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THE GEOMETRY OF MATHEMATICAL METHODS

Section 5.1 Commuting Matrices

Commuting operators play a very special role in the theory of quantum mechanics.

Definition 5.1. Commutator.

The commutator of two matrices (or operators) is defined by the combination
\begin{equation} [M,N]\equiv MN-NM\text{.}\tag{5.1.1} \end{equation}
When acting on a vector, the commutator tells you about the difference between operating in one order and operating in the other order. If the order doesn’t matter, then the commutator is zero, \([M,N]=0\text{,}\) and the operators \(M\) and \(N\) are said to commute.

Commuting operators have the same (non-degenerate) eigenvectors.

Suppose two operators \(M\) and \(N\) commute, \([M,N]\equiv MN-NM=0\text{.}\) Then if \(M\) has an eigenvector \(\vert v\rangle\) with non-degenerate eigenvalue \(\lambda_v\text{,}\) we will show that \(\vert v\rangle\) is also an eigenvector of \(N\text{.}\)
\begin{align} M\vert v\rangle \amp = \lambda_v\vert v\rangle\tag{5.1.2}\\ N\left(M\vert v\rangle\right) \amp = M\left(N\vert v\rangle\right) =\lambda_v\left(N\vert v\rangle\right)\tag{5.1.3} \end{align}
The last equality shows that \(N\vert v\rangle\) is also an eigenvector of \(M\) with the same non-degenerate eigenvalue \(\lambda_v\text{.}\) But if this is true, then \(N\vert v\rangle\) must be proportional to \(\vert v\rangle\text{,}\) i.e.
\begin{equation} N\vert v\rangle = \alpha \vert v\rangle\tag{5.1.4} \end{equation}
which is just the statement that \(\vert v\rangle\) is also an eigenvector of \(N\) with eigenvalue \(\alpha\text{.}\)
Note that if \(\lambda_v\) were a degenerate eigenvalue, then we would not have been able to assume that \(N\vert v\rangle\) is proportional to \(\vert v\rangle\text{.}\)