Invariants for Time-Dependent Harmonic Oscillators from the Real Representation of Solution

The invariants for the time-dependent one-dimensional harmonic oscillator and the time-dependent two-dimensional harmonic oscillator in a static magnetic field are derived from the real representation of the exact solution to the equation of motion. Mathematically, the orthogonal functions invariant is the angular momentum of an isotropic time-dependent two-dimensional harmonic oscillator. Based on the invariants obtained here, the wave function for time-dependent two-dimensional harmonic oscillator in a static magnetic field in cylindrical coordinate is simply derived and the dynamical and geometrical phases are easily got by expressing the wave function as the superpositions of the wave functions of time-dependent two-dimensional harmonic oscillator in rectangular coordinate. For the driven system, the driving induced dynamical phase and the geometrical phase are respectively associated with the classical Hamiltonian and de Broglie wave of the center motion of the wave function.