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d) e) and f please
Two particles of the same mass m that do not interact, with observables of position X₁ and X2, and respective observables of linear moments P₁ and P2, are subjected to the same potential energy V(X) = mw²X² a) Write the Hamiltonian operator H of the two-particle system and show that one can write in the form H = H₁ + H₂ where H₁ acts in the state space of particle 1 and H₂ in that of particle 2. Determine the possible values of the energy and their corresponding eigenvectors of the two-particle system, their degrees of degeneracy, and the wave functions for the first two energy levels. b) Does H form a CSCO? Repeat the same for the set H₁, H₂. If is denoted as P1,2) the eigenvectors common to H₁ and H₂. Write the orthonormalization and closure relations of the states 1,n2). c) Consider that at time t = 0 the system is in the state: (0)) = 2i|Po,0) + |Þ1,0) — 3|Þ0,1) + |1,1) What results can be found and with what probabilities, if the total energy of the system is measured? Repeat the same if the energy of particle 2 is measured. d) Find the state of the system at any instant of time. Determine the mean values of the energy of particle 1, particle 2 and the mean value of the energy of the system. Compare your results e) Find the mean value of the position of particle 1 and particle 2, as well as the mean value of the linear moments of each of the particles. Compare with the mean value of the linear momentum of the system of the two particles. f) If for an instant t the energy of particle 2 is measured and its second smallest energy value is found, what is the state of the system immediately after the measurement?