For every $\alpha\in\mathbb C$, the series \begin{align*} \psi_\alpha=e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\psi_n, \end{align*} converges in $L^2(\mathbb R)$ and defines the unique normalised vector in $\mathcal D(a)$ satisfying \begin{align*} a\psi_\alpha=\alpha\psi_\alpha,\qquad (\psi_0,\psi_\alpha)_{L^2}>0. \end{align*}