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DanielSank
DanielSank
12:43 AM
@TanMath Yes of course.
This was all just setup.
Note that $\langle t |D|\omega\rangle = i \omega e^{i \omega t}$.
so $D |\omega \rangle = i \omega | \omega \rangle$.
In other words, $|\omega \rangle$ is an eigenvector of $D$.
This isn't surprising, you already know that $(d/dt) e^{i \omega t} = i \omega e^{i \omega t}$
but now we understand this as an eigenvector equation.
Anyway, the point of all of this is that if we rewrite our differential equation in the $\omega$ basis we get:
$(-\omega^2 + i \beta \omega + \omega_0^2) \tilde{x}(\omega) = \tilde{J}(\omega)$.
We can rewrite this as
$\tilde{x}(\omega) = \tilde{J}(\omega) / (-\omega^2 + i \beta \omega + \omega^2)$
In other words, the response $\tilde{x}$ of the system is equal to the stimulus $\tilde{J}$ multiplied by a linear response function.
In our case, the linear response function is $(-\omega^2 + i \beta \omega + \omega_0^2)$.
We can identify each term of the response function with a term in the original equation of motion.
The $-\omega^2$ comes from the inertia term, the $i \beta \omega$ is from the friction, and the $\omega_0^2$ is from the restoring force.
 
 
2 hours later…
TanMath
TanMath
2:46 AM
@DanielSank how come? shouldn't it be 1 over that?
@DanielSank how is J measured?
 
DanielSank
DanielSank
3:09 AM
@TanMath Yes, that's what I meant.
@TanMath Well normally $J(t)$ is given
In this example where we have a damped driven harmonic oscillator, $J(t)$ would be something like a voltage or current drive.
For example, we could inject a probe signal $J(t)$ into our system and then measure how the system responds.
i.e. we supply $J$ and measure $x$.
Note that if $J$ is sinusoidal, then $\tilde{J}(\omega)$ is a delta function (or pair of delta functions, one for positive and one for negative frequency).
Therefore if we supply sinusoidal $J$ and measure the resulting $x$, we have in fact sampled the response function for a single value of $\omega$.
This is called "spectroscopy".
 
 
1 hour later…
TanMath
TanMath
4:20 AM
@DanielSank makes sense
 
DanielSank
DanielSank
4:57 AM
@TanMath Ok, well that's the basic idea of linear response theory.
We can go on to study the particular response function we have for the damped harmonic oscillator (it's a very important example).
You find the poles in the response function are of particular importance.
 
 
15 hours later…
TanMath
TanMath
7:48 PM
@DanielSank sure!
 

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