Solution Of A Quantum Evolution Equation Using A Unitary Group Of Operators

ABSTRACT

An equation describing the change of the state of a quantum system with respect to time is called a time evolution equation. One of the fundamental evolution equation of quantum mechanics is the Schrodinger equation; a dynamical system that has yielded closed form solution only in specialized and simplified cases. By observing that the time-independent Hamiltonian for the system is provided by a one parameter group of unitary operators, we cast the evolution equation within the frame work of this group of unitary operators, and thus obtained close form solutions for the time-dependent case, the equation dynamics specified a Hamiltonian that is the sum of two terms, a perturbed and the unperturbed part. The solution for the perturbed part was obtained in the form of an integral equation which can be iteratively constructed. The exact solution for the unperturbed part was obtained and probability that at time t the system can be found in a specific state from it^’s initial state  was computed.