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We prove an optimal semiclassical bound on the trace norm of the followingcommutators $[\boldsymbol{1}_{(-\infty,0]}(H_\hbar),x]$,$[\boldsymbol{1}_{(-\infty,0]}(H_\hbar),-i\hbar\nabla]$ and$[\boldsymbol{1}_{(-\infty,0]}(H_\hbar),e^{itx}]$, where $H_\hbar$ is aSchr\"odinger operator with a semiclassical parameter $\hbar$, $x$ is theposition operator and $-i\hbar\nabla$ is the momentum operator. These boundscorresponds to a mean-field version of bounds introduced as an assumption by N.Benedikter, M. Porta and B. Schlein in a study of the mean-field evolution of afermionic system.

An optimal semiclassical bound on certain commutators