This problem relates to the electronic ground state lspositronium atom, which has one electron and one positron. The mag- netic moments are equal in magnitude and of course have opposite sign. (a) Using first-order perturbation theory, calculate an expression for the energy shifts of the ground state energy levels caused by the interaction of the magnetic dipole moments of the two particles. Compute their energy shifts from the unperturbed ground state energy, and state the complete set of commuting operators associated with the energy levels. Note that the magnetic dipole-dipole interaction energy between two particles with no orbital angular momentum is given by the following expression in elec trostatic units (See Jackson's E&M textbook, Chapter 5): 87." ~ > Hold = 3 0f "3' )7+' r-: where the magnetic dipce operators of the positron and electron are, re- spectively, ,u: : igggm S :- (You may approximate the gfactor by 2 in this problem.) (b) Calculate the photon wavelength and the spontaneous emission life- time for the magnetic dipole transition from the higher of these two energy levels calculated in part (a) to the lower level. Ignore the possibility of electron-positron annihilation in this assignment