Exercises
Exercise 13A: Quantum Versus Classical Randomness- Imagine the classical version of the particle in a one-dimensional box. Suppose you insert
the particle in the box and give it a known, predetermined energy, but a random initial position
and a random direction of motion. You then pick a random later moment in time to see where
it is. Sketch the resulting probability distribution by shading on top of a line segment. Does
the probability distribution depend on energy? - Do similar sketches for the first few energy levels of the quantum mechanical particle in a
box, and compare with 1. - Do the same thing as in 1, but for a classical hydrogen atom in two dimensions, which acts
just like a miniature solar system. Assume you’re always starting out with the same fixed values
of energy and angular momentum, but a position and direction of motion that are otherwise
random. Do this forL= 0, and compare with a realL= 0 probability distribution for the
hydrogen atom. - Repeat 3 for a nonzero value ofL, say L=~.
- Summarize: Are the classical probability distributions accurate? What qualitative features
are possessed by the classical diagrams but not by the quantum mechanical ones, or vice-versa?
952 Chapter 13 Quantum Physics