## Finding the Eigenstates for the Ring Lecture (30 minutes)

Central Forces Notes Section 17

• When introducing the ring problem, it is useful to use a prop, e.g., a hula hoop, to clarify the geometry.
• It is also useful to stress that for our purposes, the ring is a toy problem that will help us build up to being able to solve the more complex hydrogen atom problem.
• If students have done particle-in-a-box and have some experience with the relationship between sines and exponentials, this derivation should be relatively quick. The only new feature is the periodic boundary conditions. Pay particular attention to the fact that whether the solutions are oscillatory or exponentially damped/growing (real exponentials) depends on the sign of the separation constant.
• Many students are still struggling with recognizing that two equations are mathematically the same if they are written in different variables. Writing the equation and solutions for particle-in-a-box in parallel with the equation for a ring will help them. Alternatively, if you have the class time, write the equation for the ring and ask a small whiteboard question asking the students to write down the solution. Compare and contrast the various answers that the students give.
• Emphasize that exponentials really are easier to use than sines once you get used to them. Weaker students need some convincing.

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