Minimum uncertainty for antisymmetric wave functions (Q1386544)

Let \(G(x)= -x\log x\). The author conjectures that for every odd \(\psi\in L^2(\mathbb{R})\) of norm \(1\) we have \[ \int\bigl[G(|\psi|^2)+ G(|\widehat\psi|^2)\bigr]\geq 2(1- \log 2). \] Some numerical evidence in support of this is presented. It is shown rigorously that a lower bound on the right cannot be greater. There are many quasimathematical speculations ``a propos'' but of little relevance to the essence.