Perturbation Approach for Nuclear Magnetic Resonance with a General Periodic Transverse Field

Quantum mechanically, when a nucleus of magnetic moment m_N and spin I interacts with a constant magnetic field H₀ in the z direction and with a weaker time t-dependent field H₁cos(wt) in the x direction, the standard Hamiltonian H for nuclear magnetic resonance is H = I_zw₀+I_xw₁cos(wt), where w₀= m_NH₀ and w₁= m_NH₁. More generally, the time-dependent field can have a more general periodic time dependence, so that the most general nuclear magnetic resonance Hamiltonian may be written as H = I_zw₀+I_xw₁f(t), where f(t) is a general periodic function of the time t with period 2p/w, which can be represented by a Fourier series expansion. Although, to our knowledge, this problem has not yet been solved exactly, we solved this problem using time-dependent perturbation theory in the interaction picture to third order in perturbation theory. Although the expression for the amplitude of a transition from the most general initial state, a linear combination of the 2I+1 quantum states |I,m> , to the final state is complicated, some numerical results valid to this order will be presented. Such results should prove useful for experimental analyses.