1. Spin: A discrete and finite system where the observables of interest are spin components \(S_n\) and energy. We use this simple system to introduce the quantum postulates and discussion of measurement.

    • Spin-1/2
    • Spin-1
    • Spin-3/2
    • Spin-s

  2. Intro to Bound States: With these systems, we introduce the position operator, the position representation of quantum states, and review all the quantum postulates.

    To connect with the previous unit on Spin, we emphasize:

    • measurement and finding the particle in a region
    • the position basis and the position operator
    • representing a quantum state with various representations
    • time evolution

    • Infinite Square Well: (i.e., Particle in Box) A 1-D potential where energy (discrete but unbounded) and position (continuous but bounded) are of interest.

    • Particle Confined to a Ring A 1-D potential where both orbital angular momentum and position are of interest. Has periodic boundary conditions and degeneracy in \(L^2\) and the energy.

    • Particle Confined to a Sphere (i.e., a quantum rigid rotor) A 2-D potential with periodic boundary conditions. Introduces the spherical harmonics. Separation of variables is needed to solve the energy eigenvalue equation.

    • Hydrogen Atom A central force potential with 3 spatial dimensions. Adds solutions to the radial equation.

  3. Raising & Lowering Operators
    • Angular Momentum
    • Harmonic Oscillator Review bound states for a 1-D system but solve for energy eigenstates using raising and lower operators.

  4. Free Particle: Normalization and the linear momentum basis.
  5. Intro to Scattering States:
    • Delta Function Barrier/Well
    • Finite Well/Barrier

    • Periodic Wells Introduces LCAO approximation (linear combination of atomic orbitals)