Consider a wave function for the circular billiard system that at t = 0 is a superposition of the two lowest stationary states: For simplicity, suppose that both a and b are real, positive, and...

Consider a wave function for the circular billiard system that at t = 0 is a superposition of the two lowest stationary states:

For simplicity, suppose that both a and b are real, positive, and non-zero.

(a) Show that the initial probability distribution |ψ|²
is not circularly symmetric. In which half of the circle is the particle most likely to be found?

(b) Show that over time |ψ|²
moves around the circle. How long does it take to return to the original probability distribution?

If you have access to the necessary computer software tools, create an animation of the dynamical behavior of |ψ|²
over time, given a = b.