Students see probability density for eigenstates and linear combinations of eigenstates for a particle on a ring. The three visual representations: standard position vs probability density plot, a ring with colormapping, and cylindrical plot with height and colormapping, are also animated to visualize time-evolution.

Students calculate probabilities for a particle on a ring using three different notations: Dirac bra-ket, matrix, and wave function. After calculating the angular momentum and energy measurement probabilities, students compare their calculation methods for notation.

Students calculate probabilities for energy, angular momentum, and position as a function of time for an initial state that is a linear combination of energy/angular momentum eigenstates for a particle confined to a ring written in bra-ket notation. This activity helps students build an understanding of when they can expect a quantity to depend on time and to give them more practice moving between representations.

Students calculate probabilities for a particle on a ring whose wavefunction is not easily separated into eigenstates by inspection. To find the energy, angular momentum, and position probabilities, students perform integrations with the wavefunction or decompose the wavefunction into a superposition of eigenfunctions.

Students calculate the expectation value of energy and angular momentum as a function of time for an initial state for a particle on a ring. This state is a linear combination of energy/angular momentum eigenstates written in bra-ket notation.

Students observe the motion of a puck tethered to the center of the airtable. Then they plot the potential energy for the puck on their small whiteboards. A class discussion follows based on what students have written on their whiteboards.

In this course, we will examine a mathematically tractable and physically useful problem - that of two bodies interacting with each other through a central force, i.e. a force that has two characteristics:

Definition of a Central Force:

1. A central force depends only on the separation distance between the two bodies,
2. A central force points along the line connecting the two bodies.

The most common examples of this type of force are those that have \(\frac{1}{r^2}\) behavior, specifically the Newtonian gravitational force between two point (or spherically symmetric) masses and the Coulomb force between two point (or spherically symmetric) electric charges. Clearly both of these examples are idealizations - neither ideal point masses or charges nor perfectly spherically symmetric mass or charge distributions exist in nature, except perhaps for elementary particles such as electrons. However, deviations from ideal behavior are often small and can be neglected to within a reasonable approximation. (Power series to the rescue!) Also, notice the difference in length scale: the archetypal gravitational example is planetary motion - at astronomical length scales; the archetypal Coulomb example is the hydrogen atom - at atomic length scales.

The two solutions to the central force problem - classical behavior exemplified by the gravitational interaction and quantum behavior exemplified by the Coulomb interaction - are quite different from each other. By studying these two cases together in the same course, we will be able to explore the strong similarities and the important differences between classical and quantum physics.

Two of the unifying themes of this topic are the conservation laws:

• Conservation of Energy
• Conservation of Angular Momentum

The classical and quantum systems we will explore both have versions of these conservation laws, but they come up in the mathematical formalisms in different ways. You should have covered energy and angular momentum in your introductory physics course, at least in simple, classical mechanics cases. Now is a great time to review the definitions of energy and angular momentum, how they enter into dynamical equations (Newton's laws and kinetic energy, for example), and the conservation laws.

In the classical mechanics case, we will obtain the equations of motion in three equivalent ways,

• using Newton's second law,
• using Lagrangian mechanics,
• using energy conservation.

so that you will be able to compare and contrast the methods. The Newtonian approach is the most straightforward and naive, but it suggests changes of coordinates that inform the other methods. The Lagrangian and energy conservation approaches are slightly more sophisticated in that they exploit more of the symmetries from the beginning.

We will also consider forces that depend on the distance between the two bodies in ways other than \(\frac{1}{r^2}\) and explore the kinds of motion they produce.

• Physics Content Learning Objectives
  1. Explain the consequences of energy and angular momentum conservation in a system of interacting particles.
  2. Use effective potential diagrams to determine properties of classical orbits.
  3. Solve for the quantum properties of a particle confined to a ring, rigid rotor, and the hydrogen atom in several different representations.
  4. Relate the state of a quantum system (ring, rigid rotor, hydrogen atom) to graphs of a wave function.
  5. Apply Schrödinger time dependence to central force systems (ring, rigid rotor, H atom).
• Mathmatics Content Learning Objectives
  1. Solve ordinary differential equations using power series methods.
  2. Use eigen expansions as an orthonormal basis.
  3. Solve the initial value problem for partial differential equations with more than one spatial variable.
• Professional Learning Objectives
  1. Communicate scientific ideas in writing and with other representations (e.g. graphs, code), using good scientific language and practices, concisely and without ambiguity.
  2. Learn to work with and communicate productively and respectfully with peers and collaborators of different backgrounds.
  3. Cite the information and ideas obtained from or with others in a clear and professional manner.
  4. Communicate in a timely and professional manner with others in the work environment when things don't go to plan.

(Quick) Purpose: Quickly recognize a consequence of central forces.

If a central force is the only force acting on a system of two masses (i.e. no external forces), what will the motion of the center of mass be?

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.

(Quick) Purpose: Recognize the definition of a central force. Build experience about which common physical situations represent central forces and which don't.

Which of the following forces can be central forces? which cannot? If the force CAN be a central force, explain the circumstances that would allow it to be a central force.

1. The force on a test mass \(m\) in a gravitational field \(\vec{g~}\), i.e. \(m\vec g\)
2. The force on a test charge \(q\) in an electric field \(\vec E\), i.e. \(q\vec E\)
3. The force on a test charge \(q\) moving at velocity \(\vec{v~}\) in a magnetic field \(\vec B\), i.e. \(q\vec v \times \vec B\)

Students consider projectile motion of an object that experiences drag force that in linear with the velocity. Students consider the horizontal motion and the vertical motion separately. Students solve Newton's 2nd law as a differential equation.