11

Optically polarized atomsOptically polarized atoms

Marcis Auzinsh, University of LatviaDmitry Budker, UC Berkeley and LBNL

Simon M. Rochester, UC Berkeley

22

A brief summary of atomic structure A brief summary of atomic structure Begin with Begin with hydrogen atomhydrogen atom TheThe SchrSchröödinger Eqndinger Eqn::

In this approximation (ignoring spin and In this approximation (ignoring spin and relativity):relativity):

Chapter 2: Atomic states

Image from Wikipedia

Principal quant. Number

n=1,2,3,…

33

Could have guessed Could have guessed me me 44//22 from dimensions from dimensions me me 44//2 2 == 11 HartreeHartree me me 44//222 2 == 1 Rydberg1 Rydberg EE does not depend on does not depend on l l or or m m degeneracydegeneracy

i.e.i.e. different wavefunction have samedifferent wavefunction have same E E

We will see that the degeneracy is We will see that the degeneracy is nn22

44

Angular momentum of the electron in the hydrogen atom

OrbitalOrbital-angular-momentum -angular-momentum quantum numberquantum number l l = 0,1,2,…= 0,1,2,…

This can be obtained, e.g., from the Schrödinger Eqn., or This can be obtained, e.g., from the Schrödinger Eqn., or straight from QM straight from QM commutation relationscommutation relations

The The Bohr modelBohr model: classical orbits quantized by requiring : classical orbits quantized by requiring angular momentum to be integer multiple of angular momentum to be integer multiple of

There is kinetic energy associated with orbital motion There is kinetic energy associated with orbital motion an upper bound on an upper bound on ll for a given value of for a given value of EEnn

Turns out: Turns out: l l = 0,1,2, …, = 0,1,2, …, nn-1-1

55

Angular momentum of the electron in the hydrogen atom

(cont’d) In classical physics, to fully specify orbital angular In classical physics, to fully specify orbital angular

momentum, one needs two more parameters (e.g., to momentum, one needs two more parameters (e.g., to angles) in addition to the magnitudeangles) in addition to the magnitude

In QM, if we know projection on one axis (In QM, if we know projection on one axis (quantization quantization axisaxis), projections on other two axes are ), projections on other two axes are uncertainuncertain

Choosing Choosing zz as quantization axis: as quantization axis:

Note: this is reasonable as we expect projection Note: this is reasonable as we expect projection magnitude not to exceed magnitude not to exceed

66

Angular momentum of the electron in the hydrogen atom

(cont’d) mm – – magnetic quantum number magnetic quantum number because because BB-field can be -field can be

used to define quantization axisused to define quantization axis Can also define the axis with Can also define the axis with EE (static or oscillating), (static or oscillating),

other fields (e.g., gravitational), or nothingother fields (e.g., gravitational), or nothing Let’s count states:Let’s count states:

m = -l,…,l m = -l,…,l i. e. i. e. 22ll+1+1 states states l l = 0,…,= 0,…,nn-1 -1 1

2

0

1 2( 1) 1(2 1)

2

n

l

nl n n

As advertised !As advertised !

77

Angular momentum of the electron in the hydrogen atom

(cont’d) Degeneracy w.r.t. Degeneracy w.r.t. m m expected from expected from isotropy of spaceisotropy of space Degeneracy w.r.t. Degeneracy w.r.t. ll, in contrast,, in contrast, is a special feature of is a special feature of 1/1/rr

(Coulomb) potential(Coulomb) potential

How can one understand why only one projection of the How can one understand why only one projection of the angular momentum at a time can be determined?angular momentum at a time can be determined?

