In condensed matter applications, the divergence problem is solved by introducing screened Coulomb potential (known as Yukawa Potential):
\begin{equation}
V(r) = \frac{e^2}{4\pi\epsilon_{0}} \frac{e^{-mr}}{r}
\end{equation}
One can get the usual (long-range) Coulomb potential back if one takes...
np. It is usually a convention to determine the sign of the exponential in Fourier transform. In physics, forward Fourier transform from time to frequency space is carried out by ##e^{-iwt}##, while forward Fourier transform from real space to momentum space contains ##e^{ikx}##.
Great work...
The Fourier transform of your function f(t) is given as:
$$ F\left[f(t)\right] = \int_{-\infty}^{\infty} dt e^{i\omega t}f(t) = \int_{-\tau}^{0} -e^{i\omega t}dt + \int_{0}^{\tau} e^{i\omega t}dt $$
In the last step, I made use of the fact that f(t) is 0 elsewhere. As a final step, one can...
Fermi-Dirac distribution f(E) gives the probability that a quantum state with energy E is occupied. So, as long as there is no symmetry breaking that lifts the degeneracy of spin up and spin down electrons, they have the same energy, thereore f(E) for spin up and spin down electrons are the...