Quantum Mechanics: Concepts and Applications · Problem 3.20

One-Dimensional Problems — Nouredine Zettili

Physics — Quantum Mechanics · compatible-observables-csco-matrices

Problem

Consider a physical system which has a number of observables that are represented by the following matrices: A=(100001010),B=(00−100i−1−i4),C=(200013031).A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}, \quad B = \begin{pmatrix} 0 & 0 & -1 \\ 0 & 0 & i \\ -1 & -i & 4 \end{pmatrix}, \quad C = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 1 & 3 \\ 0 & 3 & 1 \end{pmatrix}. (a) Find the results of the measurements of the compatible 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}\}, {A^,B^}\{\hat{A}, \hat{B}\}, {A^,C^}\{\hat{A}, \hat{C}\}, {B^,C^}\{\hat{B}, \hat{C}\} form a complete set of commuting operators?
Intuitive explanation

In quantum mechanics, observables are compatible—measurable simultaneously with no uncertainty trade-off—if and only if their matrices commute, [A,B]=AB−BA=0[A,B] = AB - BA = 0. This means they share the same eigenvectors, forming a common basis where both are diagonal. To find compatible pairs, compute commutators; then diagonalize those pairs jointly to find their shared eigenbasis. A complete set of commuting operators must commute pairwise and have enough independent eigenvectors to uniquely label all quantum states in your system.

Intuitive explanation

In quantum mechanics, observables are compatible—measurable simultaneously with no uncertainty trade-off—if and only if their matrices commute, [A,B]=AB−BA=0[A,B] = AB - BA = 0. This means they share the same eigenvectors, forming a common basis where both are diagonal. To find compatible pairs, compute commutators; then diagonalize those pairs jointly to find their shared eigenbasis. A complete set of commuting operators must commute pairwise and have enough independent eigenvectors to uniquely label all quantum states in your system.

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