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This may be better addressed as a computational question and I'm just guessing here, but let's see what happens if I just get started... To make to make the plots above, you need to map each bin in one space (say the time domain) to a intensity function in the other space (frequency in our example). This is accomplished with a limited application of the ...

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The fact that you can represent a function $f$ (of time) as a Fourier transform $\tilde{f}$ already means that you can imagine the function as a superposition of sinusoids. Focus on any one of those sinusoids $$s(t) \equiv A \sin(\omega t + \phi).$$ The time derivative (i.e. velocity) is $$\dot{s}(t) = \omega A \cos(\omega t + \phi) . \quad (*)$$ You can ...

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Your question seems to confuse a number of things. Let me try to clear everything up: There is a difference between the Kronecker delta $\delta_{ij}$ where $i,j$ are discrete indices, and the Dirac delta $\delta(x-y)$, where $x,y$ are continuous variables. Superficially, the latter can be thought of as the continuous version of the former. There is ...

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Hint: Establish first that $$\delta(xy)~=~\frac{\delta(y)}{|x|}+\frac{\delta(x)}{|y|}.$$

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It seems there is a mistake in your $\pi (x)$ expression: there must be one minus sign near $\hat{a}^{\dagger}$. The relation between classical and field is obvious since lagrangian (hamiltonian) of free $\varphi$ (it's not hard to see that $\varphi$ satisfies Klein-Gordon equation) field may be rewritten as lagrangian of free ossilator in terms of ...

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