Question

A muonic atom consists of a muon (m 106 Mev/c?) in place of an electron. For the muon in a hydrogen atom, what are the following? (a) the smallest radius in the ground state (b) the binding energy of the muon in the ground state (c) the series limit of the wavelength for the first three series ev nm (first series) nm (second series) nm (third series)

Answer 1

For muon atom electron is replaced by muon. Reduced mass mu = m_u m_p/m_p + m_u m_u = 207 m_e m_p = 1836 m_e doublerightarrow mu = 186 m_e radius of nth orbit r_n = 4 pi epsilon_0 hstrok^2 n^2/ze^2 mu for n = 1, z = 1 r_1 = mu pi epsilon_0 hstrok^2/e^2 mu = 2.84 time s10^-13 m binding energy Vertical-bar E_1 Vertical-bar = + 13.6 ev times 186 = 2529.6 ev as energy depend on mu directly \r\n as energy E_n = + z^2/n^2 E_1 = -2529.6 ev z^2/n^2 1/lambda = E_i - E_f = 2529.6 ev z^2/hc (1/n^2_f - 1/n^2_i) for lyman series n_f = 1 n_i = 2 for maximum wavelength delta = 0.653 nm for blamer series n_f = 2 n_i = 3 for maximum lambda lambda = 3.529 nm for pashen series n_f = 3, n_i = 4 for maximum lambda doublerightarrow lambda = 10.08 nm