Wigner function for noninteracting fermions in hard-wall potentials

The Wigner function $W_N({\bf x}, {\bf p})$ is a useful quantity to characterize the quantum fluctuations of an $N$-body system in its phase space. Here we study for $N$ noninteracting spinless fermions $d$-dimensional spherical hard box radius $R$ at temperature $T=0$. In large limit, local density approximation (LDA) predicts that p}) \approx 1/(2 \pi \hbar)^d$ inside finite region $({\bf plane, namely $|{\bf x}| < R$ and p}| k_F$ where $k_F$ Fermi momentum, while vanishes outside this region, or "droplet", on scale determined by fluctuations. paper investigate systematically, structure along edge droplet, called surf. one dimension, find there are three distinct regions surf compute exactly associated nontrivial scaling functions each regime. We also momentum distribution $\hat \rho_N(p)$ striking algebraic tail very momenta \rho_N(p) \propto 1/p^4$, well beyond $k_F$, reminiscent similar found interacting systems (discussed context Tan's relation). then generalize these results higher $d$ find, remarkably, close universal, i.e., independent dimension~$d$.