In analogy with In analogy with

write an write an uncertainty relation uncertainty relation between between llzz and and φφ (angle in (angle in

the x-y plane of the projection of the angular momentum the x-y plane of the projection of the angular momentum w.r.t. x axis): w.r.t. x axis):

88

Angular momentum of the electron in the hydrogen atom

(cont’d) How can one understand why only one projection of the How can one understand why only one projection of the

angular momentum at a time can be determined?angular momentum at a time can be determined? In analogy with In analogy with (*)(*)

write an write an uncertainty relation uncertainty relation between between llzz and and φφ (angle in (angle in

the x-y plane of the projection of the angular momentum the x-y plane of the projection of the angular momentum w.r.t. x axis): w.r.t. x axis):

This is a bit more complex than (*) because This is a bit more complex than (*) because φφ is is cycliccyclic With definite With definite llzz , , φφ is completely uncertain…is completely uncertain…

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Wavefunctions of the H atom A specific wavefunction is labeled with A specific wavefunction is labeled with n l m n l m :: In In polar coordinatespolar coordinates : :

i.e. separation of i.e. separation of radialradial and and angular angular partsparts

Further separation: Further separation:

Spherical functions

(Harmonics)

1010

Wavefunctions of the H atom (cont’d)

Separation into radial and angular part is possible for any Separation into radial and angular part is possible for any central potential central potential !!

Things get nontrivial for Things get nontrivial for multielectron atomsmultielectron atoms

Legendre Polynomials

1111

Electron spin and fine structure

Experiment: electron has Experiment: electron has intrinsicintrinsic angular momentum angular momentum ----spin spin (quantum number (quantum number ss))

It is tempting to think of the spin classically as a spinning It is tempting to think of the spin classically as a spinning object. This might be useful, but to a point. object. This might be useful, but to a point.

2

c

(1)

Presumably, we want finite

The surface of the object has linear velocity (2)

If we have , (1,2) = 3.9 1

L I mr

r c

L rmc

110 cm

Experiment: electron is pointlike down to ~ 10-18 cm

1212

Electron spin and fine structure (cont’d)

Another issue for classical picture: it takes a Another issue for classical picture: it takes a 44ππ rotation rotation to bring a half-integer spin to its original state. to bring a half-integer spin to its original state. Amazingly, this does happen in classical world:Amazingly, this does happen in classical world:

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

1313

Electron spin and fine structure (cont’d)

Another amusing classical pictureAnother amusing classical picture: spin angular : spin angular momentum comes from the electromagnetic field of the momentum comes from the electromagnetic field of the electron:electron:

This leads to electron sizeThis leads to electron size

Experiment: electron is pointlike down to ~ 10-18 cm

1414

Electron spin and fine structure (cont’d)

s=1/2 s=1/2

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

1515

Electron spin and fine structure (cont’d)

Both orbital angular momentum and spin have Both orbital angular momentum and spin have associated associated magnetic momentsmagnetic moments μμl l and and μμs s

Classical estimate of Classical estimate of μμl l : : current loopcurrent loop

For orbit of radius For orbit of radius rr, speed , speed p/m, p/m, revolution raterevolution rate is is

Gyromagnetic ratio

1616

Electron spin and fine structure (cont’d)

In analogy, there is also In analogy, there is also spin magnetic moment spin magnetic moment ::

Bohr magneton

1717

Electron spin and fine structure (cont’d)

The factor The factor 2 2 is important !is important ! Dirac equation for spin-1/2 predicts exactly Dirac equation for spin-1/2 predicts exactly 22 QED QED predicts deviations from 2 due to predicts deviations from 2 due to vacuum vacuum

fluctuationsfluctuations of the E/M field of the E/M field One of the most precisely measured physical One of the most precisely measured physical

constants: constants: 2=22=21.00115965218085(76)1.00115965218085(76)

Prof. G. Gabrielse, Harvard

(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)

1818

Electron spin and fine structure (cont’d)

When both When both ll and and ss are present, these are not conserved are present, these are not conserved separatelyseparately

This is like planetary spin and orbital motionThis is like planetary spin and orbital motion On a short time scale, conservation of individual angular On a short time scale, conservation of individual angular

momenta can be a good approximationmomenta can be a good approximation ll and and ss are coupled via are coupled via spin-orbit interactionspin-orbit interaction: interaction of : interaction of

the the motional magnetic field motional magnetic field in the electron’s frame with in the electron’s frame with μμss

Energy shift depends on relative orientation of ll and and ss, i.e., on , i.e., on

1919

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

2020

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

2121

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

2222

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

2323

Vector model of the atom Some people really need pictures… Recall:

We can draw all of this as (j=3/2)

0;

Expectation value of is ( 1)

x yj j

j j

2j

mj = 3/2 mj = 1/2

2424

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

2525

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 ?