One-Dimensional Waves

In the Waves Paradigm, we study waves in one dimension, beginning with classical wave phenomena (waves on ropes, electrical circuits) and following with waves in quantum systems. We introduce the basic language to describe waves, and consider standing and traveling waves, wave packets and dispersion, and superposition. We explore energy transmission, reflection and transmission at sharp boundaries, and impedance. The course features an integrated laboratory where students study the propagation of a voltage pulse in a coaxial cable, several small-group problem solving activities, and an interactive in-class demonstration to study standing waves and to introduce the concept of a dispersion relation. (more...)

Textbook: Quantum Mechanics: A Paradigms Approach—-a textbook that follows the paradigms approach. The chapters that are relevant to the Waves course are:

5. Quantized Energies:Particle in a Box

6. Unbound States

Sample Syllabus: Winter 2012

Course Contents

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)

Traveling and standing waves: Non-dispersive wave equation & initial conditions (60 minutes)

Standing waves: Physics and the measurement of a dispersion relation (xx minutes)

Reflection, transmission and impedance (xx minutes)

Energy (50 minutes)

Superposition & Fourier analysis (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 finite (square) potential energy well (xx minutes)

The infinite well & superposition, measurement, probability etc. (xx minutes)

Time Evolution (xx minutes)

Unbound states, barriers & tunneling (xx minutes)

Heisenberg Uncertainty Principle & time evolution of a Gaussian wave packet (xx minutes)

Activities Included

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