_{1}

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We present the problem of the time-dependent Harmonic oscillator with time-dependent mass and frequency in phase space and by using a canonical transformation and delta functional integration we could find the propagator related to the system. New examples of time-dependent frequencies are presented.

Recently, a great deal of attention has been paid to the subject of time dependent Hamiltonians. The importance of this problem in various areas of physics, quantum optics [

Looking through the literature one can find that an explicit expression for the propagator could not be obtained for all time varying mass-functions or frequencies because the procedure involves the solutions of non- linear differential equations. This is the reason why the literature is not reached by many exactly evaluated systems, which has many applications in physics [

In this paper we will present a way to find the propagators of the time dependent harmonic oscillators in phase space using canonical transformations and delta functional integration [

Let us present the following time dependent Hamiltonian [

The propagator corresponds to this system can be written in the phase space as

This propagator is not exactly evaluated for any arbitrary time dependent mass or frequency, because that will lead to non-linear differential equations. To deal with this system we will absorb the quadratic term of q, by taking the following transformation

where

where

and the new Hamiltonian

Since

Then

to deal with this propagator we will take the following canonical transformations

With the generating function

Then (8) will be (see [

Since

In the exponent and by integrating the first term by part, then following by the integration over q we get the following condition

which implies that

Using the formula

one can finds that

By inserting this into Equation (4) we will find the expression of the propagator related to Equation (1)

which is the desired result

We would like to present a class of time dependent Harmonic oscillators with constant mass and varied frequencies, and we will follow the way that given above to find the exact propagator of the related system. Let us present the following Hamiltonian

where

Then the propagator related to this system can has the following expression

Then we will present the following canonical transformations

where P and q are the new momentum and position. This will lead to a new expression of the propagator Equation (20)

From here it is clear why the condition

・

The related function for this frequency is

・

The related function for this frequency is

・

The related function for this frequency is

・

This frequency has a more generalized form than that given in [

where r, v and t_{0} are constants with t_{0} has the dimension of time.

The problem of the time dependent harmonic oscillator has been presented in this work. By using canonical transformations we could reach Equation (11) with the condition Equation (2), then using delta functional integration that gave us the condition Equation (13) of the momentum conservation, which can be generalized to be

We thank the Editor and the referee for their comments and support that is greatly appreciated.

B.Berrabah, (2016) Quantum Mechanical Path Integral in Phase Space and Class of Harmonic Oscillators with Varied Frequencies. Journal of Modern Physics,07,359-364. doi: 10.4236/jmp.2016.74036