Progression: Quantum Systems

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

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.

  

• Delta Function Barrier/Well

• Finite Well/Barrier

  

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

  

• Angular Momentum

• Harmonic Oscillator Review bound states for a 1-D system but solve for energy eigenstates using raising and lower operators.