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Unformatted text preview: HW 7 Solutions Problem 1: Neutrons interact with the external magnetic field via their spin ~ I = h 2 ~ where are the Pauli matrices. The Hamiltonian is similar to the electrons case up to a multiplicative constant due to a different g- factor. Indeed, the Hamiltonian reads: H =- g N e 2 m N ~ I ~ B (1) where m N is the mass of the neutron and e the electronic charge. From experiments, g N - 3 . 8. The eigenvectors are then the same as in the electrons case where the Hamiltonian is H e = e m ~ I ~ B . They are: = cos( / 2) e i sin( / 2) ! , = e- i sin( / 2)- cos( / 2) ! , (2) , for the spin being parallel and anti-parallel respectively. The Berry phase is defined as = i Z ( R h n | R n i ) dA (3) where ~ R are the parameters of the Hamiltonian and dA is the infinitesimal area in the parameter space. In our case, the parameters space is the two-dimensional sphere parametrized by , . The textbook has dealt with the spin-parallel case in which the phase was computed to be =- 2 (4) where is the solid angle swept by the magnetic field in the adiabatic process. Similarly, we can derive that for the spin-anti-parallel case by computing = 1 r e- i (1 / 2) cos(...
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