New U -matrix theory in quantum mechanics

Abstract

We have analyzed Dyson's $U$-matrix theory of solving the Schr\"odinger equation in the interaction picture and are able to express the $U$ matrix as a dominant term plus an infinite series involving multiple integrals of time. For a certain rather restrictive class of Hamiltonians, our theory is exact for a general time-dependent problem. For other Hamiltonians, we can only obtain approximate expressions for our $U$ matrix and hence the wave function. Treating a time-independent problem as a special case of the time-dependent situation with a sudden-switching process, we have shown that our $U$ matrix is exact. To demonstrate the working procedures of our theory, we apply it to study the well-known time-independent charged harmonic-oscillator problem and the more general harmonic oscillator with a time-dependent driving force. Compared with other methods, our new theory appears to lead to a result which contains more information than others due to the inclusion of noncommutability properties of operators in the operator Schr\"odinger equation. It has been shown that the classical Feynman path-integral formalism can be deduced from quantum mechanics with the use of the Green's-function operator. It is interesting to note that apart from a step function, the Green's-function operator is the same as that of our ${U}^{(s)}$ matrix, which is the $U$ matrix obtained within the regime of the Schr\"odinger picture for a time-independent Hamiltonian, as a special case of our general time-dependent treatment.