Green function for forced harmonic oscillator

[Judas503]

Homework Statement

The problem requires to solve the integration to find $G(t)$ after $G(\omega)$ is found via Fourier transform. We have $$G(\omega)= \frac{1}{2\pi}\frac{1}{\omega _{0}^2 - \omega ^2}$$

Homework Equations

As mentioned previously, the question asks to find $G(t)$

The Attempt at a Solution

It is obvious that calculus of residues is required. To account for causality ($G(t<0)=0$), the poles at $\omega=\pm \omega_{0}$ are shifted to the lower half plane by $i\epsilon$ and integrated along the contour in the lower half plane. Then,
$$G(t)=-\frac{1}{2\pi}\int_{-\infty}^{+\infty}\frac{e^{-i\omega t}}{(\omega +i\epsilon)^2 -\omega _{0} ^2}d\omega$$
After calculating the residues at $\omega =\pm \omega _{0} - i\epsilon$, I found
$$G(t)=\frac{i}{2\omega_{0}}\sin \omega_{0}t$$

Is my answer correct?

 

[vela]

The presence of $i$ suggests it's not. G(t) should be real, shouldn't it?

The sine makes sense. The system is at rest and then you impart an impulse, causing it to oscillate. It has to be sine because it starts from x=0.

 

[RUber]

It looks like the i and the 2 might both get absorbed into the $\sin \omega_0 t =\frac{e^{i \omega_0 t }-e^{-i \omega_0 t }}{2i}$ term.

 

[Judas503]

Yes, sorry! I forgot to put the "i" in the exponential form of sine. That should clear the problem.

 

[paranormal]

Your approach is correct and your answer appears to be correct as well. However, it is always important to double check your work and make sure you have accounted for all necessary factors, such as causality in this case. Additionally, it may be helpful to provide some explanation or context for the solution, especially for those who may not be familiar with the topic. Overall, your response demonstrates a strong understanding of the concept and a thorough approach to solving the problem.