# Homework Help: The allowed radii and energies of positronium

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1. May 3, 2015

### squelch

1. The problem statement, all variables and given/known data

Positronium is a hydrogen-like atom consisting of a proton and an electron revolving around each other. Find the allowed radii and energies of the system.

2. Relevant equations

See solution attempt.

3. The attempt at a solution

The allowed radii of the positron/electron system are given by $r = \frac{{4{n^2}\pi {\varepsilon _0}{\hbar ^2}}}{{\mu {e^2}}}$ where $\mu = \frac{{{m_{positron}}{m_e}}}{{{m_{positron}} + {m_e}}}$ and n={1,2,3,...}

The allowed energies of the positron/electron system are given by ${E_n} = - \frac{{\mu {e^4}}}{{2{n^2}{{(4\pi {\varepsilon _0})}^2}{\hbar ^2}}}$

It just seemed suspiciously simple. Is this all they're really asking me to state? I mean, other than calculating it all out and leaving it as a function of n times a constant.

2. May 3, 2015

### squelch

Just in case that was what I needed to do, I calculated the following allowable radii and energies, using the reduced mass:

$$\begin{array}{l} {r_n} = {n^2}(1.05835 \times {10^{ - 10}} m)\\ {E_n} = \frac{1}{{{n^2}}}( - 6.8032eV) \end{array}$$