# Harmonic oscillator in a time-dependent force

• gibbyboy
In summary, the conversation discusses the determination of the propagator in quantum mechanics, specifically in a harmonic oscillator with a time dependent force. The classical path solution to the equation of motion is a combination of a homogeneous and particular solution, and the classical action of the system can be obtained by integrating the Lagrangian with the classical path as the variable. However, there is a known problem with correctly extracting the classical action, and further knowledge and understanding of integration is needed to solve it. A potential solution is to find the propagator for the Lagrangian and determine the appropriate constants to satisfy the boundary conditions.
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.

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

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.

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

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.

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

## 1. What is a harmonic oscillator in a time-dependent force?

A harmonic oscillator in a time-dependent force is a physical system that undergoes periodic motion, where the restoring force is dependent on time. This means that the force acting on the system changes over time, causing the oscillator to oscillate with a varying amplitude and frequency.

## 2. How does a harmonic oscillator in a time-dependent force differ from a simple harmonic oscillator?

A simple harmonic oscillator has a constant restoring force, while a harmonic oscillator in a time-dependent force has a varying restoring force. This results in a more complex motion for the oscillator, with changes in amplitude and frequency over time.

## 3. What are some real-life examples of harmonic oscillators in time-dependent forces?

Some examples include a swinging pendulum in a moving car, a vibrating guitar string being plucked, and a tuning fork struck by a moving object. In all of these cases, the force acting on the oscillator changes over time, leading to a time-dependent harmonic motion.

## 4. How is the motion of a harmonic oscillator in a time-dependent force described mathematically?

The motion of a harmonic oscillator in a time-dependent force can be described using the differential equation known as the "driven harmonic oscillator equation." This equation takes into account the time-dependent force and the system's natural frequency, and can be solved to determine the motion of the oscillator at any given time.

## 5. What are the practical applications of studying harmonic oscillators in time-dependent forces?

Understanding the behavior of harmonic oscillators in time-dependent forces is crucial in fields such as engineering, physics, and chemistry. It is used to model and analyze various systems, including electronic circuits, molecular vibrations, and seismic activity. Additionally, this knowledge is essential in the development of technologies such as sensors, actuators, and energy harvesting devices.

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