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:
- A central force depends only on the separation distance between the two bodies,
- 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:
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.
- Conservation of Energy
- Conservation of Angular Momentum
In the classical mechanics case, we will obtain the equations of motion in three equivalent ways,
so that you will be able to compare and contrast the methods. The third approach is slightly more sophisticated in that it exploits more of the symmetries from the beginning.
- using Newton's second law,
- using Lagrangian mechanics,
- using energy conservation.
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.
group Small Group Activity
120 min.
assignment_ind Small White Board Question
30 min.
Angular Momentum Spin Magnetic Moment Stern-Gerlach Experiments
Students consider the relation (1) between the angular momentum and magnetic moment for a current loop and (2) the force on a magnetic moment in an inhomogeneous magnetic field. Students make a (classical) prediction of the outcome of a Stern-Gerlach experiment.group Small Group Activity
30 min.
energy conservation mass conservation collision
Groups are asked to analyze the following standard problem:
Two identical lumps of clay of (rest) mass m collide head on, with each moving at 3/5 the speed of light. What is the mass of the resulting lump of clay?
face Lecture
120 min.
ideal gas particle in a box grand canonical ensemble chemical potential statistical mechanics
These notes from week 6 of Thermal and Statistical Physics cover the ideal gas from a grand canonical standpoint starting with the solutions to a particle in a three-dimensional box. They include a number of small group activities.computer Mathematica Activity
30 min.
assignment Homework
group Small Group Activity
10 min.
groups Whole Class Activity
10 min.
face Lecture
30 min.
face Lecture
120 min.
Planck distribution blackbody radiation photon statistical mechanics
These notes from the fourth week of Thermal and Statistical Physics cover blackbody radiation and the Planck distribution. They include a number of small group activities.