This notebook has been created on 4 December by Sergei K. Suslov, Arizona State University, for simple Mathematica simulations to accompany the paper
“On the Harmonic Oscillator Group” by  R Lopez-Medina, SK Suslov and JM Vega-Guzman.
See:  http://arxiv.org/abs/1111.5569 

EXAMPLE 1
The simplest (dynamic ground state) wavefunction has the form:

Ψ[{x,t}]=Exp[I*(x*Cos[2*t]-(1/4)*Sin[4*t]-t)]*Exp[-(1/2)*(x-Sin[2*t])^2]

This wavefunction (which cannot be obtained by the separation of variables) does satisfy the time-dependent Schrõdinger equation for the harmonic oscillator. The corresponding eigenvalue problem for the quadratic invariant holds.

{Ψ=Exp[I*(x*Cos[2*t]-(1/4)*Sin[4*t]-t)]*Exp[-(x-Sin[2*t])^2/2],
DTΨ=D[Exp[I*(x*Cos[2*t]-(1/4)*Sin[4*t]-t)]*Exp[-(x-Sin[2*t])^2/2],t],
DXΨ=D[Exp[I*(x*Cos[2*t]-(1/4)*Sin[4*t]-t)]*Exp[-(x-Sin[2*t])^2/2],x],
DXXΨ=D[Exp[I*(x*Cos[2*t]-(1/4)*Sin[4*t]-t)]*Exp[-(x-Sin[2*t])^2/2],{x,2}]};
{Schroedinger=I*DTΨ+DXXΨ-x^2*Ψ,
QuadraticInvariant=-DXXΨ+2*I*Cos[2*t]*DXΨ+x^2*Ψ-2*Sin[2*t]*x*Ψ};
Factor[%]
FullSimplify[%]

The probability density is given by

ρ[{x,t}]=Exp[-(x-Sin[2*t])^2]

Please run the following (one and two period) animations for this “dynamic ground state”:

ListAnimate[Table[Plot[(E^(-(((x-Sin[(3.1415926/1000) (-1+T)]))^2))),{x,-3.5,3.5},Filling->Bottom],{T,2001}]]

ListAnimate[Table[Plot[(E^(-(((x-Sin[(3.1415926/1000) (-1+T)]))^2))),{x,-3.5,3.5},Filling->Bottom],{T,4001}]]

(It is rotating, sorry oscillating!)