Functions of Operators

Let $f$ be a (complex) function with Taylor expansion \[f(z) = \sum_{n=0}^{\infty} a_n z^n,\] and radius of convergence $|z| \leq r$.

We define the function $f$ as a function of operators as \[f(\hat{A}) = \sum_{n=0}^{\infty} a_n \hat{A}^n.\]

It is possible to prove that this series converges if \[\sup_{\{\ket{\psi}| \| \psi \| \}} \Abs{\sand{\psi}{\hat{A}}{\psi}} leq r.\]

Since $[\hat{A}+\hat{B},\hat{C}] = [\hat{A},\hat{C}] + [\hat{B},\hat{C}]$ and $[\hat{A}^{n},\hat{B}] = 0$ whenever $[\hat{A},\hat{B}] = 0$, the fact that functions are defined in terms of power series means that

• If $[\hat{A},\hat{B}] = 0$ then $[f(\hat{A}),\hat{B}] = 0$ for any function $f$.
• In particular, since $[\hat{A},\hat{A}] = 0$, we have $[f(\hat{A}),g(\hat{A})] = 0$ for any functions $f$ and $g.

Since $(\hat{A} + \hat{B})^{\dagger} = \hat{A}^{\dagger} + \hat{B}^{\dagger}$ and $\left ( \hat{A}^n \right )^{\dagger} = \left ( \hat{A}^{\dagger} \right )^n$, for a function \[f(\hat{A}) = \sum_{n=0}^{\infty} a_n \hat{A}^n,\] we have \[f(\hat{A})^{\dagger} = \sum_{n=0}^{\infty} a_n^* \hat{A^{\dagger}}^n.\]

Note, if $\hat{A}$ is a Hermitian operator then $f(\hat{A})$ is a Hermitian operator if and only if the expansion coefficients $a_n$ are real.