Abstract: Quantum mechanics is a physical theory which describes the behavior of microscopic matter. Moreover, it is the foundation of many physical branches. In fact, under the influence of external time-dependent perturbation, microscopic particles can undergo quantum transitions between different stationary states, which has attracted great research interest. According to time-dependent perturbation theory, the probability of microscopic particles transitioning from one stationary state to another per unit time is called the transition rate. For the constant perturbation, transition rate formula derived by the first-order perturbation approximation is time independent, which is the so-called Fermis golden rule and is widely used in theoretical research in physics, chemistry, biology, and other disciplines, especially in the field of quantitative computing. In traditional quantum mechanics text books, the high-order perturbation corrections are often ignored, and the transition probability or transition rate is derived by the low-order perturbation approximation. In the multi-levels systems, if the first-order perturbation is zero, or the Fermi golden rule is no longer applicable. For this complex situation, it is necessary to go beyond the Fermis golden rule and consider the impact of high-order time-dependent perturbation on the transition rate. On the basis of the time-dependent perturbation theory, both constant perturbation and periodic time-dependent perturbation are considered in this paper and the influence of high-order correction terms on the transition rate is attained. Furthermore, it is shown that intermediate states which satisfy certain conditions between the initial and final states can have an impact on the transition rate when the high-order perturbation approximation are considered. Therefore, the results can reflect the actual quantum transition process more comprehensively.

Key words: quantum mechanics, time-dependent perturbation theory, fermi's golden rule, higher order perturbation approximation