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

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

3. 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

4. Angular momentum of the electron in the hydrogen atom
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
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

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 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

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
Can also define the axis with E (static or oscillating), other fields (e.g., gravitational), or nothing
Let’s count states:
m = -l,…,l i. e. 2l+1 states
l = 0,…,n-1 
As advertised !

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. 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):

8. Angular momentum of the electron in the hydrogen atom (cont’d)
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…

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

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

11. Electron spin and fine structure
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.
Experiment: electron is pointlike down to ~ cm

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:
from Feynman's 1986 Dirac Memorial Lecture
(Elementary Particles and the Laws of Physics, CUP 1987)

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

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

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

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

17. Electron spin and fine structure (cont’d)
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 (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, (2006)
Prof. G. Gabrielse, Harvard

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

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

20. 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

21. Electron spin and fine structure (cont’d)
1/137  relativistic corrections are small
~ 10-5 Ry
E  cm-1 or 10.9 GHz for 2P3/2 , 2P1/2
E  cm-1 or 3.24 GHz for 3P3/2 , 3P1/2

22. 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 2S1/2 , 2P1/2: E  cm-1 or 1057 MHz

23. Vector model of the atom
Some people really need pictures…
Recall:
We can draw all of this as (j=3/2)
mj = 3/2
mj = 1/2

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

25. 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 ?