Question

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.

Answer 1

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.