We present the exactsolution of the Schr dinger equation of an oscillator with mass,frequency and centripetal barrierall vorying with time though a canonical transformation and give a correct representation for the ground state energy.
The time-dependent Schrdinger equation for the harmonic oscillator with time-dependent frequency ω=ω(t) and mass m=m(0)ω(0)ω(t)exp[2c∫t0ω(t)dt] can be solved by means of simple transformation and separation of variables.The wave functions are obtained for c21,c2=1 and c21,respectively.When |x|→∞,the wave function for c21 tends to zero but it is not so for c2≥1.Finally,some special cases are given for the time-dependent harmonic oscillator and one of them is similar to a damped harmonic oscillator.