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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
- The Schrödinger Eqn
- In this approximation (ignoring spin and

relativity)

Principal quant. Number n1,2,3,

- 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

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

Angular momentum of the electron in the hydrogen

atom (contd)

- 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

Angular momentum of the electron in the hydrogen

atom (contd)

- 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 - Lets count states
- m -l,,l i. e. 2l1 states
- l 0,,n-1 ?

As advertised !

Angular momentum of the electron in the hydrogen

atom (contd)

- 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 (contd)

- 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 generalized uncertainty

relation - between lz and f (azimuthal angle of the e)
- This is a bit more complex than () because f is

cyclic - With definite lz ,

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 (contd)

Legendre Polynomials

- Separation into radial and angular part is

possible for any central potential ! - Things get nontrivial for multielectron atoms

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 (contd)

- Another issue for classical picture it takes a

4p 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)

Electron spin and fine structure (contd)

- 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 (contd)

- s1/2 ?
- Spin up and down should be used with

understanding that the length (modulus) of the

spin vector is gt?/2 !

Electron spin and fine structure (contd)

- 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

Electron spin and fine structure (contd)

Bohr magneton

- In analogy, there is also spin magnetic moment

Electron spin and fine structure (contd)

- 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 ?22?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

Electron spin and fine structure (contd)

Electron spin and fine structure (contd)

- 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

electrons frame with µs - Energy shift depends on relative orientation of l

and s, i.e., on

Electron spin and fine structure (contd)

- 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 ? ls
- s1/2 ? j l ? ½ for l gt 0 or j ½ for l 0

Electron spin and fine structure (contd)

- Spin-orbit interaction is a relativistic effect
- Including rel. effects
- Correction to the Bohr formula ??2
- The energy now depends on n and j

Electron spin and fine structure (contd)

- ??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 (contd)

- 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

Electron spin and fine structure (contd)

- Example n2 and n3 states in H (from C. J.

Foot)

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 (j3/2)

Vector model of the atom (contd)

- 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

Vector model of the atom (contd)

- 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 (contd)

- 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 ?
- Energies depend on orbital ang. momenta

Gross structure, LS coupling

- Individual electron sees nucleus and other es
- Approximate total potential as central f(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 (contd)

- 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

Gross structure, LS coupling (contd)

- Beyond central-field approximation (cfa)
- 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

Gross structure, LS coupling (contd)

- 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

(l?s) interaction, energies of states do not

depend on spins

WRONG !

Gross structure, LS coupling (contd)

- 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 ? - es are fermions ? Pauli exclusion principle

no two es 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 (contd)

- Exchange Interaction
- The value of the total spin S affects the

symmetry of the spin wavefunction - Since overall ? 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 2S1L
- 2S1 is called multiplicity
- Example He(g.s.) 1s2 1S

Gross structure, LS coupling (contd)

- 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 goodsuperpositions
- Precession rate hierarchy
- l1, l2 around L and s1, s2 around S
- residual Coulomb interaction
- (term splitting -- fast)
- L and S around J
- (fine-structure splitting -- slow)

jj and intermediate coupling schemes

- Sometimes (for example, in heavy atoms),

spin-orbit interaction gt 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(contd)

- 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 - L and S rules approximate only hold within

LS coupling - J, mJ rules 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 Ne3s - Term 2S1L ? State 2S1LJ
- 2S1 multiplicity (another inaccurate

historism) - Complete designation of a state e.g., Ba

(g.s.) Xe6s2 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.)
- Splitting ? with n

Hyperfine structure of atomic states

- Nuclear spin I ? magnetic moment
- Nuclear magneton
- Total angular momentum

Hyperfine structure of atomic states (contd)

- Hyperfine-structure splitting results from

interaction of the nuclear moments with fields

and - gradients produced by es ?
- Lowest terms
- M1 E2
- E2 term B?0 only for I,Jgt1/2

Hyperfine structure of atomic states

- A nucleus can only support multipoles of rank

??2I - E1, M2, . moments are forbidden by P and T
- B?0 only for I,Jgt1/2
- Example of hfs splitting (not to scale)

85Rb (I5/2)

87Rb (I3/2)