The exact solution of magnetic resonance for arbitrary spin using an operator-based formalism

The time evolution of a spin precessing in a large static field with a perpendicular rotating field is a classic problem in magnetic resonance that appears in nearly all quantum textbooks. It can be solved exactly by going to the rotating frame. Often, however, the final transition probabilities are determined only for the spin 1/2 case. In this work, we show how an operator-based approach naturally produces a time-ordered product via a Trotter product formula and then by employing simple operator identities such as the Hadamard lemma, the Baker–Campbell–Hausdorff formula, and the exponential disentangling identity, the problem can be directly solved for arbitrary spin.