A Hermitian operator Aˆ has only three normalized eigenfunctions ψ1, ψ2, ψ3, with corresponding eigenvalues a1 = 1, a2 = 2, a3 = 3, respectively. For a particular state Φ of the system, there is a 50% chance that a measure of A produces a1 and equal chances for either a2 or a3. (a) Calculate hAi. (b) Express the normalized wave function Φ of the system in terms of the eigenfunctions of Aˆ.
a) "<A> = 7\/4"
b) "\\phi=\\dfrac{1}{\\sqrt2}\\psi_1+\\dfrac{1}{2}\\psi_2+\\dfrac{1}{2}\\psi_3"
See the detailed derivation here: https://www.grandinetti.org/resources/Teaching/Chem4300/Au2017-Exam2-KEY.pdf page 9.
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