Topic: Quantum Mechanics Hi. Could someone please help me with Problem 7? I also included problem 6 as reference. Proble

[answerhappygod]

Topic: Quantum Mechanics
Hi. Could someone please help me with Problem 7? I
also included problem 6 as reference.
Problem 6 has been resolved by many people but not problem
7. Thank you in advance

Topic Quantum Mechanics Hi Could Someone Please Help Me With Problem 7 I Also Included Problem 6 As Reference Proble 1 (326.75 KiB) Viewed 301 times

6. Consider the system composed of two spin 1/2's, S₁ and S₂, and the basis of four vectors |±, ±) defined in Complement Div. The system at time t = 0 is in the state: 1 | 3³ (0)) = { 1 + +) + 2 | + −) + 3/2 -) Reference exercise for exercise 7 a. At time t = 0, S₁z is measured; what is the probability of finding -ħ/2? What is the state vector after this measurement? If we then measure Siz, what results can be found, and with what probabilities? Answer the same questions for the case where the measurement of S12 yielded +ħ/2. b. When the system is in the state (0)) written above, Siz and S2% are measured simultaneously. What is the probability of finding opposite results? Identical re- sults? c. Instead of performing the preceding measurements, we let the system evolve under the influence of the Hamiltonian: 7. Consider the same system of two spin 1/2's as in the preceding exercise; the state space is spanned by the basis of four states |±, ±). a. Write the 4 x 4 matrix representing, in this basis, the Siy operator. What are the eigenvalues and eigenvectors of this matrix? b. The normalized state of the system is: |) = a + +) + ß| + −) + y − +) + 8| − - - -) where a, 3, 7 and 8 are given complex coefficients. S1 and S2y are measured simultaneously; what results can be found, and with what probabilities? What happens to these probabilities if ) is a tensor product of a vector of the state space of the first spin and a vector of the state space of the second spin? c. Same questions for a measurement of S₁y and S2y. d. Instead of performing the preceding measurements, we measure only S2y. Calcu- late, first from the results of b and then from those of c, the probability of finding -ħ/2.