the wave functions px and dxz are linear combinations of the spherical harmonic functions, which are eigenfunctions of the operators H, 12, Iz for rotation in three dimensions. The combinations have been chosen to

Show the relation LxL = iħL for the quantum mechanical angular momentum operator L

The normalized wavefunction is (1/4*pi*a^2) e^-(x^2)/(2a^2)

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.