Index for 5½ Examples in Quantum Mechanics

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  1. Falling: Motion in a Linear Potential

    1. One Dimensional
      1. classical motion: PE, turning points, z(t), etc.
      2. length and energy scales
      3. Schrödinger's equation solution: Airy function
      4. graphs of solutions
      5. connecting classical variables and the wavefunction: WKB approximation
      6. QM "motion", Heisenberg uncertainty relation
      7. mathematical details of the above: superposition, time dependence
      8. approximation methods: WKB, Rayleigh-Ritz (variational), perturbation theory

    2. Two Dimensional
      1. classical ballistic (projectile) motion
      2. separation of variables
      3. QM force-free motion: a moving lump of probability density
      4. visualizing the moving lump of probability density
      5. falling motion (again)
      6. fire the quantum cannon

    3. Problems

  2. Simple Harmonic Oscillator

    1. One Dimensional
      1. classical motion: PE, turning points, x(t), etc.
      2. length and energy scales
      3. Schrödinger's equation solution: Hermite polynomials
      4. graphs of solutions
      5. connecting classical variables and the wavefunction: WKB approximation
      6. QM "motion", Heisenberg uncertainty relation
      7. mathematical details of the above: superposition, time dependence
      8. raising and lowering operators

    2. Two Dimensional
      1. Schrödinger's equation solutions: xy 1D products, r: Laguerre polynomials
      2. degeneracy
      3. QM "motion": an orbiting wavefunction

    3. Three Dimensional
      1. Schrödinger's equation solutions: xyz 1D products, r: Laguerre polynomials
      2. visualizing the wavefunctions in 3D
      3. WKB approximation in 3D
      4. second order perturbation theory & degeneracy

    4. Problems

  3. Hydrogen Atom
    1. classical motion: PE, turning points, Kepler's Laws, r(t), etc.
    2. length and energy scales
    3. Schrödinger's equation solutions: Laguerre polynomials
    4. radial plots of wavefunctions
    5. visualizing the wavefunctions in 3D
    6. two electron atoms
    7. exchange symmetry and spin-statistics
    8. spin-statistics theorem
    9. Aufbau: building up multi-electron atoms
    10. Stark effect
    11. crystal field theory
    12. chemical bonding

    13. problems

  4. Square Wells

    1. One Dimensional
      1. classical motion: PE, turning points, x(t), etc.
      2. length and energy scales
      3. Schrödinger's equation solution: infinite square well
      4. finite square well: bound state solutions
      5. scattering from finite square well: reflection, transmission
      6. square barrier: tunneling

    2. Two Dimensional
      1. infinite rectangular square-well: xy separation
      2. round square-well: Bessel functions
      3. round finite square-wells: bound states
      4. reflection and refraction
      5. scattering from round infinite barrier: phase shifts, cross-section
      6. scattering from round finite square-well: rainbows, resonance

    3. Three Dimensional
      1. infinite 3D rectangular square-well: xyz separation
      2. spherical square-well: spherical Bessel functions
      3. visualizing the wavefunctions in 3D
      4. spherical finite square-wells: bound states
      5. scattering from spherical infinite barrier: solid angle, phase shifts, cross-section
      6. scattering from spherical finite square-well: glory, resonance

    4. Problems

  5. Delta Function Potentials
    1. delta function basics
    2. dimensionless units for delta function potential
    3. isolated "atom"
    4. diatomic "molecule": bound states
    5. finite crystal: bound states
    6. theory for one cell after another
    7. bands in 1D crystal
    8. quantum motion in finite crystal
    9. scattering from a finite crystal

    10. Problems

  6. Electrons in a Lattice: Band Structure
    1. lattice & lattice vectors; classical motion: PE, turning points, etc.
    2. Block wavefunctions; Schrödinger's equation; length and energy scales
    3. reciprocal lattice space
    4. grids: approximation to Schrödinger's equation in real space
    5. band structure in an empty lattice (i.e., free particle)
    6. numerical and exact solution to grided empty lattice
    7. numerical band structure for various potentials
    8. wavefunctions for various potentials
    9. Plane wave expansion aproximation
    10. discussion: relationship to classical considerations

    11. Problems

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