3. Let us assume that a physical system can be completely described in the three- dimensional...

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3. Let us assume that a physical system can be completely described in the three- dimensional space defined by the orthonormal basis (), |u), u3). On this basis, the Hamiltonian and the two observables A and B are represented by the matrices 0 1 0 0 =E020 1 0 0 A=a00 1 y B=b1 0 0 0 0 2 0 1 0 00 1 where E, a and b are real and positive constants. At time zero, the state of the system is given by the state vector 1 1 ||y(0)) = = (a) Choose among the three previous operators a complete set of commuting operators (CSCO). (b) Find a common vector basis of the CSCO. (c) If at t = 0 the energy of the system is measured, which are the values that could be obtained and with what probability? What if we measured instead of ? (c) If at t = 0 the energy of the system is measured, which are the values that could be obtained and with what probability? What if we measured instead of ? (d) Calculate | y(t)), ()(t) y (B)(t). (e) Which are the results that could be obtained and with what probabilities in case of measuring A and B at an instant t? (f) Suppose that at an instant t = t, the result of measuring the energy is E. Can the result of measuring immediately after t = t be predicted? If yes, what would that result be? What would the state of the system be after having measured ? And if after measuring A, we measure B, what would the result of the measurement and the state of the system be after the measurement? With what probability? (g) Same as case (f), but, after having obtained the energy E, we measure first and then A. What would the result of the measurement and the state of the system be after the measurement of B and then ? With what probabilities? (h) Same as (f), but the result of measuring the energy is 2E instead of E. (i) Same as (g), but the result of measuring the energy is 2E instead of E. 3. Let us assume that a physical system can be completely described in the three- dimensional space defined by the orthonormal basis (), |u), u3). On this basis, the Hamiltonian and the two observables A and B are represented by the matrices 0 1 0 0 =E020 1 0 0 A=a00 1 y B=b1 0 0 0 0 2 0 1 0 00 1 where E, a and b are real and positive constants. At time zero, the state of the system is given by the state vector 1 1 ||y(0)) = = (a) Choose among the three previous operators a complete set of commuting operators (CSCO). (b) Find a common vector basis of the CSCO. (c) If at t = 0 the energy of the system is measured, which are the values that could be obtained and with what probability? What if we measured instead of ? (c) If at t = 0 the energy of the system is measured, which are the values that could be obtained and with what probability? What if we measured instead of ? (d) Calculate | y(t)), ()(t) y (B)(t). (e) Which are the results that could be obtained and with what probabilities in case of measuring A and B at an instant t? (f) Suppose that at an instant t = t, the result of measuring the energy is E. Can the result of measuring immediately after t = t be predicted? If yes, what would that result be? What would the state of the system be after having measured ? And if after measuring A, we measure B, what would the result of the measurement and the state of the system be after the measurement? With what probability? (g) Same as case (f), but, after having obtained the energy E, we measure first and then A. What would the result of the measurement and the state of the system be after the measurement of B and then ? With what probabilities? (h) Same as (f), but the result of measuring the energy is 2E instead of E. (i) Same as (g), but the result of measuring the energy is 2E instead of E.