The hamiltonian for a spin-3 particle placed in a static magnetic field aligned along the z axis is given by h ws, where w, = ^b2 is the larmor frequency of the spin, and y is its gyromagnetic ratio. from lectures, we know that a spin initialised in one of the eigenstates of this hamiltonian (i. e. |s, m) = |2, +2) 2il remain in that state. however, in quantum computing or quantum sensing applications, we can control the state of the spin using an oscillating field applied along an axis perpendicular to z (which we take here to be the x axis without loss of generality), as described by or e-iwts + h = where w= b, and b and w are the strength and frequency of the oscillating field, respectively, and st stisy are the spin ladder operators (a) determine whether or not the full hamiltonian, h = h, h, is hermitian. (b) the time-dependent state vector a(t) b(t) x(t) is governed by the time-dependent schrödinger equation, ih(t) = hx(t). determine the coupled system of differential equations for a(t) and b(t). by eliminating b(t) decouple the system and find the second order differential equation describing the evolution of the upper component a(t) alone. (c) if s initialised in the state, i. e. (6) x(0) the exact solution for the evolution of a(t) is given by i(w-ws -itcos t + a(t) 20 vw4 (w-ws)2. where (i) find expressions for the time-dependent probabilities p4 (t) and p_(t) of finding the system in the ,+2) and ,2)states, respectively. (ii) using your answer to (i), examine the behaviour of p4 (t) and p_ (t) for the two cases: on-resonance (w = wz) and far off-resonance (jw - w^ > > comment on the physical significance of tuning the frequency of the oscillating field to the spin's larmor frequency, compared to the off-resonance case. this is the fundamental physics underpinning magnetic resonance-based control of spin systems (iii) suppose our spin- system is a qubit in a quantum computer and that we wish to prepare the system in an equal superposition of |, +2) and |2,-4) states, with amplitude a(t) exactly 1/v2. in terms of applied frequency, relative strengths of the magnetic fields b2 and bz, and amount of time for which we would need to 'switch on' h, determine how to achieve this superposition state from an initial state of |, +). explain prepare the system in a general state |v) = a, +}) + b|} where the magnitudes of the complex amplitudes a and b are specified (and the phases can be arbitrary) how data: - ( ) 0 1 1 0 0 1 0 0