  We consider a six-parameter family of the square integrable wave functions
for the simple harmonic oscillator, which cannot be obtained by the standard
separation of variables. They are given by the action of the corresponding
maximal kinematical invariance group on the standard solutions. In addition,
the phase space oscillations of the electron position and linear momentum
probability distributions are computer animated and some possible applications
are briefly discussed. A visualization of the Heisenberg Uncertainty Principle
is presented.
