User:Wing gundam/sandbox

$$ \color{Salmon} \begin{array}{lcl} \nabla\cdot\mathbf{E} = \frac {\rho} {\varepsilon_0} \\ \nabla\cdot\mathbf{B} = 0 \\ \nabla\times\mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t} \\ \nabla\times\mathbf{B} = \mu_0\mathbf{J} + \mu_0\varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} \\ \\ \partial_{\mu}A^\mu = 0 \\ \partial^\nu\partial_\nu A^\mu  = \mu_0 J^\mu \end{array} $$

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physics

 * $$U(r)=\hbar c \frac{\ {g_e}^2}{4\pi r}=\frac{\hbar c \alpha}{r}$$
 * $$F(r)=\hbar c \frac{\ {g_e}^2}{4\pi r^2}=\frac{\hbar c \alpha}{r^2}$$

where:


 * $$g_e$$ is the electromagnetic gauge coupling constant, related to the fine structure constant by $$\alpha=\frac{{g_e}^2}{4\pi}$$. It's the value of an elementary charge when electric charge is nondimensionalized, and more fundamental than $$\alpha$$.

For compound charges,
 * $$U(r)=\hbar c \frac{\ q_1 q_2}{4\pi r} =\hbar c \alpha \frac{n_1 n_2}{r}$$
 * $$F(r)=\hbar c \frac{\ q_1 q_2}{4\pi r^2}=\hbar c \alpha \frac{n_1 n_2}{r^2}$$

where:
 * $$q_i = g_e n_i$$
 * and $$n_i$$ is the integer number of charges


 * $$0 = T^{\mu \nu}{}_{;\nu} = \nabla_{\nu} T^{\mu \nu} = T^{\mu \nu}{}_{,\nu} + T^{\sigma \nu} \Gamma^{\mu}{}_{\sigma \nu} + T^{\mu \sigma} \Gamma^{\nu}{}_{\sigma \nu}$$

physics 2

 * $$z=\sum_{k=0}^\infty \sum_{\{\mathbf{x}_1,\cdots,\mathbf{x}_k\}\,\subset\, \mathbb R^3} \frac{1}{k!}c_{\mathbf{x}_1\cdots\mathbf{x}_k} \theta_{\mathbf{x}_1}\cdots\theta_{\mathbf{x}_k}$$

derivative:


 * $$i \hbar c \gamma^\mu \partial_\mu \psi = m c^2 \psi $$


 * $$i \hbar c \gamma^\mu (\partial_\mu + i e A_\mu) \psi = m c^2 \psi $$


 * $$- (\hbar c)^2 (\partial_0 + i e A_0)^2 \psi = - (\hbar c)^2 (\partial_i + i e A_i)^2 \psi + (m c^2)^2 \psi $$


 * $$- (\frac 1 c \partial_t + i e A_0)^2 \psi = - \nabla^2 \psi + {\lambda\!\!\!\!-}_c^2 \psi $$

free:


 * $$- \frac 1{c^2} \partial_t^2 \phi = - \nabla^2 \phi + {\lambda\!\!\!\!-}_c^2 \phi $$


 * $$- \frac 1{c^2} \partial_t^2 (e^{-i\omega_c t} \psi) = - \nabla^2 (e^{-i\omega_c t} \psi) + {\lambda\!\!\!\!-}_c^2 (e^{-i\omega_c t} \psi) $$


 * $$- \frac 1{c^2} \cancel{e^{-i\omega_c t}}(-i\omega_c + \partial_t)^2 \psi = - \cancel{e^{-i\omega_c t}}\nabla^2 \psi + \cancel{e^{-i\omega_c t}} {\lambda\!\!\!\!-}_c^2 \psi $$


 * $$- \frac 1{c^2} (-i\omega_c + \partial_t)^2 \psi = - \nabla^2 \psi + {\lambda\!\!\!\!-}_c^2 \psi $$


 * $$- \frac 1{c^2} (-\omega_c^2 - 2 i \omega_c \partial_t + \partial_t^2) \psi = - \nabla^2 \psi + {\lambda\!\!\!\!-}_c^2 \psi $$


 * $$- \frac 1{c^2} (\cancel{-\omega_c^2} - 2 i \omega_c \partial_t + \partial_t^2) \psi = - \nabla^2 \psi + \cancel{{\lambda\!\!\!\!-}_c^2 \psi} $$


