Positronium transition, energy of photon emitted

[yxgao]

What is the energy of the photon emitted when a positronium atom goes from the n=3 state to the n=1 state?

Edit:
Nevermind, I figured it out.

$$E_n = -\frac{\mu}{2{\hbar}^2} (\frac{Ze^2}{4\pi\epsilon_0})^2 \frac{1}{n^2}$$

For hydrgen, the effective mass is approximately the mass of the electron.
$$\mu=m_e$$

However, for positronium,
$$\mu=\frac{m_pm_e}{m_p+m_e}=\frac{m_em_e}{m_e+m_e}=\frac{m_e}{2}$$

So the length of the energy is only half that of the hydrogen atom.

Therefore,
$$E_3-E_1=(\frac{1}{3^2}-1)(-6.8 eV)=-\frac{8}{9}*-6.8 eV = 6.04 eV$$


grn. 31.

 

[Evalesco]

8 eV