Construction of a sequence associated to the Heisenberg uncertainty principle.

We consider the function with the form $$ f=c\cdot e^{-(k_1 x_1^2+k_2 x_2^2+x_3^2+\cdots+x_n^2)/2} $$ and then adjust the coefficients. First, since $$ \Vert f\Vert_{L^2}=1$$ we have \begin{align*} 1 &=c^2\cdot \Big(\int e^{-k_1 x_1^2}\, dx_1\Big)\Big(\int e^{-k_2 x_2^2}\, dx_2\Big)\Big(\int e^{-x_3^2}\, dx_3\Big)\cdots \Big(\int e^{- x_n^2}\, dx_n\Big) \\ &= c^2\cdot \big(\frac{\pi}{k_1}\big)^{1/2}\cdot \big(\frac{\pi}{k_1}\big)^{1/2}\cdot \pi^{(n-2)/2} \end{align*} thus, we can take $$c=\Big(\frac{k_1 k_2}{\pi^n}\Big)^{1/4}. $$ Furthermore, we have \begin{align*} \int x_1^2 |f|^2\,dx &=c^2\Big(\int x_1^2 e^{-k_1 x_1^2}\,dx_1\Big)\cdot \Big(\int e^{-k_2 x_2^2}\,dx_1\Big)\Big(\int e^{-x_3^2}\, dx_3\Big)\cdots \Big(\int e^{- x_n^2}\, dx_n\Big)\\ &=\Big(\frac{k_1 k_2}{\pi^n}\Big)^{1/2}\cdot \frac{\pi^{1/2}}{2k_1^{3/2}}\cdot \frac{\pi^{1/2}}{k_2^{1/2}}\cdot \pi^{(n-2)/2}\\ &=\frac{1}{2k_1}, \end{align*} and \begin{align*} \int \xi_2^2|\hat{f}|^2\,dx &=\int |\partial_{x_2}f|^2\,dx\\ &=\int |k_2x_2\cdot f|^2\,dx\\ &=k_2^2\cdot \frac{1}{2k_2}\\ &=\frac{k_2}{2} \end{align*} thus $$\Big(\int x_1^2|f|^2\,dx\Big)\Big(\int \xi_2^2|\hat{f}|^2\,dx\Big)=\frac{k_2}{4k_1} $$ so, what we need is to take suitable coefficients $k_1,k_2$ such that $$\frac{k_2}{k_1}\to 0.$$