I believe that the Hamiltonian generated by the external magnetic field in the neutron interferometry experiment effectively leads to an increase in the neutron's potential energy.
Why is this explained merely as a Hamiltonian in the textbooks such as Sakurai MQM or Griffiths' one? This has led me to search and find various sources.
As with the neutron quantum interference experiment induced by gravity, this external magnetic field must ultimately increase the neutron's potential energy for a consistent explanation. This seems evident.
On Wikipedia, it is also indicated that -u * B is explained as potential energy U in classical electromagnetic theory, which makes me certain.
I would appreciate your correct feedback on this matter.
