Exact wave function of the coupled harmonic oscillator with time-dependent mass and frequency

In this paper we apply Wegner s flow equation method to study nonlinear harmonic oscillators and nonlinear-coupled harmonic oscillators systems.

Normal mode of coupled harmonic oscillator is obtained by means of algebra, the procedure is simple and the physical meaning is clear.

Divided energy of the double momentum coupling harmonic oscillator;

The energy splitting of the coupling harmonic oscillator in non-commutative spaces are discussed.

Exact solution for non-identical n modes coupled harmonic oscillators;

For the coupled harmonic oscillators,the energy spectrum of system relates with the coupling items and exists a minimum value,when(λp(0)/ωj(0))1 and(gi(0)/ωj(0))1,λp(0) and gi(0) are the anharmonic parameter,ωj(0) is the frequency of the harmonic oscillators,where i,j=a,b;p=1,2),the analytic equation is gained.