The question is: Prove that, for a 1-D harmonic oscillator, every conserved observable is a function of the energy. Find, for the 3-D harmonic oscillator, some conserved observable not a function of the energy and angular momentum. My first problem with this question is I'm not sure what a "conserved observable" is. I know that the state of a 1-D harmonic oscillator can be expressed by only one position component and one velocity component, and from that you can get a function of energy, but I don't know how to write a proof for this. Thanks.