A Schrodinger-type equation in homogeneous canonical formalism and protective measurement

The stationary-state problems of neutrons passing through a magnetic field with a slightly non-uniform structure are studied in relation to the problem of the protective measurement proposed by Aharonov, Anandan and Vaidman. For this purpose, we use homogeneous canonical formalism that treats energy-eigenstate equations in quantum mechanics as Schrodinger-type equations with respect to a space variable; then, the problem of neutrons in a non-uniform magnetic field is on common ground with that of a time-dependent Hamiltonian in ordinary quantum mechanics. With this formalism, we derive the structure of dynamical and geometrical phases acquired by neutron states subject to the magnetic field. The result is analyzed from the viewpoint of the protective measurement of the spin states by means of the particle's momentum.