Before learning the quantum harmonic oscillator, you should
understand the classical harmonic oscillator, reviewed here. See
page 402 in the text.
Consider a spring of unit force constant connecting two masses, m1 and m2. They are at rest at unit distance separation. At time zero, they are pulled apart a fractional distance X and released. The top figure animates their motion in time. The red cross locates the center of mass of the system. The black crosses locate the rest positions of the masses. Mass m1 is on the left, and m2 is on the right. The green spots locate their positions at t = 0. Select values for the masses and for X, then click the graph to animate the oscillation. Try to predict where the center of mass will move as you change the masses, and try to predict how the masses will move.