Let $\psi(x)$ be a quantum wavefunction on $\mathbb{R}$, that is, a complex function such that $\int_{-\infty}^{\infty} dx |\psi(x)|^2 = 1$. Let $\widetilde{\psi}(p)$ be its Fourier transform: $\widetilde{\psi}(p) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} dx e^{-ipx} \psi(x)$. Let $$ \langle x^d \rangle = \int_{-\infty}^{\infty} dx |\psi(x)|^2 x^d, $$ $$ \langle p^d \rangle = \int_{-\infty}^{\infty} dp |\widetilde{\psi}(p)|^2 p^d. $$ If we are only interested in $d=1,2$ then a complete list of constraints is known, namely $$\langle x^2 \rangle \geq \langle x \rangle^2$$ $$\langle p^2 \rangle \geq \langle p \rangle^2$$ $$(\langle x^2 \rangle - \langle x \rangle^2 ) ( \langle p^2 \rangle - \langle p \rangle^2 ) \geq \frac{1}{4},$$ the last of which is Heisenberg's uncertainty principle. It is not obvious that this set of constraints is complete, i.e. that any list of four numbers $\langle x \rangle, \langle x^2 \rangle, \langle p \rangle, \langle p^2 \rangle$ satisfying the above three inequalities is achievable by some normalized wavefunction $\psi(x)$. But this is proven in the quantum optics literature.
My question is: what if, in addition to $\langle x \rangle, \langle x^2 \rangle, \langle p \rangle, \langle p^2 \rangle$ we also include $\langle x^4 \rangle$ in our list of "moments". Can we find a complete list of constraints for these?
