Activities
In this unit, you will explore the most common partial differential equations that arise in physics contexts. You will learn the separation of variables procedure to solve these equations.
Motivating Questions
- How are partial differential equations (PDEs) different from ordinary differential equations (ODEs)?
- What new kinds of physics can we learn from solving partial differential equations?
- What can we learn about physics and geometry from the separation of variables procedure?
Key Activities/Problems
- Problem: Laplace's Equation in Polar Coordinates
- Activity: https://paradigms.oregonstate.edu/activity/2084
Unit Learning Outcomes
At the end of this unit, you should be able to:
- Identify and classify several common PDEs.
- Identify the number and types of boundary and initial conditions that are appropriate to different PDEs.
- Identify the conditions where the separation of variables is appropriate and useful.
- Solve simple partial differential equations through the separation of variables procedure.
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
- Activity: Working with Representations on the Ring
- Problem: Ring Table
- Activity: Visualization of Quantum Probabilities for a Particle Confined to a Ring
- Activity: Time Dependence for a Quantum Particle on a Ring
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
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
- Drawing Equipotential Surfaces
- Electrostatic Potential Due to a Pair of Charges (with Series)
- Linear Quadrupole
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.
In this unit, you will explore the classical mechanics of central forces, especially gravitational orbits like the earth going around the sun.
Motivating Questions
- What shapes can the orbits have?
- What are Kepler's laws and why are they true?
- What is an effective potential diagram and how can it be used to predict the shape of an orbit?
Key Activities/Problems
- Problem: Undo Formulas for Center of Mass (Geometry)
- Activity: Acting Out Effective Potentials (In class, only)
- Activity: Effective Potentials
- Problem: Hockey
- Problem: Scattering
Unit Learning Outcomes
At the end of this unit, you should be able to:
- List the properties that define a central force system.
- Calculate a reduced mass for a two-body system and describe why it is important.
- Use the solution (algebraic or geometric) to a reduced mass system to describe the motion of the original system.
- Describe the role that conservation of energy and angular momentum play in a central force system. In particular, where do these properties appear in the solutions of the equations of motion?
- Use an effective potential diagram to predict the possible orbits in a central force system: which orbits are bound or unbound? which are closed or open? where will the turning points be?
Equation Sheet for This Unit