[NOTES?QM-06009] Functions of Operators

Function of an Operator

Let $\widehat{X}$ be an operator which has eigenvalues and eigenvectors \( \{\lambda_k, \ket{u_k}| k=1,2,...\}\). Let us further assume that the span of eigenvectors of $\widehat{X}$ is entire  vector space. Then a function \(\widehat{F}(X)\) of the operator \(\widehat{X}\) is defined by specifying its action on the basis formed by the eigenvectors \(\mathscr{B}=\{\ket{u_k}| k=1,2,...\}\) \begin{equation} \widehat{F}(X) \ket{u_k} \equiv F(\lambda_k) \ket{u_k}, \quad k=1,2, \ldots. \end{equation} The action of the function \(\widehat{F}(X)\) on an arbitrary vector $\ket{\psi}$ is obtained, as usual, by expanding the vector \(\ket{\psi}\) in the basis \(\mathscr{B}\): \begin{eqnarray} \ket{\psi} &=& \sum _k c_k \ket{u_k}\\ \widehat{F}(X) \ket{\psi} &=& \sum c_k F(\lambda_k) \ket{u_k}. \end{eqnarray}

As the hermitian operators and unitary operators have a complete set of orthonormal eigenvectors, their functions are defined by the method outlined above. It is a simple exercise to show that if an operator \(\widehat{Y} \) commutes with \(\widehat{X}\), it commutes with every function of \(\widehat{X}\).

Complete commuting set
A set \(\mathscr{S}\) of operators is called {\it complete commuting set} if it is a commuting set and if any operator which commutes with every member of the set \(\mathscr{S}\) can be written as function of the operators in the set \(\mathscr{S}\). The concept of complete commuting set is important in choosing a basis and working with representations as against with abstract vector space.