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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., 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
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
How can one understand why only one projection of the
angular momentum at a time can be determined?
In analogy with
write an uncertainty relation between lz and φ (angle in
the x-y plane of the projection of the angular momentum
w.r.t. x axis):
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Angular momentum of the electron
in the hydrogen atom (cont’d)
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How can one understand why only one projection of the
angular momentum at a time can be determined?
In analogy with
(*)
write an uncertainty relation between lz and φ (angle in
the x-y plane of the projection of the angular momentum
w.r.t. x axis):
This is a bit more complex than (*) because φ is cyclic
With definite lz , φ is completely uncertain…
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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
Presumably, we want  finite
The surface of the object has linear velocity r  c
If we have L
, (1,2)  r 
mc
=
c
(1)
(2)
 3.9  1011 cm
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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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
~ 10-5 Ry
 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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Vector model of the atom
Some people really need pictures…
jx  j y  0;
 Recall:
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Expectation value of j2 is j( j  1)
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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 ?
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