Unit: Classical waves in non-dispersive systems: ropes and coaxial transmission lines
This unit illustrates standing and traveling waves in the context of classical mechanics (waves in a rope) and classical electromagnetism (charge and voltage oscillations in a coaxial cable). The idea of a dispersion relation is introduced. Propagation, reflection and transmission at an abrupt boundary, superposition, attenuation, and energy are explored.
Basic concepts (50 minutes)
- Basic Language (Lecture, 10 minutes)
- Standing & Traveling Waves (Lecture, 30 minutes)
Traveling and standing waves: Non-dispersive wave equation & initial conditions (60 minutes)
- The (non-dispersive) wave equation & separation of variables (Lecture, 30 minutes)
- Initial Conditions for a function of 2 variables (Small Group Activity, 30 minutes)
Standing waves: Physics and the measurement of a dispersion relation (xx minutes)
- Dispersion Relation for Waves on a String (Interactive Demo, 15 minutes)
- Newton's Law applied to a Rope under Tension (Lecture, xx minutes)
Reflection, transmission and impedance (xx minutes)
- Reflection and Transmission (Lecture, 35 minutes)
- Reflection and Transmission Animation (Simulation, 15 minutes)
- Creating a Pressure-like wave in a Coaxial Cable (Kinesthetic Activity, 15 minutes)
- Wave Propagation in a Coaxial Cable (Laboratory Activity, 2 hours)
- Displacement vs. Force Waves (Simulation, 15 minutes)
- Attenuation (Lecture, 30 minutes)
Energy (50 minutes)
- Energy Density (Lecture, 15 minutes)
- Energy Density of Waves on a String (Small Group Activity, 35 minutes)
Superposition & Fourier analysis (xx minutes)
- Fourier Superposition (Lecture, xx minutes)
- Boundary Conditions and Fourier Superposition (Small group activity, xx minutes)
- Wave packets (Class Activity, xx minutes)
Unit: Quantum waves: The Schrödinger equation
This unit examines quantum systems in the context of the Schrödinger equation, which is an example of a dispersive wave equation. We discuss mostly bound states, but also unbound states. The students draw on their knowledge of quantum systems from the Spins paradigm, and extend it to the case of a continuum of observables (position), which allows a discussion of probability density and the probability of locating a particle in a particular region of space. The other lessons from the quantum postulates encountered in the Spins paradigm (measurement, superposition, time evolution etc.) are re-examined in this new language.
The wave function (xx minutes)
- The wave function (Lecture, 10 minutes)
- Probability density (Lecture, 10 minutes)
- Operators (Lecture, 5 minutes)
- Operators and functions (Small Group Activity)
The finite (square) potential energy well (xx minutes)
- Eignevalues and eigenfunctions of the finite potential energy well (Small Group Activity)
- The finite well (Lecture, xx minutes)
The infinite well & superposition, measurement, probability etc. (xx minutes)
- Superposition & Measurement (Lecture, xx minutes)
Time Evolution (xx minutes)
- The time-dependent Schrödinger equation (Lecture, xx minutes)
- Time evolution of infinite well solutions(Activity, Maple Worksheet)
Unbound states, barriers & tunneling (xx minutes)
- Barriers and Tunneling, unbound states (Lecture, xx minutes)
Heisenberg Uncertainty Principle & time evolution of a Gaussian wave packet (xx minutes)
- Heisenberg Uncertainty Principle (Lecture, xx minutes)
- Time evolution of a Gaussian wave packet (Maple Worksheet)
