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Book: Cohen Cap. 6 Ex.2
2. Consider an arbitrary physical system whose four-dimensional state space is spanned by a basis of four eigenvectors |j, m₂ > common to J² and J₂ (j = 0 or 1; -j≤ m₂ < +j), of eigenvalues jj + 1)ħ² and mh, such that: Jt|j₁m₂ >=ħ√√j(j + 1) − m (m₂ ± 1) | j, m₂ ± 1) J₂|j, j = J_ | j, -j> = 0 a. Express in terms of the kets |j, m.), the eigenstates common to J² and J, to be denoted by j, m, >. b. Consider a system in the normalized state: |v> = x | j = 1, m₂ = 1 ) + B | j = 1, m₂ = 0) + y | j = 1, m, = measured -1) + 8 | j = 0, m₂ = 0) (i) What is the probability of finding 2ħ² and ħ if J² and Jare simultaneously? (ii) Calculate the mean value of J, when the system is in the state | >. and the probabilities of the various possible results of a measurement bearing only on this observable. (iii) Same questions for the observable J² and for J (iv) J is now measured; what are the possible results, their proba- bilities, and their mean value?