| In this work, we present the time-independent wavepacket (TIW) and the time-dependent wavepacket (TDW) theories for quantum dynamics in which the fundamental equations are the time-independent wavepacket Lippmann-Schwinger (TIWLSE), the time-independent wavepacket Schrodinger equation (TIWSE), and the time-dependent wavepacket equation expressed in terms of path integrals (PI). We show that the TIW theory leads to new insight into fundamental quantum theory, where the source is located at a finite distance from target. It retains the principal advantage of wavepacket propagation that the scattering information at many energies is obtained from a single wavepacket, while replacing the temporal-spatial evolution by a purely spatial evolution of wavepacket.; We also present the distributed approximating functional (DAF) representation for kinetic energy operator and free particle evolution operator. Physically, the DAF can be considered as a filter in momentum space and results in a filtered representation for the operators. The approach offers a very promising tool for quantum dynamics, because on a uniform grid, the discretized DAF leads to highly banded Toeplitz matrix representation.; The polynomial expansion (PE) method for Green and Green-like operators is developed. One of the main advantages of the PE method is the separation of the dependence on the energy and Hamiltonian.; The theoretical equations and method based on them are illustrated through numerical calculations of bound and resonance states, Monte Carlo path integration, and quantum scattering, especially atom-diatom reactive scattering. |
