Recall that for a system of two particles, we can construct states [s,m> that are eigenstates of both Sz = S1z + S2z and S² = (S1 + S2)² These states must satisfy s2s,ms (s+1) ħ 2 |s,m> equation (1) Sz |s,m> = mh|s,m> equation (2) Recall that these states [s,m> can be found as linear combinations of the direct product states [S1, m1> >|s2, m2>. Consider the system of two spin ½ particles, so that s₁ = s2 = 1/2. As found in class, the two particles form a system with either spin 1 or spin 0. In class and in the textbook, we found the states [1,1>, [1,0>, |1,-1>, and 10,0> by guessing that they were the same linear combinations of |1/2,1/2≥1 |1/2,1/2>2, |1/2,1/2>1 |1/2,-1/2>2, |1/2,-1/2>1|1/2,1/2>2, and 1/2,-1/2>1 as eigenstates of the spin-spin Hamiltonian, and then verifying that these linear combinations satisfied equations (1) and (2) for the appropriate values of's and m. In this problem, you will find the same result in a different way. (a) Using the rule that m = m1 + m2, find the state [1,-1> as a direct product of single-particle states. Verify by checking that your result satisfies equations (1) and (2) above. |1/2,-1/2>2 that we had previously found (b) By applying the raising operator to the state [1,-1> from (a), find the state [1,0> in terms of single-particle states. Verify by checking that your result satisfies equations (1) and (2) above. (c) By applying the raising operator to the state |1,0> from (b), find the state [1,1>. in terms of single-particle states Verify by checking that your result satisfies equations (1) and (2) above. (d) By using the requirements that m = m1 + m2 and that [1,0> and 10, 0> must be orthogonal, find the state 10,0>. Verify by checking that your result satisfies equations (1) and (2) above.

Recall that for a system of two particles, we can construct states [s,m> that are eigenstates of both Sz = S1z + S2z and S² = (S1 + S2)² These states must satisfy s2s,ms (s+1) ħ 2 |s,m> equation (1) Sz |s,m> = mh|s,m> equation (2) Recall that these states [s,m> can be found as linear combinations of the direct product states [S1, m1> >|s2, m2>. Consider the system of two spin ½ particles, so that s₁ = s2 = 1/2. As found in class, the two particles form a system with either spin 1 or spin 0. In class and in the textbook, we found the states [1,1>, [1,0>, |1,-1>, and 10,0> by guessing that they were the same linear combinations of |1/2,1/2≥1 |1/2,1/2>2, |1/2,1/2>1 |1/2,-1/2>2, |1/2,-1/2>1|1/2,1/2>2, and 1/2,-1/2>1 as eigenstates of the spin-spin Hamiltonian, and then verifying that these linear combinations satisfied equations (1) and (2) for the appropriate values of's and m. In this problem, you will find the same result in a different way. (a) Using the rule that m = m1 + m2, find the state [1,-1> as a direct product of single-particle states. Verify by checking that your result satisfies equations (1) and (2) above. |1/2,-1/2>2 that we had previously found (b) By applying the raising operator to the state [1,-1> from (a), find the state [1,0> in terms of single-particle states. Verify by checking that your result satisfies equations (1) and (2) above. (c) By applying the raising operator to the state |1,0> from (b), find the state [1,1>. in terms of single-particle states Verify by checking that your result satisfies equations (1) and (2) above. (d) By using the requirements that m = m1 + m2 and that [1,0> and 10, 0> must be orthogonal, find the state 10,0>. Verify by checking that your result satisfies equations (1) and (2) above.