text: The Hamiltonian for a single spin  evolving under a chemical shift of angular frequency  is
  which means that in an ensemble of many such spins with slightly different chemical shifts, there is a dephasing of the magnetisation in the - plane. Consider the pulse sequence
  where  is a time interval. Starting in an equilibrium state with all the polarisation along the -axis, the evolution of an individual spin in the ensemble is
  Hence this sequence refocuses the transverse magnetisation produced by the first pulse, independent of the value of the chemical shift.
  As an indication of the utility of the formalism, suppose instead that we tried to reach the same result using states only and therefore the Schrödinger time evolution operators. This amounts to trying to simplify the unitary propagator  taking the initial state to the final state  as , where explicitly
  Essentially we want to find the propagator in the form , that is as a single exponential of a combination of operators, because that gives the effective Hamiltonian acting during the sequence. Since the arguments of the exponentials in the original form of the propagator do not commute, this amounts to solving a specific example of the Baker–Campbell–Hausdorff (BCH) problem. In this relatively simple case we can solve the BCH problem using the fact that  for unitary operator , operator  and function , as well as the mathematical similarity of the spin operators with the physical rotation generators, which allow us to write
  Hence  and only the effect of the 180° pulse remains, which agrees with the product operator treatment. For larger sequences of pulses this state treatment quickly becomes even more unwieldy, unless more advanced methods such as exact effective Hamiltonian theory (which gives closed-form expressions for the entangled propagators via the Cayley-Hamilton theorem and eigendecompositions) are used.