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Activities

This activity gives links to some external resources (2 simulations and 1 video) that allow students to explore circle trigonometry. There are no prompts and nothing specific to turn in.

Find the rectangular coordinates of the point where the angle \(\frac{5\pi}{3}\) meets the unit circle. If this were a point in the complex plane, what would be the rectangular and exponential forms of the complex number? (See figure.)

Problem

5 min.

Circle Trigonometry

On the following diagrams, mark both \(\theta\) and \(\sin\theta\) for \(\theta_1=\frac{5\pi}{6}\) and \(\theta_2=\frac{7\pi}{6}\). Write one to three sentences about how these two representations are related to each other. (For example, see: this PHET)

This handout lists Motivating Questions, Key Activities/Problems, Unit Learning Outcomes, and an Equation Sheet for a Unit on Classical Mechanics Orbits. It can be used both to introduce the unit and, even better, for review.
  • Found in: Central Forces course(s)

In this unit, you will explore the quantum mechanics of a simple system: a particle confined to a one-dimensional ring.

Motivating Questions

  • What are the energy eigenstates, i.e. eigenstates of the Hamiltonian?
  • What physical properties of the energy eigenstates can be measured?
  • What other states are possible and what are their physical properties?
  • How do the states change if this system and their physical properties depend on time?

Key Activities/Problems

Unit Learning Outcomes

At the end of this unit, you should be able to:

  • Describe the energy eigenstates for the ring system algebraically and graphically.
  • List the physical measurables for the system and give expressions for the corresponding operators in bra/ket, matrix, and position representations.
  • Give the possible quantum numbers for the quantum ring system and describe any degeneracies.
  • For a given state, use the inner product in bra/ket, matrix, and position representations, to find the probability of making any physically relevant measurement, including states with degeneracy.
  • Use an expansion in energy eigenstates to find the time dependence of a given state.

Equation Sheet for This Unit

  • Found in: Central Forces course(s)

In this unit, you will explore the electrostatic potential \(V(\vec{r})\) due to one or more discrete charges and the gravitational potential \(\Phi(\vec{r})\) due to one or more discrete masses. How does the potential vary in space? How do equipotential surfaces and the superposition principle help you answer these questions graphically? How does the value of the potential fall-off as you move away from the charges? How do power series approximations help you answer these questions algebraically?

Key Activities/Problems

At the end of this unit, you should be able to:

  • Describe the important similarities and differences between the electrostatic potential and the gravitational potential.
  • Sketch the potential due to a small number of discrete charges or masses, showing important regions of interest and qualitatively depicting the correct spacing between equipotential surfaces (or curves).
  • Compute power and Laurent series expansions from a real-world problem using simple, memorized power series.
  • Truncate a series properly at a given order by keeping all the terms up to that order and none of the terms of higher order.
  • Discuss in detail the relationship between the graphical and algebraic representations of the potentials.

  • Found in: Static Fields course(s)
Students move their left arm in a circle to trace out the complex plane (Argand diagram). They then explore the rectangular and exponential representations of complex numbers by using their left arm to show given complex numbers on the complex plane. Finally they enact multiplication of complex numbers in exponential form and complex conjugation.

Small Group Activity

30 min.

Visualization of Curl
Students predict from graphs of simple 2-d vector fields whether the curl is positive, negative, or zero in various regions of the domain using the definition of the curl of a vector field at a point as the maximum circulation per unit area through an infinitesimal box surrounding that point. Optionally, students can use computer algebra to verify their predictions.
  • Found in: Static Fields, AIMS Maxwell, Vector Calculus II, Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Geometry of Vector Fields Sequence sequence(s)

Small Group Activity

30 min.

Visualization of Divergence
Students predict from graphs of simple 2-d vector fields whether the divergence is positive, negative, or zero in various regions of the domain using the geometric definition of the divergence of a vector field at a point as flux per unit volume (here: area) through an infinitesimal box surrounding that point. Optionally, students can use computer algebra to verify their predictions.
  • Found in: Static Fields, AIMS Maxwell, Vector Calculus II, Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Geometry of Vector Fields Sequence, Flux Sequence sequence(s)