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### Optically polarized atoms

Marcis Auzinsh, University of Latvia

Dmitry Budker, UC Berkeley and LBNL

Simon M. Rochester, UC Berkeley

Chapter 2: Atomic states

- A brief summary of atomic structure
- Begin with hydrogen atom
- TheSchrödinger Eqn:
- In this approximation (ignoring spin and relativity):

Principal quant. Number

n=1,2,3,…

Could have guessed me 4/2 from dimensions

- me 4/2 =1Hartree
- me 4/22 =1 Rydberg
- E does not depend on lor m degeneracy

i.e.different wavefunction have same E

- We will see that the degeneracy is n2

Angular momentum of the electron in the hydrogen atom

- Orbital-angular-momentum quantum numberl = 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 lfor a given value of En
- Turns out: l = 0,1,2, …, n-1

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

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 !

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

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

- How can one understand restrictions that QM puts on measurements of angular-momentum components ?
- The basic QM uncertainty relation(*) leads to (and permutations)
- We can also write a generalizeduncertainty relation

between lzand φ(azimuthal angle of the e):

- This is a bit more complex than (*) because φis cyclic
- With definite lz , φis completely uncertain…

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)

Wavefunctions of the H atom (cont’d)

- Separation into radial and angular part is possible for any central potential !
- Things get nontrivial for multielectron atoms

Legendre Polynomials

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 ~ 10-18 cm

Electron spin and fine structure (cont’d)

- Another issue for classical picture: it takes a 4πrotation 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)

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 ~ 10-18 cm

Electron spin and fine structure (cont’d)

- s=1/2
- “Spin up” and “down” should be used with understanding that the length (modulus) of the spin vector is >/2 !

Electron spin and fine structure (cont’d)

- Both orbital angular momentum and spin have associated magnetic momentsμl and μs
- Classical estimate of μl : current loop
- For orbit of radius r, speed p/m, revolution rate is

Gyromagnetic ratio

Electron spin and fine structure (cont’d)

- In analogy, there is also spin magnetic moment :

Bohr magneton

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=21.00115965218085(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

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 sare 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

Electron spin and fine structure (cont’d)

- QM parlance: states with fixed ml and ms are no longer eigenstates
- States with fixed j, mjare 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

Electron spin and fine structure (cont’d)

- Spin-orbitinteraction is a relativistic effect
- Includingrel. effects :
- Correction to the Bohr formula 2
- The energy now depends on n and j

Electron spin and fine structure (cont’d)

- 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

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 0.035 cm-1 or 1057 MHz

mj= 1/2

Vector model of the atom- Some people really need pictures…
- Recall: for a state with given j, jz
- We can draw all of this as (j=3/2)

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 jxor jy that can be measured ?
- A:

that might be inferred from the picture is wrong…

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 ?
- Complicated
- Still wrong…

Vector model of the atom (cont’d)

- A compromise :
- 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

Multielectron atoms

- 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
- Energiesdepend onorbital ang. momenta

Gross structure, LS coupling

- Individual electron “sees” nucleus and other e’s
- Approximate totalpotential 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

Gross structure, LS coupling (cont’d)

- 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 orto spins degeneracy
- If we go beyond the central-field approximation some of the degeneracies will be lifted
- Also spin-orbit (ls) interaction lifts some degeneracies
- In general, both effects need to be considered, but the former is more important in light atoms

Gross structure, LS coupling (cont’d)

Beyond central-field approximation (cfa)

- Non-centrosymmetric part of electron repulsion (1/rij) = residual Coulomb interaction (RCI)
- The energy now depends on how li andsi combine
- Neglecting (ls) 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

Gross structure, LS coupling (cont’d)

Beyond central-field approximation (cfa)

- 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 (ls) interaction, energies of states do not depend on spins

WRONG !

Gross structure, LS coupling (cont’d)

The role of the spins

- 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

Gross structure, LS coupling (cont’d)

Exchange Interaction

- The value of the total spinS affects the symmetry of the spin wavefunction
- Since overall ψhas to be antisymmetric symmetry of spatial wavefunction is affected this affects Coulomb repulsion between electrons effect on energies
- Thus, energies depend on Land S. Term: 2S+1L
- 2S+1 is called multiplicity
- Example: He(g.s.): 1s2 1S

Gross structure, LS coupling (cont’d)

- Within present approximation, energies do not depend on (individually conserved) projections of L and S
- This degeneracy is lifted by spin-orbit interaction (also spin-spin 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

Vector Model

- Example: a two-electron atom (He)
- Quantum numbers:
- J, mJ “good” no restrictions

for isolated atoms

- l1, l2 , L, S “good” in LS coupling
- ml ,ms , mL , mS “not good”=superpositions
- “Precession” rate hierarchy:
- l1, l2 around L and s1, s2 around S:

residual Coulomb interaction

(term splitting -- fast)

- Land S around J

(fine-structure splitting -- slow)

jj and intermediate coupling schemes

- 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

jj and intermediate coupling schemes(cont’d)

- 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
- Land S rules: approximate; only hold within LS coupling
- J, mJrules: strict; hold for any coupling scheme

Notation of states in multi-electron atoms

Spectroscopic notation

- Configuration (list of ni and li )
- ni – integers
- li – code letters
- Numbers of electrons with same n and l – superscript, for example: Na (g.s.): 1s22s22p63s = [Ne]3s
- Term 2S+1L State2S+1LJ
- 2S+1 = multiplicity (another inaccurate historism)
- Complete designation of a state [e.g., Ba (g.s.)]: [Xe]6s21S0

Fine structure in multi-electron atoms

- LS states with different J are split by spin-orbit interaction
- Example: 2P1/2-2P3/2splitting in the alkalis
- Splitting Z2(approx.)
- Splitting with n

Hyperfine structure of atomic states

- NuclearspinI magnetic moment
- Nuclear magneton
- Total angular momentum:

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

- E2 term: B0 only for I,J>1/2

Hyperfine structure of atomic states

- A nucleus can only support multipoles of rank κ2I
- E1, M2, …. moments are forbidden by P and T

B0 only for I,J>1/2

- Example of hfs splitting (not to scale)

85Rb (I=5/2)

87Rb (I=3/2)

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