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You can use the fact $$ \int e^{-iqr \cos\theta} \sin \theta d\theta = -\int e^{-iqr c} dc $$ where $c$ is the integration variable that replaced $\cos\theta$. The integral is now elementary. …

Given the normally distributed random variable $Z\sim {N}(\mu,\sigma^2)$, it is possible to show that its moment-generating function is $$\langle \exp({tZ})\rangle=\exp({\mu t + \sigma^2 t^2/2})=\exp …

In molecular dynamics you have to evolve a large number of particles, so efficiency is important. The Verlet algorithm provides numerical stability, as well as other properties that are important in …

You may also wonder why we use the concept of density of a material despite that material is made of molecules, or atoms, or in general quantized entities. This is the basis of hydrodynamics and solid …

Consider $z=x+i y$ and $\bar z=x-iy$ ($x$ and $y$ are real). It is easy to show that: $$ dz\land d\bar z=(dx+i\,dy)\land(dx-i\,dy) =dx\land dx+i \, dy\land dx -i \, dx\land dy-dy\land dy. $$ However, …