Questiоn 1: 25 pоints 1. An electrоn is аt rest аt the origin, in the presence of а magnetic field whose magnitude (B0) is constant but whose direction rides around at constant angular velocity ω on the lip of a cone of opening angle α: B(t) = B0[sin(α) cos(ωt)ˆi + sin(α) sin(ωt)ˆj + cos (α)ˆk] (1) 1a) Write the formula for the Hamiltonian for a particle with spin coupled to a magnetic field. (You only need to write the coupling Hamiltonian.)[5 points] 1b) Construct the 2 × 2 Hamiltonian matrix for this system. [7 points] 1c) Use first-order time-dependent perturbation theory to calculate the approximate probability of a transition from spin up (the initial state) to spin down, as a function of time. [8 points] 1d) State the criterion on the strength of the field that determines whetherperturbation theory is applicable in this case. [5 points]