Exact solution and perturbation theory in a general quantum system

Abstract

By splitting a Hamiltonian into two parts, using the solvability of eigenvalue problem of one part of the Hamiltonian, proving a useful identity and deducing an expansion formula of power of operator binomials, we obtain an explicit and general form of time evolution operator in the representation of solvable part of the Hamiltonian. Further we find out an exact solution of Schrödinger equation in a general time-independent quantum system, and write down its concrete form when the solvable part of this Hamiltonian is taken as the kinetic energy term. Comparing our exact solution with the usual perturbation theory makes some features and significance of our solution clear. Moreover, through deriving out the improved forms of the zeroth, first, second and third order perturbed solutions including the partial contributions from the higher order even all order approximations, we obtain the improved transition probability. In special, we propose the revised Fermi's golden rule. Then we apply our scheme to obtain the improved forms of perturbed energy and perturbed state. In addition, we study an easy understanding example to illustrate our scheme and show its advantage. All of this implies the physical reasons and evidences why our exact solution and perturbative scheme are formally explicit, actually calculable, operationally efficient, conclusively more accurate. Therefore our exact solution and perturbative scheme can be thought of theoretical developments of quantum dynamics. Further applications of our results in quantum theory can be expected.