[NOTES/QM-06007] Compatible Dynamical Variables

Let  \(A\) and \(B\) be two dynamical variables, \(\hat{A}, \hat{B}\) be the corresponding operators, \(\alpha_j, j=1,2,\ldots\) and \(\beta_k, k=1,2,\ldots\) be their eigenvalues. Let us assume that \(A\) and \(B\) can be measured simultaneously. This means there are states in which these variables have definite values \(\alpha_j, \beta_k, j,k =1,2,\ldots\).The corresponding vectors \(\ket{\alpha_j,\beta_k}\) must then be simultaneous eigenvectors of the the two operators. \begin{equation} \hat{A}\ket{\alpha_j,\beta_k} =\alpha_j\ket{\alpha_j,\beta_k}, \qquad \hat{B}\ket{\alpha_j,\beta_k} =\beta_k\ket{\alpha_j,\beta_k} \end{equation}

In order that the probability of getting pair of values \(\alpha_j,\beta_k\) for all pairs \(j,k\) be given by the postulate III, it should be possible to write an arbitrary vector \(\ket{\psi}\) as linear combination of these vectors \( \mathscr{B}=\{\ket{\alpha_j,\beta_k}| j=1,2,..., k=1,2,... \}\) and these states must form a basis.Now it is easy to show that the action of \(\hat{A}\hat{B}-\hat{B}\hat{A}\) on
each of these vectors in the set \(\mathscr{B}\) is zero. In fact
\begin{eqnarray}
  \big(\hat{A}\hat{B} - \hat{B} \hat{A}\big)\ket{\alpha_j,\beta_k}
  &=& \hat{A}\hat{B} \ket{\alpha_j,\beta_k}- \hat{B}
                   \hat{A}\ket{\alpha_j,\beta_k}\\
  &=& \hat{A}\beta_k\ket{\alpha_j,\beta_k} -
            \hat{B}\alpha_j\ket{\alpha_j,\beta_k}\\
   &=& \alpha_j\beta_k  \ket{\alpha_j,\beta_k} - \beta_k\alpha_j
           \ket{\alpha_j,\beta_k}\\
   &=& 0.
\end{eqnarray}
Thus we have proved that the action of  commutator \([\hat{A},\hat{B}]\) on every element of  basis \(\mathscr{B}\) results in zero. This implies that \([\hat{A},\hat{B}]=0\) and the two operators \(\hat{A}, \hat{B}\) must commute.

Conversely, if two hermitian operators commute, one can select a basis of orthonormal vectors which are simultaneous eigenvectors of the two operators.The above considerations generalize to several dynamical variables.

A set of operators \(\{\hat{A}_k,k=1,2,\ldots\}\) is called commuting set if every pair of operators \( \hat{A}_\ell, \hat{A}_m\) commute, i.e.
\begin{equation}
  [\hat{A}_\ell, \hat{A}_m]=0 \quad \text{for all pairs} \ell, m.
\end{equation}

A set of dynamical variables \(\{A_k, k=1,2,\ldots\}\) is called a compatible set if the corresponding set of operators  \(\{\hat{A}_k, k=1,2,\ldots\}\) is a commuting set of operators.