated system
So far, we have been ignoring spins
One might think that since we have neglected (ls)
interaction, energies of states do not depend on spins
WRONG !
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Gross structure, LS coupling (cont’d)
The role of the spins
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Not all configurations are possible. For example, U has 92
electrons. The simplest configuration would be 1s92
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
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
So how does spin affect energies (of allowed configs) ?
 Exchange Interaction
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Gross structure, LS coupling (cont’d)
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Exchange Interaction
The value of the total spin S affects the symmetry of
the spin wavefunction
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+1L
2S+1 is called multiplicity
Example: He(g.s.): 1s2 1S
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Gross structure, LS coupling (cont’d)
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Within present approximation, energies do not depend on
(individually conserved) projections of L and S
This degeneracy is lifted by spin-orbit interaction (also spinspin and spin-other orbit)
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
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Vector Model
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Example: a two-electron atom (He)
Quantum numbers:
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“good” no restrictions
for isolated atoms
l1, l2 , L, S
“good” in LS coupling
ml , ms , mL , mS “not good”=superpositions
J, mJ
“Precession” rate hierarchy:
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l1, l2 around L and s1, s2 around S:
residual Coulomb interaction
(term splitting -- fast)
L and S around J
(fine-structure splitting -- slow)
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jj and intermediate coupling schemes
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Sometimes (for example, in heavy atoms),
spin-orbit interaction > residual Coulomb  LS coupling
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 
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jj and intermediate coupling schemes
(cont’d)
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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
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L and S
J, mJ
rules: approximate; only hold within LS coupling
rules: strict;
hold for any coupling scheme
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Notation of states in multi-electron atoms
Spectroscopic notation
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Configuration (list of ni and li )
ni – integers
 li – code letters
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Numbers of electrons with same n and l – superscript, for
example: Na (g.s.): 1s22s22p63s = [Ne]3s
Term 2S+1L  State 2S+1LJ
2S+1 = multiplicity (another inaccurate historism)
Complete designation of a state [e.g., Ba (g.s.)]:
[Xe]6s2 1S0
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Fine structure in multi-electron atoms
LS states with different J are split by spin-orbit
interaction
 Example: 2P1/2-2P3/2 splitting in the alkalis
 Splitting Z2 (approx.)
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Splitting  with n
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Hyperfine structure of atomic states
Nuclear spin I  magnetic moment
 Nuclear magneton
 Total angular momentum:
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Hyperfine structure of atomic states (cont’d)
Hyperfine-structure splitting results from
interaction of the nuclear moments with fields and
gradients produced by e’s 
 Lowest terms:
M1
E2
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E2 term: B0 only for I,J>1/2
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Hyperfine structure of atomic states
A nucleus can only support multipoles of rank
κ2I
 E1, M2, …. moments are forbidden by P and T
B0 only for I,J>1/2
 Example of hfs splitting (not to scale)
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85Rb
87Rb
(I=5/2)
(I=3/2)
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