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Question 13 In the Coulomb model of a hydrogen atom, the energy eigenfunctions take the form Unim (r, 0, 0) = Rnt (r) Yim (0, 0). 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 1 1/2 R2,1 (r) = re-¹/2a0 and Y₁,1(0, 0) == sin etio 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, 0). 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, 0) is subject to a small perturbation C SH p3: 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 Fe25+. 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): L une-u/cdu= n!c+1 for c> 0 and n = 1, 2, 3,.... = (²³) [4] [3] [3] [4] [3]