2. An operator A, corresponding to an observable a, has two eigenfunctions 1 and 2, with eigenvalues a₁ and a2 respectively. A second operator B, corre- sponding to an observable ẞ, has two normalised, orthogonal eigenfunctions X1 and X2, with eigenvalues b₁ and b₂ respectively. The eigenfunctions are related by: Φι (X1 +3X2) √10 Φ2 (3X1 - X2) √10 (a) Show that the eigenfunctions 1 and 2 are normalised and orthogonal. (b) What is the probability of obtaining a value b₁ if ẞ is measured for a state described by ₁? (c) If a similar measurement is made on the system returned to the state described by 1, what must the probability of obtaining a value of b₂ be? Demonstrate that this is the case. (d) If a second measurement of a is made immediately after a measurement of a giving a value a₁, what will be the result of the second measurement? Briefly explain your reasoning. (e) If, following the first measurement of a in part (d), ẞ is then measured, followed by a again, show that the probability of obtaining a₁ a second time is 82/100.

2. An operator A, corresponding to an observable a, has two eigenfunctions 1 and 2, with eigenvalues a₁ and a2 respectively. A second operator B, corre- sponding to an observable ẞ, has two normalised, orthogonal eigenfunctions X1 and X2, with eigenvalues b₁ and b₂ respectively. The eigenfunctions are related by: Φι (X1 +3X2) √10 Φ2 (3X1 - X2) √10 (a) Show that the eigenfunctions 1 and 2 are normalised and orthogonal. (b) What is the probability of obtaining a value b₁ if ẞ is measured for a state described by ₁? (c) If a similar measurement is made on the system returned to the state described by 1, what must the probability of obtaining a value of b₂ be? Demonstrate that this is the case. (d) If a second measurement of a is made immediately after a measurement of a giving a value a₁, what will be the result of the second measurement? Briefly explain your reasoning. (e) If, following the first measurement of a in part (d), ẞ is then measured, followed by a again, show that the probability of obtaining a₁ a second time is 82/100.