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Atomic}Positronium


The positronium atom ground-state energy can be deduced from recalling the reduced mass. \mu=m/2, since one has a positron and electron orbiting each other, and thus the energy, which is related to mass, is half of that of Hydrogen. E_{positronium}=-13.6/2 eV = -6.8eV

The Bohr formula still applies, and thus E=E_1 \left(1/n_f^2 - 1/n_i^2 \right)=8E_1/9, which is choice (A).

See below for user comments and alternate solutions! See below for user comments and alternate solutions!
Alternate Solutions
FutureDrSteve
2011-11-07 12:22:35		Oops, this was supposed to be a separate post, not a reply....

If you realize that the ground state energy of positronium is roughly -6.8eV (which I think you have to in order to solve this problem), you don't have to do any math. An electron transitioning down from n=\infty would have initial potential energy of 0, and drop down to final energy -6.8eV, for a max energy change of 6.8eV, eliminating (C), (D), and (E). Since the electron is only falling down from n=3, it stands to reason that the photon would be less energetic than 6.8eV, leaving only answer (A).		Alternate Solution - Unverified
Comments
bill92
2014-10-27 13:17:41		I just took the exam and there was no positronium question! All my studying was for naught!

Just kidding, I think it went well. Thanks to Yosun for this very helpful site!		NEC
FutureDrSteve
2011-11-07 12:22:35		Oops, this was supposed to be a separate post, not a reply....

If you realize that the ground state energy of positronium is roughly -6.8eV (which I think you have to in order to solve this problem), you don't have to do any math. An electron transitioning down from n=\infty would have initial potential energy of 0, and drop down to final energy -6.8eV, for a max energy change of 6.8eV, eliminating (C), (D), and (E). Since the electron is only falling down from n=3, it stands to reason that the photon would be less energetic than 6.8eV, leaving only answer (A).		Alternate Solution - Unverified
Tommy Koulax
2007-10-31 17:48:57		Why is the reduced mass m/2 ?
nick1234
2007-11-02 17:16:11		 \mu = \frac{m_{1} \times m_{2}}{m_{1} + m_{2}}

For the hydrogen atom, m_{proton} dominates the denominator, and \mu becomes \frac{m_{e} \times m_{p}}{m_{p}} = m_{e}

For positronium, m_{1} = m_{2} = m{e}, so \mu = \frac{m_{e}^{2}}{2 m_{e}} = \frac{m_{e}}{2}
NEC
rinnie
2007-03-26 21:27:49		Hydrogen levels E3 - E1 = -1.5 + 13.6 = 12.1/2 = 6.05.
Energy level of positronium is half those of Hydrogen.
FutureDrSteve
2011-11-07 12:20:53		 If you realize that the ground state energy of positronium is roughly -6.8eV (which I think you have to in order to solve this problem), you don't have to do any math. An electron transitioning down from n=\infty would have initial potential energy of 0, and drop down to final energy -6.8eV, for a max energy change of 6.8eV, eliminating (C), (D), and (E). Since the electron is only falling down from n=3, it stands to reason that the photon would be less energetic than 6.8eV, leaving only answer (A).
NEC

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