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In Feynman's path integral, the propagator is sum over all
paths that particle go from one point to another. By the
relation on propagator and wave function, we can prove the
equivalence of path integral and Schrodinger equa- tion. When we
derived the Schrodinger equation, we discussed the situation
that the potential term in Lagrangian is velocity dependent or
not. From this,we can see the problems of ordering that
representation in quantum mechanics here. If the Lagrangian is
quadratic form, we can separate the propagator in to two parts,
one term is the amplitude that as a function of time only,
another is the phase factor which depend on slassical action.
Because of the prooperty that amplitude is as function of time
only, the complete form of propagator can be worked out
accurately, including amplitude and phase factor of the
propagator. Finally, by using the path integral and image
method, consider the particle traveling into the potential
well and potential barrier,the paths and phase problems for
particle will discuss. We firstly consider the potential is
infinite, then consider it is finite in the same treatment.
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