(a) Assuming that the system of Problem 3.4 is initially in the state |\phi _{3}〉, what values for the energy and the observable A will be obtained if we measure: (i)H first then A, (ii) A first then H?
(b) Compare the results obtained in (i) and (ii) and infer whether \hat{H} and \hat{A} re compatible. Calculate [\hat{H},\hat{A}]|\phi _{3}〉.
(a) (i) The measurement of H first then A is represented by \hat{A} \hat{H}|\phi _{3}〉. Using the relations \hat{H}|\phi _{n}〉=n^{2} \varepsilon _{0} |\phi _{n}〉 and \hat{A}|\phi _{n}〉=na_{0}|\phi _{n}+1〉, we have
\hat{A}\hat{H}|\phi _{3}〉=9\varepsilon _{0}\hat{A}|\phi _{3}〉=27\varepsilon _{0}a_{0}|\phi _{4}〉. (3.174)
(ii) Measuring A first and then H, we will obtain
\hat{H}\hat{A}|\phi _{3}〉=3a_{0}\hat{H}|\phi _{4}〉=48 \varepsilon _{0}a_{0}|\phi _{4}〉. (3.175)
(b) Equations (3.174) and (3.175) show that the actions of \hat {A} \hat{H} and \hat{H}\hat{A} yield different results. This means that \hat{H} and \hat {A} do not commute; hence they are not compatible. We can thus write
[\hat{H},\hat{A}]|\phi _{3}〉=(48-27)\varepsilon _{0}a_{0} |\phi _{4}〉=17\varepsilon _{0}a_{0}|\phi _{4}〉. (3.176)