Quantum Hermite functions and Fourier Transform of operators

We construct operator analogues of Hermite functions which form an orthonormal basis for the Hilbert space ${\mathcal{S}}_2$ of Hilbert-Schmidt operators on $L^2({\mathbb{R}}^n)\mathrm{.}$ We use this orthonormal basis to define Fourier transform on ${\mathcal{S}}_2$ and study some of its basic properties.