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QM This problem relates to some quantum aspects of muon catalysis of the fusion reaction d+t → a+n. a) Explain (no derivation needed) what needs to be done to the textbook hydrogen-atom problem with V(r) = -ke²/|r| (where k depends on your choice of units) to take into account that the electron and nucleus both move about their common center of mass. Recall that for the textbook fixed-nucleus problem the hydrogen ground-state energy is E = -13.6 eV = -mek²e¹/2h². b) A μ can bind with any of the nuclei p, d, t to form a muonic atom. Which one will have the lowest ground-state energy? c) Compute the energy difference (in eV's) between the ground states of the muonic atoms du and tu. Now suppose that a muonic molecule composed of the three particles dt is formed. Since the is significantly lighter than either of the nucleons, in this part of the question we will ignore the nuclear motion and use the Born-Oppenheimer approximation in which the nuclei are held fixed a distance R apart, and a Schrödinger equation is solved to find the ground- state energy E(R). The actual nuclear separation will be that R which makes E(R) a minimum. d) Write down the Schrödinger equation that must be solved to find E(R). e) Let (r) be the minimum-E(R) ground-state eigenfunction for the electron in the dte molecule. Adapt the same function to solve the dtu problem. Hence find the ratio of bond lengths and the ratio of the chemical binding energy Ebinding = min{E(R)} for the dtu molecule compared to the dte molecule. f) For the fusion reaction to occur the two nuclei, d and t, must be in the total J = 3/2 nuclear spin state. Assuming that the spins of the two nuclei are initially unpolarized, what is the probability that J = 3/2? charge spin mass (MeV) -1 1/2 muon -1 1/2 proton P +1 1/2 deuteron d +1 1 triton t +1 1/2 particle electron 0.511 106 938 1876 2791