(1) The Hamiltonian operator for a two-state system is given by H = a(|1> ...

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(1) The Hamiltonian operator for a two-state system is given by H = a(|1><1| - |2><2| + |1><2| + |2><1|), where a is a number with the dimension of energy. Find the energy eigenvalues and the corresponding energy eigenkets. (2) Consider the space of polynomials of degree less than or equal to n. Any polynomial p(x) = a_0 + a_1 x + ... + a_n x^n can be written as a vector (a_0, a_1, ..., a_n), where the a_i are complex numbers; this is the representation of p(x) in the basis x^0, x^1, ..., x^n. What is the matrix representation of the operator d/dx in this basis?