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(b) Let us assume that the general wavefunction of the system upon which \( \hat{A} \) operates is given by: \( \psi=0.707 \psi_{1}+0.316 \psi_{2}+0.632 \psi_{3} \) with \( \hat{A} \psi_{i}=a_{i} \psi_{i} \) and \( i=1,2,3 \). The ensemble of \( \psi_{i} \) forms an orthonormal basis. (i) Describe the meaning of orthonormality. [2] (ii) What is the probability that a measurement of the dynamical variable associated with the operator \( \hat{A} \) on the wavefunction \( \psi \) will give 3 as a result? [3] (iii) What is the mean value of the operator \( \hat{A} \) applied on the wavefunction \( \psi \) ? [4] (iv) Can the mean value ever be obtained as a measurement of the dynamic variable associated with the operator \( \hat{A} \) ? [3] (v) If we have another operator \( \hat{B} \) that commutes with \( \hat{A} \), what can be said about the eigenfunctions and eigenvalues of \( \hat{B} \) ? [2]

(i) Orthonormality refers to the property of a set of vectors such that they are all mutually orthogonal (i.e., perpendicular) and have unit length. I