Chapter 4 - Quantum Mechanics in Three Dimensions

Semiclassical formulation: The probabilistic Kepler problem

  • A curious point
  • Kepler's laws
  • 3rd law from circular case
  • Kepler applet
  • Virial theorem
  • Generalized
  • Circumference of an ellipse
  • Numerical solution
  • Radial probabilities
  • Conservation laws
  • The radial derivative
  • Position probability density

    • Average values of powers of r

  • Kepler average values from SHO solution
  • Kepler average values from azimuthal distribution
  • Laplace integral
  • Legendre polynomials
  • Classical values for rk
  • Quantum values for rk
  • Classical and QM
  • n dependence of powers
  • Quantum defect method
  • Summary
  • Atomic perturbations

    • The EKB Kepler problem

  • Spherical polar coordinates
  • Generalized momenta
  • Action formulation
  • Nonrel angle-action
  • Classical and QM observables
  • EBK ellipses and probabilities
  • EBK orbits, n=1,2,3
  • EBK orbits, n=4
  • Periapsis and apoapsis
  • nd orbitals
  • Balmer series
  • Electron configurations

    Contour integration (Griffiths pp. 410-411)

  • Cauchy's integral theorem
  • Poles of the Kepler problem
  • The Balmer energy
  • The angular libration integral

    • EBK and QM radial wave functions

  • EBK Summary
  • Y(1s), Y(2s),Y(2p)
  • Y(n=40,l=20)
  • Orbital distributions
  • QM Distributions
  • Y(3s), Y(3p),Y(3d)
  • Morphologies
  • Eta Carina

  • More morphologies
  • High res
  • Yet another morphology
  • The flying spaghetti monster

    • Exercises: numerical solution

  • Numerical solution
  • Kepler Exercise

  • Curtis & Ellis, Am. J. Phys. 72, 1521 (2004).
  • Curtis & Ellis, Eur. J. Phys. 27, 485 (2006)
  • Larkoski, Ellis & Curtis, Am. J. Phys. 74, 572 (2006)

    Quantum Mechanical Formulation

    Solution in spherical polar coordinates (Griffiths pp. 131-144)

  • Separation of variables
  • Solutions for 4 equations
  • Quantization and operators
  • Legendre equation
  • Legendre polynomials
  • Associated Legendre polynomials
  • Spherical harmonics
  • Uncertainty relations

    • Coulomb Potential (Griffiths pp. 145-159)

  • Natural units
  • Ansatz for 2-term recursion
  • Laguerre solution
  • Laguerre polynomials

    The Isotropic Harmonic Oscillator

  • Connection to isotropic SHO
  • Solution in series

    • Rigid Rotors

  • Vibration and rotation
  • Spherical, symmetric, and asymmetric tops
  • Energy levels
  • Transitions

  • French mathematicians

    Angular Momentum

    Elementary examples

  • Precession of a top
  • Billiard cushions
  • Billiards with impulse
  • Billiards w/o forces
  • Bowling with friction
  • Bowling alleys w/o forces
  • Physical pendulum
  • Walking

    The Schrödinger approximation (Griffiths pp. 160-170)

  • Two relativistic limits of the Dirac equation
  • Wave equations
  • Approximations

  • Kepler Exercise
    Compute: EBK avg values, & compare/exact

    Electron spin and magnetic moment (Griffiths pp. 171-172)

  • Angular momentum and magnetic moment
  • "Classical" g-factor
  • Tangential speed
  • Zitterbewegung
  • Foldy

    • Pauli spin matrices (Griffiths pp. 173-178)

  • Pauli spin matrices
  • Philippine wine dance
  • Spin and rotation
  • Low-order spin doctors

    Mixed eigenstates

  • Port Lawrence and Vistula Meadows
  • Cherry Street Transformation
  • K-meson regeneration
  • Impulsive excitation quantum beats
  • continued
  • Entanglement

    Internal magnetic fields

  • J, L and S
  • Strengths of internal fields
  • Nuclear vs electronic

    • Coupling approximations (Griffiths pp. 184-188, 210-218)

  • LS and jj coupling
  • 6-j diagrams
  • Addition of angular momentum
  • Spectroscopic nomenclature
  • Application to helium
  • Application to 1s2p and 1s3d
  • Singlet-triplet Grotrian diagram

    • The connection between spin and statistics (Griffiths pp. 203-210)

  • Adding two spin 1/2 momenta
  • The exclusion principle
  • Exchange symmetry

    • Field effects in bulk matter

  • The origins of para- and diamagnetism
  • Ferromagnetism - exchange on the half-shell
  • Hund's rule, exclusion and condensation
  • The field vectors D, E, H, B