Positronium

Here are a bunch of unchecked equations for positronium. We will not be covering it this term due to lack of time. I'll try to clean this up later.

$$\bgroup\color{black}E_n={1\over 2}\alpha^2 \mu c^2{1\over{n^2}} \qquad \mu={m_e\over 2}\egroup$$

Relativistic Correction

$$\bgroup\color{black}\vec{r}\equiv\vec{r}_1-\vec{r}_2 \egroup$$

$$\bgroup\color{black}\vec{p}=\mu\dot{\vec{r}} = { m\dot{\vec{r}}_1 - m\dot{\vec{r}}_2\over 2}\egroup$$

$$\bgroup\color{black} = \vec{p}_1-\vec{p}_2 = \vec{p}_1 \egroup$$

$$\bgroup\color{black}H_{()}=-{1\over 8}{p^4_1+p^4_2\over{m^3c^... ...}{p^4\over{m^3c^2}} = {-1\over{32}}{p^4\over{\mu^3c^2}}\egroup$$

$$\bgroup\color{black}={-1\over{4\mu c^2}}\left({p^2\over{2\mu}}\right)^2\egroup$$

$$\bgroup\color{black}H_{s0}=-\vec{\mu}_{e^-}\cdot\vec{B}_+ - \vec{\mu}_{e^+}\cdot\vec{B}_-\egroup$$

$B_{\pm}$ comes from $\vec{L}$ motion we worked out in $e^+$ rest frame ...??

$$\bgroup\color{black} KE\longrightarrow {1\over 2}KE_H\qquad\Rightarrow mv^2 ={1\over 2}{1\over 2}mv^2_H \egroup$$

$$\bgroup\color{black}\Rightarrow v={1\over 2}v_H\egroup$$

$$\bgroup\color{black}v_{rel}=V_{rel_H}\egroup$$

$$\bgroup\color{black}\Rightarrow B(r) = B(r)_H\egroup$$

$$\bgroup\color{black}H_s(-r)=H_{s0}(r)_H\egroup$$

$$\bgroup\color{black}B=-{1\over{c^2}}\vec{v}\times\vec{E}\egroup$$

same also

$$\bgroup\color{black}W_{s0}={e^2\over{2m^2c^2}}{1\over{r^3}}\vec{L}\cdot\vec{S}\egroup$$

again ${1\over 4}$ of H

$s=0,1$ should go with hyperfine.

$$\bgroup\color{black}\Delta E_{rel}={1\over 8}\alpha^4 \mu c^2... ...} \left[ {3\over n}-{1\over {\ell +{1\over 2}}} \right]\egroup$$

$$\bgroup\color{black}\Delta E_{s0}= {1\over 8}\alpha^4 \mu c^2{1\over {n^3}} \left[ stuff \right] \egroup$$

basically $1/4$ of Hydrogen

Hyperfine

$$\bgroup\color{black}{4\over 3} 2 \alpha \mu c^2{1\over {n^3}}\left( {\vec{S}_1 - \vec{S}_2\over{\hbar^2}}\right)\egroup$$

$$\bgroup\color{black}\Delta E = {2\alpha^2\mu c^2\over{n^3}}\l... ...\matrix{-1\cr +1/3}\right\}\qquad \matrix{s=0 \cr s=1} \egroup$$

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