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Using the commutation relation for spin, namely that [S_x,S_y]=iS_z (and cyclic premutations), prove that [S.X , S]=iS×X , where X is a vector.

Using the commutation relation for spin, namely that
[Sx, Sy] = iSz (and cyclic permutations), prove that
[Ŝ - X, Ŝ) = iŝ x X,
(1.75)
where X is a vector.

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Consider a system of two particles, one with spin 1 and the other with spin 1/2. The particles interact with one another via the Hamiltonian H = Eo [S(1). S(2)] + E₁ [ S:(1) + S. (2)] ħ " where Eo and E₁ are constants with units of energy, S is the spin operator and the labels (1) and (2) indicate the particle on which it operates. (a) Find the energy spectrum of this system, i.e. the eigenvalues of the Hamiltonian H and identify the ground state and first excited state. For numerical purposes assume Eo = 0.2eV and E₁ = 0.1eV. (b) If the two particle system is known to be in a configuration such that the total spin is 1/2, what values a mea- surement of S% of the spin 1 particle would produce and what are the probabilities associated with each one of those values?
The system described by the Hamiltonian Ho has just two orthogonal energy eigenstates [1> and 12>, with =1=, =0= The two eigenstates have the same energy eigenvalue Eo: Holi> = Eoli>, i=1,2 Now suppose the Hamiltonian for the system is changed by the addition of the term V, giving H = Ho+V The matrix elements of V are =V12=, =0= where V12 is real. a. Find the eigenvalues of the new Hamiltonian, H, in terms of the above quantities. b. Find the normalized eigenstates of H in terms of 11>, 12> and the other given expressions. Hint: Write Ho, V and H as 2x2 matrices and the states as column vectors.
Please obtain the same result as in the book.
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