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Problem 1: A system of two spin-half particles 1.1) What are the possible basis states in terms of the states of the individual particles? Show that z, -z) = |S = 1,m, = -1). 1.2) Using the above, use the raising operator to find |S = 1,m, = 0). Verify it is indeed IS 1, ms 0) by showing it is an eigenket of both S2 and S,. 1.3) Determine the matrix representation of S2 using the m, = 0 states as a basis (i.e., using |z, -z) and |- z, +z)) 1.4) 1.5) Find the eigenvectors and eigenvalues of the matrix from the previous subproblem. With these, determine the eigenket |S = 0,m, = 0). Problem 2: A system of two spin-one particles Use the previous problem as an inspiration for how to solve this one There are 9 possible states. What are they? 2.1) For convenience, use the following shorthand S 1,m m)1 S = 1,m, = m2)2 = \m1, m2) Write the state 1, 1) as |S = X, m = Y) by finding the eigenvalues of 52 and Sz 2.2) 2.3) Determine the representation of IS = 2, m, = 0) in terms of the spin states of the individual particles using the previous results

Problem 1: A system of two spin-half particles 1.1) What are the possible basis states in terms of the states of the individual particles? Show that z, -z) = |S = 1,m, = -1). 1.2) Using the above, use the raising operator to find |S = 1,m, = 0). Verify it is indeed IS 1, ms 0) by showing it is an eigenket of both S2 and S,. 1.3) Determine the matrix representation of S2 using the m, = 0 states as a basis (i.e., using |z, -z) and |- z, +z)) 1.4) 1.5) Find the eigenvectors and eigenvalues of the matrix from the previous subproblem. With these, determine the eigenket |S = 0,m, = 0). Problem 2: A system of two spin-one particles Use the previous problem as an inspiration for how to solve this one There are 9 possible states. What are they? 2.1) For convenience, use the following shorthand S 1,m m)1 S = 1,m, = m2)2 = \m1, m2) Write the state 1, 1) as |S = X, m = Y) by finding the eigenvalues of 52 and Sz 2.2) 2.3) Determine the representation of IS = 2, m, = 0) in terms of the spin states of the individual particles using the previous results

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Problem 1: A system of two spin-half particles 1.1) What are the possible basis states in terms of the states of the individual particles? Show that z, -z) = |S = 1,m, = -1). 1.2) Using the above, use the raising operator to find |S = 1,m, = 0). Verify it is indeed IS 1, ms 0) by showing it is an eigenket of both S2 and S,. 1.3) Determine the matrix representation of S2 using the m, = 0 states as a basis (i.e., using |z, -z) and |- z, +z)) 1.4) 1.5) Find the eigenvectors and eigenvalues of the matrix from the previous subproblem. With these, determine the eigenket |S = 0,m, = 0). Problem 2: A system of two spin-one particles Use the previous problem as an inspiration for how to solve this one There are 9 possible states. What are they? 2.1) For convenience, use the following shorthand S 1,m m)1 S = 1,m, = m2)2 = \m1, m2) Write the state 1, 1) as |S = X, m = Y) by finding the eigenvalues of 52 and Sz 2.2) 2.3) Determine the representation of IS = 2, m, = 0) in terms of the spin states of the individual particles using the previous results
Transcribed Image Text:Problem 1: A system of two spin-half particles 1.1) What are the possible basis states in terms of the states of the individual particles? Show that z, -z) = |S = 1,m, = -1). 1.2) Using the above, use the raising operator to find |S = 1,m, = 0). Verify it is indeed IS 1, ms 0) by showing it is an eigenket of both S2 and S,. 1.3) Determine the matrix representation of S2 using the m, = 0 states as a basis (i.e., using |z, -z) and |- z, +z)) 1.4) 1.5) Find the eigenvectors and eigenvalues of the matrix from the previous subproblem. With these, determine the eigenket |S = 0,m, = 0). Problem 2: A system of two spin-one particles Use the previous problem as an inspiration for how to solve this one There are 9 possible states. What are they? 2.1) For convenience, use the following shorthand S 1,m m)1 S = 1,m, = m2)2 = \m1, m2) Write the state 1, 1) as |S = X, m = Y) by finding the eigenvalues of 52 and Sz 2.2) 2.3) Determine the representation of IS = 2, m, = 0) in terms of the spin states of the individual particles using the previous results
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