Since the original 1926 Schroedinger's paper, there was a misconception that the “simple” harmonic oscillator can be solved only by the separation of variables, which results in a traditional “static” electron density distribution.It is not entirely accurate and a nontrivial oscillator hidden symmetry group, found by Niederer in 1973, provides "dynamic solutions". The phase space oscillations of the electron position and linear momentum probability distributions are computer animated and some possible applications to quantum optics are briefly discussed.
A visualization of the Heisenberg Uncertainty Principle is presented.
In addition, these "dynamic harmonic states" possess the nontrivial Berry phase, which may be use for their identification. The corresponding phase is evaluated in terms of elementary functions.
In view of importance of the simple harmonic oscillators in numerous applications, these results will be interesting to everybody who is going to study and/or teach quantum mechanics --- It may help better understand a general concept of quantum motion.