2. Two particles of the same mass m, with positions X and X2 and momenta P...

 

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2. Two particles of the same mass m, with positions X₁ and X2 and momenta P₁ and P2, are subject to a harmonic force. The particles do not interact with each other. Write the Hamiltonian of the system. Show that I can be written as H = H₁ + H2, where H₁ and H₂ act respectively only in the state space of particle (1) and in that of particle (2). Calculate the energies of the two-particle system, their degrees of degeneracy and the corresponding wave functions. (b) Does H form a C.S.C.O.? Same question for the set {H₁, H₂}. Write the orthonormalization and closure relations for the states On₁,n₂). (c) Consider a system which, at t=0, is in the state: 1 |V(0)) = (1,1) + 0,0) + [91,0) + [90,1)) (1) What results can be found and with what probabilities, if at this time one measures: (i) The total energy of the system. (ii) The energy of particle (1). 2. Two particles of the same mass m, with positions X₁ and X2 and momenta P₁ and P2, are subject to a harmonic force. The particles do not interact with each other. Write the Hamiltonian of the system. Show that I can be written as H = H₁ + H2, where H₁ and H₂ act respectively only in the state space of particle (1) and in that of particle (2). Calculate the energies of the two-particle system, their degrees of degeneracy and the corresponding wave functions. (b) Does H form a C.S.C.O.? Same question for the set {H₁, H₂}. Write the orthonormalization and closure relations for the states On₁,n₂). (c) Consider a system which, at t=0, is in the state: 1 |V(0)) = (1,1) + 0,0) + [91,0) + [90,1)) (1) What results can be found and with what probabilities, if at this time one measures: (i) The total energy of the system. (ii) The energy of particle (1).