Chapter 2
Semiclassical Conceptual Models

Part I: Classical Position Probability Densities (Sect. 2.1)

Probabilistic formulation (Sect. 2.1.1)

Announcement
Källén Prize

Molecules in a room (Sect. 2.1.2)

Microscopic origin of pressure
continued
van der Waals equation
Multipoles by differentiation
Dipole-dipole interaction
Force or energy?
Exclusion principle

Dwell time formulation

Probability formulation
EBK box quantization
Classical and BSW
Mean position
Mean square position
Variance and standard deviation
Electron in a cell
Energy levels
Measurements

Harmonic oscillator (Sect. 2.3.3)

Generalized formulation
SHO average values
Summary
Action quantization
Classical and BSW
Anharmonic well
Perturbation expansion
Accepted QM solution

Semiclassical Quantization (Sect. 2.3)

Semiclassical quantization
Chaos
Einstein, Deutsche Physikalische Gesellschaft 19,(1917) (translation)
EBK Quantization
EBK References
J.B. Keller, Ann.Phys. (NY) 4, 180 (1958).
Sommerfeld and Caustics
Caustic animation

Part II: The Kepler-Coulomb Potential (Sect. 2.4)

The probabilistic Kepler problem (Sect. 2.4.1)

Numerical solution
Kepler's laws
Kepler applet
perimeter
Kepler in probabilities
Conservation laws
The radial derivative
Kepler distribution

Average values of powers of r (Sect. 2.4.3)

Kepler average values
Azimuthal alternative
Laplace integral
Legendre polynomials
Classical values for rk
Symmetry about k=-3/2
Expansion of Rydberg's formula
Quantum values for rk
Atomic perturbations

The EKB Kepler problem (Sect. 2.4.4)

Spherical polar coordinates
Generalized momenta
Action formulation
Nonrel angle-action
Classical vs QM results
  • Cauchy's integral theorem
  • Poles of the Kepler problem
  • The angular libration integral
    EKB orbits, n=4
    EKB orbits, n=1,2,3
    Periapsis and apoapsis
    nd orbitals

    • EBK and QM radial wave functions (Sect. 2.4.5)

    Y(1s), Y(2s),Y(2p)
    Y(3s), Y(3p),Y(3d)
    Y(n=40,l=20)
    QM distributions
    Classical vs QM
    continued

    Recent publications

  • Curtis & Ellis, Use of the EBK action quantization, Am.J.Phys. 72, 1521 (2004)
  • Curtis & Ellis, Probabilities as a Bridge between Classical & QM Treatments, Eur.J.Phys. B 27 /B , 485 (2006)>
  • Larkoski, Ellis, & Curtis, Numerical implementation of the EBK quantization for arbitrary potentials, Am.J.Phys. 74, 572 (2006)

    Part III: Applications to Atoms and Planets (Sect. 2.4.7)

    Perturbed energy and Precession

    Mechanics and electrodynamics of moving bodies

    Maxwell's equations
    Woldemar Voigt's 1887 discovery
    Voigt bio
    Einstein, Ann. Phys, Chim 17, 890-921 (1905).
    English translation
    Length contraction and time dilation
    Model for a current in a wire
    Force between a moving charge and a current
    Reverse direction

    Relativistic corrections to the kinetic energy (Sect. 2.4.8)

    E=mc2
    Average value of KE-squared
    EBK values


    Magnetic moments

    Torque on a current loop
    Magnetic moment of an orbiting charge
    Magnetic moment of a Dirac electron
    Speed of an electron of finite radius
    Zitterbewegung of a point charge

    Relativistic corrections to the potential energy (Sect. 2.4.9)

    Spin-orbit fine structure
    Gen.Rel. and spin-orbit
    EKB and QM formulations
    Classical and QM k operators
    Strengths of internal fields
    Thomas precession g-1, not g/2

    Combining relativistic corrections

    Combining relativistic mass and spin-orbit
    Explicit calculations for spin 1/2
    Expansion of the Dirac equation

    Core polarization model (Sect. 2.4.10)

    Hydrogen, hydrogenlike, and Rydberg atoms
    The two-center problem
    Multipole expansion
    Multipole energy corrections
    Classical formulation
    EBK expectation values
    Na-like P IV example
    Polarization plot

    Planetary perturbations (Sect. 2.4.11)

    The two-center problem
    Potential due to a ring of charge
    Perturbations of the Planets
    Masses and orbital data
    Effects on the period of Mercury
    Test of relativity

    Part IV: Self-Consistent Fields for Many-Electron Atoms (Sect. 2.4.12)

    Semiclassical Self-Consistent Field Calculations

    Potential energy with screening
    1-D radial probability distribution
    Newton-Raphson method
    Schematic
    EBK for the Many Body Problem
    Y(QM) and Y(EBK)
    Comparison with HF for Na seq
    Artifact in Cu seq

    Relativistic Semiempirical Formulation (Sect. 2.4.13)

    Relativistic radial momentum
    In terms of alpha
    Repeat SCSCF
    Collapse of the Maslov Perihelion
    Calculated examples of turning points
    Redistribution
    Relativistic angle-action

    HOMEWORK ASSIGNMENT #1: Due Monday 24 September 2007

  • EBK quantization program
  • Average value program
  • Compare with exact solution
  • Curtis & Ellis, Probabilities as a Bridge between Classical & QM Treatments, Eur.J.Phys. 27, 485 (2006)
  • Larkoski, Ellis & Curtis, Numerical implementation of the EBK quantization for arbitrary potentials, Am.J.Phys. 74, 572 (2006)
  • SHO example
  • losning

    Part V: Time-Dependent Processes (Sect. 2.5)

    Semiempirical formulation of the decay meanlife

    Maxwell Radiation
    Classical vs QM

    Wien's model (Sect. 2.5.1)

    Correspondence limit
    Wien model
    Compare with experiment
    Expelling two action quanta
    Compare to QM
    Large n limit
    Worst case value
    Blending
  • Wien's paper (1)
  • Wien's paper (2)
  • Wien's paper (3)
    Noble employment
    Review

    • Oscillator stength (Sect. 2.5.2)

    Branched decay
    Express as oscillators
    Relationships between absorption and emission

    Chapt. 1
    Chapt. 3
    Chapt. 4
    Chapt. 5
    Chapt. 6
    Chapt. 7
    Chapt. 8
    Chapt. 9
    Chapt. 10
    Chapt. 11
    Chapt. 12
    Chapt. 13

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    Trojan asteroids

    Trojan asteroids (animation)
    Trojan asteroids (still)
    Wave packets of high Rydberg states
    Animation
    Whole number of heartbeats per cycle
    Box, SHO, Kepler
    Wavelengths of light emitted