# answer%20HW7 - HW 7 Solutions Problem 1 Neutrons interact...

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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 ~σ where σ are the Pauli matrices. The Hamiltonian is similar to the electron’s 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 electron’s case where the Hamiltonian is H e = e m ~ I · ~ B . They are: χ ↑ = cos( θ/ 2) e iφ sin( θ/ 2) ! , χ ↓ = e- iφ sin( θ/ 2)- cos( θ/ 2) ! , (2) , for the spin being parallel and anti-parallel respectively. The Berry phase is defined as γ = i Z ( ∇ R × h ψ n |∇ R ψ 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 θ,φ . The textbook has dealt with the spin-parallel case in which the phase was computed to be γ ↑ =- Ω 2 (4) where Ω 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 ∇ χ ↓ = 1 r e- iφ (1 / 2) cos(...
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answer%20HW7 - HW 7 Solutions Problem 1 Neutrons interact...

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