Positronium is a bound state of an electron and a positron (a particle with the same mass as an electron but positive charge). If the ground state energy and the first Bohr radius of a normal hydrogen atom are -13.6 eV and a₀ respectively, what will be the ground state energy and the first Bohr radius of positronium? (Assume the proton mass in hydrogen is infinitely large compared to the electron mass).

Step-by-step solution

In a two-body system where the nucleus is not infinitely massive, the effective mass of the orbiting particle is replaced by the reduced mass of the system. For positronium, both particles have mass m_e . The reduced mass is: = m_e m_e m_e + m_e = m_e 2 The energy of a Bohr orbit is directly proportional to the reduced mass ( E ). Therefore, the ground state energy of positronium is: E = E_H ( m_e ) = -13.6 eV 1 2 = -6.8 eV The radius of a Bohr orbit is inversely proportional to the reduced mass ( r 1 ). Therefore,