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1. We begin with a two state system with states labeled by |1) and [2). This may seem unphysical; however, there are many two state systems in quantum mechanics such spin 1/2 particles. The Hamiltonian we consider is (a) Compute the eigenvalues of H (b) Compute the eigenvectors of H, normalize them, and express them both as column vectors and in terms of | 1〉 and |2) (c) Denoting the two eigenvectors as lva) and |Vb), compute l/a) <>a and lvb)(hb (d) Verify that where 1 is the identity matrix. (e) Suppose at t = 0, the system is in state 1: |ψ(t = 0)〉 = |1〉, Express |s(t 0) in terms of |ψ.) and using (f) Compute lb(t)) using (g) What is the probability at time t that the system is in state |1)? (Hint: compute

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