Quantum Mechanics: Concepts and Applications · Problem 3.10

One-Dimensional Problems — Nouredine Zettili

Physics — Quantum Mechanics · compatible-observables-csco

Problem

Consider a physical system which has a number of observables that are represented by the following matrices: A=(500012021),B=(100003030),C=(030302020),D=(10000−i0i0).A = \begin{pmatrix} 5 & 0 & 0 \\ 0 & 1 & 2 \\ 0 & 2 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 3 \\ 0 & 3 & 0 \end{pmatrix}, \quad C = \begin{pmatrix} 0 & 3 & 0 \\ 3 & 0 & 2 \\ 0 & 2 & 0 \end{pmatrix}, \quad D = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0 \end{pmatrix}. (a) Find the results of the measurements of these observables. (b) Which among these observables are compatible? Give a basis of eigenvectors common to these observables. (c) Which among the sets of operators {A^}\{\hat{A}\}, {B^}\{\hat{B}\}, {C^}\{\hat{C}\}, {D^}\{\hat{D}\} and their various combinations, such as {A^,B^}\{\hat{A}, \hat{B}\}, {A^,C^}\{\hat{A}, \hat{C}\}, {B^,C^}\{\hat{B}, \hat{C}\}, {A^,D^}\{\hat{A}, \hat{D}\}, {A^,B^,C^}\{\hat{A}, \hat{B}, \hat{C}\}, form a complete set of commuting operators (CSCO)?

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