text: As an extension of the refocussing pulse treated above, consider a set of two pulses with arbitrary flip angles  and , that is sequence
  where again  is a time interval. Liberally dropping irrelevant terms, the evolution for a single spin with offset  up to just after the second pulse is
  Now consider an ensemble of spins in a magnetic field that is sufficiently inhomogeneous to completely dephase the spins in the interval between the pulses. After the second pulse, we can decompose the remaining terms into a sum of two spin populations differing only in the sign of the  term, in the sense that for an individual spin we have
  where we used the identities  and .
  It is the spins in the new population that has been generated by the second pulse, namely the one with , that will lead to the formation of an echo after evolution for the next  interval. Therefore, remembering to include the  introduced by the first pulse, the amplitude of the resulting Hahn echo relative to that produced by an ideal 90°—180° refocussing pulse sequence is roughly
  Note that this is not an exact result, because it considers only the refocussing of polarisation that was transverse immediately before the second pulse. In reality there will be further transverse components originating from the tipping of the longitudinal magnetisation that remained after the first pulse. However, for many tip angles, this is a good rule of thumb.
  To instead arrive at this result using the state formalism, we would have had to non-trivially evaluate the rotation propagator as
  and then evaluate a transition probability by considering the result of applying this to a state representing polarisation in the transverse plane.