July 23rd, 2009

Some Interesting Equations

Here’s a collection of random, important equations in advanced physics. Find one you like, and read up on it.

Einstein’s Field Theory

G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8 \pi G_N}{c^4} T_{\mu\nu}

Maxwell’s equations

\frac{4\pi}{c}j^\beta = \partial_{\alpha}F^{\alpha\beta} + \Gamma^{\alpha}_{\mu\alpha}F^{\mu\beta} + \Gamma^{\beta}_{\mu\alpha}F^{\alpha\mu}

0 = \partial_{\gamma}F_{\alpha\beta} + \partial_{\beta}F_{\gamma\alpha} + \partial_{\alpha}F_{\beta\gamma}

Schrodinger Equation

-\frac{\hbar^2}{2m} \nabla^2 \Psi (\vec{r},t) + U(\vec{r}) \Psi (\vec{r},t) = i\hbar \frac{\partial}{\partial t} \Psi (\vec{r},t)

i\hbar \partial_t |\Psi\rangle = H|\Psi\rangle

More Quantum Mechanical Equations

1 = \int \Psi^{*}\Psi dV = \langle\Psi|\Psi\rangle = \Psi_i \Psi^i

|\Psi\rangle = Ae^{i(\vec{r}\cdot\vec{k} - \omega t)}

Hawking Radiation
T = \frac{\hbar c^3}{8 \pi G_N M_{BH} k_B}

P = \frac{\hbar c^6}{15360 \pi G^{2}_{N} M^{2}_{BH}}

Feynman integral in curved spacetime

\langle q_{j+1} | \exp\left( {- {i \over \hbar } \hat H \delta t} \right) |q_j\rangle = \exp\left( {- {i \over \hbar } V \left( q_j \right) \delta t} \right) \int { dp \over 2\pi } \exp\left( {- {i \over \hbar } { { p}^2 \over 2m} \delta t} -{i\over \hbar} p \left( q_{j+1} - q_{j} \right) \right)

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