Exact Solution For A Harmonic Oscillator With A Time-dependent Inverse Square Potential By Path Integral Method And Its Geometry Phase

Harmonic oscillators are important models in physics for its wide applications in various areas and provid an exactly solvable example.Especially for time-dependent harmonic oscillator,it is used in such as molecular physics,quantum chemistry,quantum optics,plasma physics.and quantum field theory,in which many quantum-mechanical effects are treated phenomeno-logically by means of the time-dependent parameters in the Hamiltonian of a general time-dependent oscillator.Recently,much more attention was paied to the research of geometry phase which acquired from time-dependent quantum system evolving along a closed cycle.it maybe reveal the basic characteristic inherited in quantum field.In this paper ,we considered a kind of important time-dependent harmonic oscillator and finally gave its propagator and exact wave function by means of time-dependent coordinate transformation and path integral methods based on works some people had done before.The same result also has been obtained via diverse methods which prove our result's correction.Furthermore,we find that the total time derivative of a function in Lagrangian has some relationship with phase factor of the wave function.However,we know that it never result in any effect on motive equation with the same Lagrangian within the field of classic mechinics.For the purpose of finding whether this phase factor is arbitary we tried to compute the geometry phase using A-A formula with the help of Legendre transformation which transformed Lagrangian into Hamiltonian.Finally,we quoted a special case which partly proved our supposition.