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Problem 1 Consider a hypothetical system where the particle can be in two positions ₁ and x2. The normalized eigenstates of the particle being in these two positions is denoted as |x₁) and |x2). The two together form an orthonormal basis. (a) What is the matrix X representing the position operator X for this particle in the basis {\x1),|x₂}}? Suppose the system has two energy levels and the Hamiltonian (energy) operator of the system is given ÎI = E (|x₁) (x1| — |X2) (X2| + |X1) (X2| + |X2) (X1|) (1) where is a constant with the dimension of energy. (b) What is the matrix H representing Î with respect to the basis {\x1), \x₂)}. (c) Can the particle have definite position and energy at the same time? Please explain. (d) Find the energy eigenvalues and energy eigenstates (as linear combinations of x₁) and [x₂)). (e) Suppose we made a position measurement and found the particle to be at position ₁. Immediately after that, we now want to measure the particle's energy. What is the probability to find the particle to be in the excited state, i.e. the state with higher energy? by