Five Examples¶
Lennard-Jones potential:¶
$$
LJ(r) = 4\epsilon( ({\frac{\sigma}{r}})^{12} - ({\frac{\sigma}{r}})^{6} ) \tag{1}
$$
Application of Hooke's Law to vibrational frequency:¶
$$
\displaystyle \widetilde{\nu}={\frac{1}{2\pi c}}{\sqrt {\frac {k_{force}}{\mu}}} \tag{2}
$$
Redox Reaction:¶
$$
[Fe(CN)_{6}]^{3-} + e^{-} \rightleftharpoons [Fe(CN)_{6}]^{4-} \hspace{0.5cm} E_{cell} = 0.356\hspace{0.1cm} V\tag{3}
$$
Heisenberg's Uncertainty Principle:¶
$$
\delta \widehat{x} \delta \widehat{p} = \sqrt{ \langle\widehat{x}^{2}\rangle - \langle\widehat{x}\rangle^{2} }\sqrt{ \langle\widehat{p}^{2}\rangle - \langle\widehat{p}\rangle^{2} } \geq \frac{\hbar}{2} \tag{4}
$$
Volume of Sphere in Spherical Coordinates:¶
$$
{\displaystyle V=\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{r}\rho^{2}\sin\theta \,d\rho\,d\theta \,d\varphi =2\pi\int_{0}^{\pi}\int_{0}^{r}\rho^{2}\sin\theta \,d\rho\,d\theta =4\pi\int_{0}^{r}\rho^{2}\,d\rho\ ={\frac{4}{3}}\pi r^{3}.} \tag{5}
$$