LaTeX

Energy levels for particle in 1-D box

$E = \frac {\pi^2 \hbar^2 n^2} {2 m a^2}$

Where n is an integer, $n= 1,2,3,\dots$

and $\hbar = \frac h {2 \pi}$

so $E = \frac {\pi^2 h^2 n^2} {2 m a^2 4 \pi^2} = \frac {h^2 n^2} { 8 m a^2}$

Relativistic energy

$\ {E^2=p^2 c^2 + m_0^2 c^4}$

One-dimensional Schrödinger wave equation

$- \frac {\hbar^2} {2m} \frac {d^2 \psi (x)} {dx^2} + U(x)\psi(x) = E\psi(x)$

Magnetic field around a single wire

$r = \frac {\mu _{0}I} {2 \pi B}$

Probability of finding an object on the other side of a quantum barrier

$P = \exp \left( -2d \sqrt { \frac {2m(V-E)} { \hbar^2} } \right)$

Uncertainty in energy of a state with lifetime $\tau\$

${\delta}E = \frac h {2 \pi\ } \tau\$

Van Der Waals Gas

$\left ( p + \frac {an^2} {V^2} \right ) (V - nb) = nRT$

Diffraction Experiment

$F(hkl) = \sum _{j} f _j (\theta) \cdot \exp[-8\pi^2U_j\sin^2 \frac {\theta} {\lambda^2}] \cdot \exp [2 \pi i (hx_j + ky_j + i z_j]$

Eyring constant

$k= \frac {k_B T} n k^ \ddagger\$

Difference Map

$\Delta \rho (xyz) = \frac 1 {V} \sum _{hkl} ( \left| F _{obs} \right| - \left| F_{calc} \right|) \exp[-2 \pi i(hx+hy+lz-\alpha _{hkl})]$

Quadratic formula

$x_{0,1} = \frac {-b \pm \sqrt { b^2 - 4ac } } {2a}$

Arithmetic mean

$r = \frac {1} {N} \sum_{n=1}^N r_n$

Geometric mean

$r = [ ( 1 + r_1 ) ( 1 + r_2 ) \dots ( 1 + r_N ) ] ^\frac { 1 } { N } - 1$

$r = (\sum_{n=1}^N 1 + r_n)^\frac {1}{N} - 1$

Mexican Hat

$\psi(t) = {2 \over {\sqrt {3\sigma}\pi^{1 \over 4}}} \left( 1 - {t^2 \over \sigma^2} \right) e^{-t^2 \over 2\sigma^2}$