Summary: This paper deals with the derivation of a sharp estimate on the difference of traces of the one-parameter Schrödinger semigroup associated with the quantum isotropic harmonic oscillator. Denoting by \(H_{\infty,\kappa}\) the self-adjoint realization in \(L^2(\mathbb R^d)\), \(d\in\{1,2,3\}\) of the Schrödinger operator \(-\frac{1}{2}\Delta+\frac{1}{2}\kappa^2|\mathbf x|^2\), \(\kappa>0\) and by \(H_{L,\kappa}\), \(L>0\) the Dirichlet realization in \(L^2(\Lambda_L^d)\) where \(\Lambda_L^d:=\{\mathbf x\in\mathbb R^d:-\frac{L}{2}<x_l<\frac{L}{2},l=1,\dots,d\}\), we prove that the difference of traces \(\mathrm{Tr}_{L^2(\mathbb R^d)}\mathrm e^{-tH_{\infty,\kappa}}-\mathrm{Tr}_{L^2(\Lambda_L^d)}\mathrm e^{-tH_{L,\kappa}}\), \(t>0\) has for \(L\) sufficiently large a Gaussian decay in \(L\). Furthermore, the estimate that we derive is sharp in the two following senses: its behavior when \(t\downarrow 0\) is similar to the one given by \(\mathrm{Tr}_{L^2(\mathbb R^d)}\mathrm e^{-t H_{\infty,\kappa}}=(2\sinh(\frac{\kappa}{2}t))^{-d}\) and the exponential decay in \(t\) arising from \(\mathrm{Tr}_{L^2(\mathbb R^d)}\mathrm e^{-tH_{\infty,\kappa}}\) when \(t\uparrow\infty\) is preserved. For illustrative purposes, we give a simple application within the framework of quantum statistical mechanics.