a quantum system can exist in two states 0 and 1 which are normalised eigen states of an observa

(a) The time evolution operator for a quantum system is given by the exponential of the Hamilton

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A quantum system can exist in two states |0⟩ and |1⟩, which are normalized eigenstates of an observable Â with eigenvalues 0 and 1 respectively. The Hamiltonian operator is defined by Ĥ |0⟩ = α|0⟩ + β|1⟩ Ĥ |1⟩ = β|0⟩ + α|1⟩ where α, β are real. (a) Determine the time evolution operator for this system and its matrix in the |0⟩, |1⟩ basis. (b) If the system is in the state |1⟩ at time t = 0, write down the state of this system at time t = T. (c) The same system is made to evolve from the state |0⟩ at the time t=0 and is further subjected to a sequence of measurements. First, the observable Â is measured at t = T, but the value is lost. The system is then allowed to further evolve for an additional period of T and then the system energy is measured. Calculate the expectation value of this energy measurement. What is the relation between this expectation value and the average value of energy obtained in energy measurements of an ensemble of systems, identically prepared as this one?

          A quantum system can exist in two states |0⟩ and |1⟩, which are normalized eigenstates of an observable Â with eigenvalues 0 and 1 respectively. The Hamiltonian operator is defined by
Ĥ |0⟩ = α|0⟩ + β|1⟩
Ĥ |1⟩ = β|0⟩ + α|1⟩
where α, β are real.

(a) Determine the time evolution operator for this system and its matrix in the |0⟩, |1⟩ basis.
(b) If the system is in the state |1⟩ at time t = 0, write down the state of this system at time t = T.
(c) The same system is made to evolve from the state |0⟩ at the time t=0 and is further subjected to a sequence of measurements. First, the observable Â is measured at t = T, but the value is lost. The system is then allowed to further evolve for an additional period of T and then the system energy is measured. Calculate the expectation value of this energy measurement. What is the relation between this expectation value and the average value of energy obtained in energy measurements of an ensemble of systems, identically prepared as this one?

Added by Barbara W.

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