Harmonic oscillator in a time-dependent force

[gibbyboy]

In quantum mechanics, one of the major concerns is the propagator determination of the system. The propagator is completely expressed in terms of its classical in the Van-Vleck Pauli Formula.
In a harmonic oscillator in a time dependent force, the Lagrangian is given by
L=1/2m(dx/dt)^2-1/2mw^2x^2-e(t)x (eqn.1)
the equation of motion according to Lagrange equation is
(d/dt)(dx/dt)+w^2x=-1/m[e(t)] (eqn.2)
the general solution (classical path) to this equation consists of a homogeneous plus the particular solution.
In order to get the classical action of the system, i need to integrate the Lagrangian (eqn.1) from 0 to T with the classical path now as the x. But it seems that i cannot correctly get the classical action needed for the system. Do i need to substitute the general solution to the Lagrangian or i just have to substitute the homogeneous solution to the simple harmonic oscillator part and then substitute the general solution to the potential term the e(t)x. I'm really bothered regarding with this matter. Can someone enlightened me with this? An answer to this is very much appreciated. Thank you very much.


Gibby.

 

[Avodyne]

I believe the first procedure is the correct one, and should work.

 

[gibbyboy]

I also think of that as the correct answer, but i can't really extract the correct classical action of the system from the general solution of the classical path. I guess i do need more knowledge when it comes to integration following a long and tedious job.

 

[per.sundqvist]

This is a known problem and has been discussed here before. I think the link can give you some hints:

https://www.physicsforums.com/showthread.php?t=288742

The idea is to find the propagator (kernel) for this Lagrangian and then you have he solution for all times. Generally in finding Green functions you add homogeneous solution (classical harmonic oscillator) and the particular solution, like A*x_hom+B*x_part, but you must determine A and B so that you satisfy the boundary conditions.

 

[gibbyboy]

Thank you very much for your brilliant insights and big help. It really helped a lot.
God bless you.