Tae Jun ParkDepartment of Chemistry, Dongguk University, Seoul 100-715, KoreaReceived September 19, 2002We solve the Schrodinger equation analytically for systems whose potentials have a certain time-dependence(which is monotonously decreasing) and general coordinate-dependences. Only a few time-dependent systemshave been reported to be analytically solved whose potentials are constant, linear, and quadratic functions ofcoordinate with arbitrary time-dependences. From a different perspective, we focus on the time-dependentsystems whose potentials are monotonously decreasing functions of time with arbitrary coordinate-dependences. Time-dependent potential of any coordinate-dependence can be handled analytically bytransforming it to a time-independent potential of known solutions if its time-dependence is monotonouslydecreasing. We do this by a unitary transformation of the wavefunction and variable transformations to changethe Schrodinger equation to be time-independent in new variables. These variables are then determined bysolving a set of simple differential equations. This way we are able to find and to obtain analytical solutions fortime-dependent potentials which we mention above. Key Words : Time-dependent, Eckart, BarrierIntroductionTime-dependent Schrodinger equations are not generallysolvable in closed forms even in one-dimension despitemuch theoretical attention has been paid.