Comparison of the path-integral and the U -matrix theories
P. C. W. Fung, C. C. Lam · Physical Review A · 1984
We have previously developed a basic $U$-matrix formalism and have been able to obtain explicit expressions for the $U$ matrix and the wave function of a quantum-mechanical system by solving the operator Schr\"odinger equation directly within the regime of the interaction picture [Phys. Rev. A 27, 1760 (1983)]. In our approach, we introduce the switching function ${S}_{w}$ which describes how the interaction Hamiltonian is introduced to the system. In the present investigation, we list the corresponding $U$ matrix and wave function pertaining to the Schr\"odinger picture. Under the special case where the unperturbed Hamiltonian and the Lagrangian commute, our wave function can be reduced to the form identical to that obtained from the path-integral method. A comparison of the path-integral theory and our $U$-matrix theory is presented. The difference in the wave functions of the two theories lies in the fact that in the $U$-matrix approach the Schr\"odinger equation is treated as an operator equation, whereas the key equation in the path-integral theory is taken as algebraic.