The Autocorrelation Function In Some Exactly Solvable Models

In quantum mechanics,there exist two parallel branches.One is about the static properties of a system,namely the eigenvalues and eigenstates of the Hamiltonian.The other is about the dynamics of a system,that is,how the wave function or the expectation values of various physical quantities evolve.Actually,many unexpected research results have been achieved for the latter.One of them is that in the quenching dynamics of the one-dimensional tight binding model,it is found that the probability of finding the particle in the initial state is an unsmooth curve with respect to time,specifically,they show cusps periodically in time.In this paper,in order to explain this strange dynamic phenomenon,the following two works have been done.First,we solve an exactly solvable toy model,and the nonsmooth change curves of the autocorrelation function and survival probability are obtained,which are consistent with the known results.In addition,the reason why the model is strictly solvable is analyzed.Second,the previous work was well explained by Fourier analysis,and tested by the infinite square well model.In addition,we find that the autocorrelation function has a fractional power law behavior in the short-time regime,which is closely related to the quantum Zeno effect and quantum anti-Zeno effect.With the help of Mellin transform,we have studied theoretically the short-time behavior and obtained innovative research results.Our research clearly shows the link between the autocorrelation function and Fourier analysis and Mellin transform.They respectively explains the strange phenomena in the quenching dynamics and the short-time behavior of the autocorrelation function,which helps to explore the unknown field of dynamics of more models.