A time-convolutionless expression for the quantal equation of motion for the linear response of a system in contact with a heat bath to an external, time-varying driving field is derived from the quantal Liouville equation of motion for the total system. This is done by renormalizing the memory effect by the method of Shibata, Takahashi and Hashitsume. It has a form convenient for the perturbational expansion. It is shown that this equation coincides with the time-convolution equation of Argyres and Kelley in their Markoffian approximation in the narrowing limit and in the lowest Born approximation for the system-bath interaction. This equation is expanded up to fourth order in powers of the system-bath interaction.
Time-convolution and time-convolutionless expressions for the quantal equation of motion for the linear response of a system under random perturbations to an external, time-varying driving field are derived from the stochastic quantal Liouville equation for the total system. They have forms convenient for the perturbational expansions in powers of the random perturbations. One is compared with another for two initial conditions. It is shown that the time-convolutionless form of equation prevails over the time-convolution form, especially when the effects of the random perturbations are discussed except in the narrowing limit or when the effects of interference of the external driving field and the random perturbations are discussed. The higher order expansions of these equations are mentioned.