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C. Tensor Products 1. 6 Let d-level atoms A and B be non-interacting and have A. Hamiltonian Evolution local Hamiltonians H A and H B respectively, {EA.IEA) The eigendecomposition of a time independent and {EB,IEB) are the respective nondegenerate Hamiltonian is given by H = P , EjEjihEj|. You may assume eigenenergies and eigenstates. For the combined (non- interacting) system of A and B, write down the combined that it is non-degenerate. Hamiltonian and work out the joint eigenenergies and eigenstates. 1. By using the series expansion of the exponential function, show that the unitary evolution generated by H^ is U(t) = = (N = 1). Hint: Take care when rearranging sums. 2. 2 marks Consider an initial state 1yoi = P c|Eji expressed in the energy eigenbasis of the above Hamiltonian. Write down the evolved state under unitary evolution for time t, = U(t)| yoi. B. Projective Measurement A three-level system is initially in the state = 1 1 Let 0° = |0ih0|-|1ih1|+|2ih2 be an observable with eigenvalues +1. An ideal (incomplete) projective measurement is made. 1. What are the probabilities of obtaining the eigenvalues +1 or - -1 and what is the expectation value 2. In the case of obtaining the result +1, what is the conditional (normalized) state of the system postmeasurement?

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