Suppose we have two spin-1/2 particles, with associated spin operators and 2. The simultaneous eigenstates of...

  

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Suppose we have two spin-1/2 particles, with associated spin operators and 2. The simultaneous eigenstates of S1 and S2z are m, m2), which we can write as It. t), It), H), H). The spins interact with each other, such that the two-particle Hamiltonian is = a1 - $2, where a is a real constant (with units of energy divided by h). (a) Does the Hamiltonian (2.2) have rotational symmetry? Justify your answer. (2.1) (2.2) (b) Show that t, ) and ) are eigenstates of the Hamiltonian (2.2), and determine their energy eigenvalues. (c) Determine the remaining energy eigenstates that involve the states |,) and ), along with the corresponding energy eigenvalues. (d) For a > 0, identify the ground state and discuss whether or not it is entangled. How do your results change when a < 0? Suppose we have two spin-1/2 particles, with associated spin operators and 2. The simultaneous eigenstates of S1 and S2z are m, m2), which we can write as It. t), It), H), H). The spins interact with each other, such that the two-particle Hamiltonian is = a1 - $2, where a is a real constant (with units of energy divided by h). (a) Does the Hamiltonian (2.2) have rotational symmetry? Justify your answer. (2.1) (2.2) (b) Show that t, ) and ) are eigenstates of the Hamiltonian (2.2), and determine their energy eigenvalues. (c) Determine the remaining energy eigenstates that involve the states |,) and ), along with the corresponding energy eigenvalues. (d) For a > 0, identify the ground state and discuss whether or not it is entangled. How do your results change when a < 0?