Optically polarized atoms

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)

Principal quant. Number n1,2,3,
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 (contd)

• 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 (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 !
7
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
• 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 f
  (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 (contd)

• 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 f
  (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 f is
  cyclic
• With definite lz , f 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 (contd)
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 10-18
cm
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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)
13
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
14
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 !

15
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
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Electron spin and fine structure (contd)
Bohr magneton

• In analogy, there is also spin magnetic moment

17
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
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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

19
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 gt l-s j lt ls
• s1/2 ? j l ? ½ for l gt 0 or j ½ for l 0

20
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

21
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

22
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

23
Vector model of the atom

• Some people really need pictures
• Recall
• We can draw all of this as (j3/2)

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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

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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 ?