Question

Positronium is a hydrogen-like atom that consists of a positron (anti-electron) and an electron revolving around one another. The positron has the same mass as an electron but the opposite charge (same magnitude, but positive). a. Use Bohr's theory and the reduced mass (see problem lb and Krane Sec. 6.8) of the positron-electron system to show that 6.8 eV En= --"2_ , for positronium. b. The n -1 to n-2 transition in positronium has been measured to be roughly 1.2336 x 1015Hz[ the precision stated here? Explain why or why not. [I]. Does this agree with the prediction of Bohr's model to within

Answer 1

Total energy of H-like atom is given by (using Bobs theory) E_n = -me^4/2 epsilon^2_0 h^3 c [z^2 hc/n^2] = -R infinity hc/n^2 z = 1 E_n = 1.0974 times 10^7 times 6.6 times 10^-34 times 3 times 10^8/n^2 = -13.6/n^2 eV Part - a For Positronium atom reduced mass = me middot me/me + me = me/2 So energy of nth state of positronium = -13.6/n^2 times 1/2 cv = -6.8/n^2 eV part - b For n = 1 to n = 2 transition 1/lambda = R infinity [1/n^2_f - 1/n^2_f] 1/lambda = 109670 [1/1 - 1/4] times 10^-8 = > lambda = 1215 times 10^-10 m Frequency gamma = c/lambda = 3 times 10^8/1215 times 10^-10 = 3000 times 10^15/1215 freq. = 2.47 times 10^-15 Hz So this freq. is in close agreement with 1.2336 times 10^15 Hz in order some difference may arises due to following reasons (1) Binding energy is -6.8/x^2 i.e half