Question

Observables with more than two possible results (2 pts each part) A. Consider a quantum system with an observable A that has three possible measurement results a1, a2, and az. Write down orthogonality, normalization, and completeness relations for the three basis kets corresponding to possible results of the A measurement, la), la2), and las). Consider a state: It) = C (2 laj-31a2) + V3i las)). Normalize this state (that is, find C), and B. write this state in matrix notation. C. Reproduce McIntyre's Fig. 1.16 for measurement of A in this state. Find the probabilitythatthesystemismeasuredtobeïnthefinalstate»-「In)+zł2) D.

Answer 1

A. The orthononality, normalization relation < a_i vert a_j > = delta_ij The completeness relation: Sum_i vert ai > < ai vert = b B. vert psi > = C (2 vert a_1 > - 3 vert a_2 > + squareroot 3 i vert a_3 >) Normalization gives < psi vert psi > = 1 1 = vert c vert^2 {2 < a_1 vert - 3 < a^2 vert - squareroot 3i < a_3 vert} {2 vert a_1 > - 3 vert a_2 > + squareroot 3 i vert a_3 > } = vert C vert_^2 {4 + 9 + 3} vert C vert^2 middot 16 Therefore vert C vert^2 = 1/16 rightwardsdoublearrow C = 1/4 Therefore Normalized state: vert psi > = 1/4 (2 vert a_1 > - 3 vert a_2 > + squareroot 3 i vert a_3 >) C. vert psi > A vert a_1 > rightwardsdoublearrow 1/4 a_2 > rightwardsdoublearrow 9/16 vert a_3 > rightwardsdoublearrow 3/16 D. The probability of finding the state in vert phi > = 1/squareroot 2 vert a_1 > + 1/squareroot 2 vert a_2 > is vert < phi vert psi > vert^2