A method for measuring small changes in the energy of a neutron has been proposed on the basis of the anomalous behavior of the dispersion of the neutron in the crystal near Bragg "resonance." A high sensitivity of the method allows the observation of the acceleration of the neutron in the alternating magnetic field. It has been found that the small difference between the energies of two spin states of the neutron (parallel and antiparallel to the magnetic field) leads to significant spatial splitting of wave packets and, correspondingly, to the depolarization of the neutron beam.