# e Hamiltonian for a spin I system is given by HI=AS+B(S²-S). Solve this problem exactly to find the normalized energy eigenstates and eigenvalues. (A spin-dependent Hamiltonian of this kind actually appears in crystal physics.) Is this Hamiltonian invariant under time reversal? How do the normalized eigenstates you obtained transform under time reversal?

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The eigenfunctions of Hermitian Operators are orthonormal (orthogonal and normalizable). a. Prove the eigenfunctions of the Hamiltonian Operator for a particle in a box that extends from x = 0 to x = a: плх y.(x) = are orthonormal by integrating a pair of functions, y,(x) and y.(x), with n = m in one case and n m in another. b. For the ground state of a particle in a box, use the momentum and position operators to show that the expectation values are 0 for momentum and - for position, by evaluating the resulting integrals. c. Calculate the uncertainty in x and p (0x and Op) for the lowest energy state of a particle in a box by taking the standard deviation of x and p as a measure of their uncertainty: Ar =0, = (x*)-(x)* and Ap=o,=Kp*)-(p)* Is the product you obtain for OxOp consistent with the Heisenberg Uncertainty Principle?