A one-dimensional classical harmonic oscillator with translation total energy E = 1/ 2m (p2) + 1/2 kx**2: A) Find the partition function. b) Find the average energy. c) Find the Helmholtz free energy. d) Find its entropy.

A qubit is in state |) = o|0) +₁|1) at time t = 0. It then evolves according to the Schrödinger equation with the Hamiltonian Ĥ defined by its action on the basis vectors: Ĥ0) = 0|0) and Ĥ|1) = E|1), where E is a constant with units of energy. a) Solve for the state of the qubit at time t. b) Find the probability to observe the qubit in state 0 at time t. Explain the result by referring to the way that the time-evolution transforms the Bloch sphere.