Answer Happy

3. Consider the following conversation between Andy and Caroline about the above Hamiltonian op- erator: • Andy: Ĥ is essentially S, except for some multiplicative constants. Therefore, the eigenstates of Ŝ will also be the eigenstates of ĤI. • Caroline: No. The presence of magnetic field will make the eigenstates of $; and Ĥ different. The eigenstates of Ĥ will change with time in a non-trivial manner. • Andy: I disagree. If the magnetic field had a time dependence, e.g., B = Bo cos(wt)k, the eigen- states of Ĥ will change with time in a non-trivial manner but not for the present case where B is constant. With whom do you agree? (a) Andy (b) Caroline (c) Neither 239 4. If the eigenstates of S, and Ĥ, | 1), and ), are chosen as the basis vectors, which one of the following is their matrix representation? (a) () and (1) (b) (1) and (8) (c) 1/√2 (1) and 1/√2 (1) (d) 1/√2 (1) and 1/√2 (¹) 5. If we choose| †), and | ↓), as the basis vectors for the two dimensional spin space, which one of the following is the correct expression for a general state [x)? (a) |x) = al 1): +b| 1) where a + b = 1. (b) |x)= a 1)₂ + b ), where |a|² + |b|² = 1. (c) |x) = a 1)₂ + b ), where a and b can be any integers. (d) |x) = a 1)₂ x b| 1), where a and b can be any integers.