Show that 1/2 are eigenfunctions of the one-spin Hamiltonian when it is written in angular frequency units

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Show that ψ±1/2 are eigenfunctions of the one-spin Hamiltonian when it is written in angular frequency units

hence find the corresponding eigenvalues. You should assume that

i.e. that ψ±1/2, are eigenfunctions of z with eigenvalues ±1/2.

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