 * $$- \frac 1{c^2} (-2 i \omega_c \partial_t + \partial_t^2) \psi = - \nabla^2 \psi $$


 * $$\frac {2 i {\lambda\!\!\!\!-}_c}{c} \partial_t\psi - \cancel{\frac 1 {c^2} \partial_t^2 \psi} = - \nabla^2 \psi $$


 * $$(\hbar c)^2\frac {2 i \omega_c}{c^2} \partial_t\psi = - (\hbar c)^2 \nabla^2 \psi $$


 * $$\frac 1 {2\hbar \omega_c} \frac {2 i {\lambda\!\!\!\!-}_c}{c} \partial_t\psi' = - \frac 1 {2\hbar \omega_c} \nabla^2 \psi' $$


 * $$\frac {i}{\hbar c^2} \partial_t\psi' = - \frac 1 {2\hbar \omega_c} \nabla^2 \psi' $$


 * $$i\hbar\partial_t\psi' = - \frac {\hbar^2 c^2} {2\hbar \omega_c} \nabla^2 \psi' $$


 * $$i\hbar\partial_t\psi' = - \frac {\hbar^2} {2m} \nabla^2 \psi' $$

again:


 * $$- (\hbar c)^2 (\frac 1 c \partial_t + i e A_0)^2 \phi = - (\hbar c)^2 \nabla^2 \phi + E_0^2 \phi,\quad A_{0,0}=0$$


 * $$- (\hbar c)^2 (\frac 1 {c^2} \partial_t^2 + 2 i e A_0 \frac 1 c \partial_t - e^2 A_0^2) \phi = - (\hbar c)^2 \nabla^2 \phi + E_0^2 \phi $$

subtitute, factor, divide,


 * $$\left\{- (\hbar c)^2 (\frac 1 {c^2} (-\omega_c^2 - 2 i \omega_c \partial_t + \partial_t^2) + 2 i e A_0 \frac 1 c (-i\omega_c + \partial_t) - e^2 A_0^2) \psi = - (\hbar c)^2 \nabla^2 \psi + E_0^2 \psi\right\}\frac 1{2E_0} $$

cancel,


 * $$\left\{- (\hbar c)^2 (\frac 1 {c^2} (\cancel{-\omega_c^2} - 2 i \omega_c \partial_t + \partial_t^2) + 2 i e A_0 \frac 1 c (-i\omega_c + \partial_t) - e^2 A_0^2) \psi = - (\hbar c)^2 \nabla^2 \psi + \cancel{E_0^2} \psi\right\}\frac 1{2E_0} $$


 * $$- \frac{(\hbar c)^2}{2E_0} (\frac 1 {c^2} (- 2 i \omega_c \partial_t + \partial_t^2) + 2 i e A_0 \frac 1 c (-i\omega_c + \partial_t) - e^2 A_0^2) \psi = - \frac{\hbar^2}{2m} \nabla^2 \psi $$


 * $$ i\hbar\partial_t\psi - \frac{\hbar^2}{2E_0} \partial_t^2\psi - \frac{(\hbar c)^2}{2E_0}(2 i e A_0 \frac 1 c (-i\omega_c + \partial_t) - e^2 A_0^2) \psi = - \frac{\hbar^2}{2m} \nabla^2 \psi $$


 * $$ i\hbar\partial_t\psi - \frac{\hbar^2}{2E_0} \partial_t^2\psi - \frac{(\hbar c)^2}{E_0}i e A_0 \frac 1 c (-i\omega_c + \partial_t)\psi + \frac{(\hbar c)^2}{2E_0} e^2 A_0^2 \psi = - \frac{\hbar^2}{2m} \nabla^2 \psi $$


 * $$ i\hbar\partial_t\psi - \cancel{\frac{\hbar^2}{2E_0}\partial_t^2\psi} - \hbar c e A_0 \psi - \cancel{\frac{\hbar^2}{m c}i e A_0 \partial_t\psi} + \cancel{\frac{\hbar^2}{2m} e^2 A_0^2 \psi} = - \frac{\hbar^2}{2m} \nabla^2 \psi $$


 * $$ i\hbar\partial_t\psi = - \frac{\hbar^2}{2m} \nabla^2 \psi + \hbar c e A_0 \psi\quad$$ as expected. e is dimensionless coupling, $$A_0\propto\frac{1}{L}$$