Applications Of Path Integral In Study Of The Green Function Of One Dimensional System

Semiclassical approximation has comprehensive applications in modern physics. Among these approaches, WKB method and path integral approach are successful methods. WKB method is successful in dealing with the problem of barrier penetration, evaluating bound state energies. Path integral approach is another version of quantum mechanics equivalent to the standard scheme. In some literatures, another equivalence between the WKB method and the path integral approach has been discussed, a set of rules of constructing semiclassical green functions with respect to the position and energy which is the fourier transformation of the semiclassical path integral propagator with respect to position and time was established.We apply these set of rules to study three kind of typical quantum problems including harmonic oscillator, double well potential and impulse barrier penetration. In the harmonic oscillator and double well potential cases, we study the density of states of these two cases along with the trace formula which has extensive applications in the study of quantum chaos. In the impulse barrier penetration problem, we use the set of rules of constructing the semiclassical green function to obtain the transmission coefficient which is identical to the results evaluated via standard methods.