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Question 13 In the Coulomb model of a hydrogen atom, the energy eigenfunctions take the form nim(r,0,0) = Rnt(r) Yim (0,6). We consider here a state with n = 2, 1 = 1 and m= +1 for which the normalized radial function and spherical harmonic are 1/2 R2,1(r) = (215) re-r/2ao and Y₁,1(0,0) = - - (2) ¹² 1/2 3 sin eti 24a (a) If the atom makes a radiative transition from the state described by 2,1,1(r. 0, 0) to a state with n = 3, what is the energy of the absorbed photon? What are the possible I and m quantum numbers of the final state of the atom after the transition? (b) Write down an expression for the probability of finding an electron proton separation in the small range between r and r + dr in the state described by 2,1,1(r, 0,6). Hence find the most probable value of the electron-proton separation in this state. (c) Find the expectation value of the electron-proton separation in the state described by 2.1.1 (r,0,0). (d) Suppose that a hydrogen atom in the unperturbed state 2,1,1 (r, 0,6) is subject to a small perturbation SĤ = 7-31 where is a constant. Use perturbation theory to find the first-order correction to the energy of the state due to this perturbation. (e) Let C-Era²a, where a = 1/137. Is the perturbation theory of part (d) valid? Suppose instead of hydrogen, we consider the hydrogenic ion 2Fe25+. Use a scaling argument to determine if the first order perturbation theory method remains valid (assume that C does not scale with Z or A and ignore any reduced mass effects). You may find the following integral useful in parts (c) and (d): Su une-u/cdu= n!c"+1 for e> 0 and n = 1, 2, 3, ....