Let $\psi\colon\mathbb R\to\mathbb C$, and set $$ F[\psi]:=\pi\ \frac{\displaystyle\int_{\mathbb R} |x|\;|\psi(x)|^2\,\mathrm dx}{\displaystyle\int_{\mathbb R}|\psi(x)|^2\,\mathrm dx}\frac{\displaystyle\int_{\mathbb R} |p|\;|\tilde\psi(p)|^2\,\mathrm dp}{\displaystyle\int_{\mathbb R}|\tilde\psi(p)|^2\,\mathrm dp} $$ with $\tilde\psi$ the Fourier transform of $\psi$. In this physics.SE post we tried to minimise $F[\psi]$ (over some "reasonable" function space, e.g., Schwartz), but we didn't get as far as we would like.

Note that if we were to replace $|x|,|p|\to |x|^2,|p|^2$, we would have $F[\psi]\ge \pi/2$, corresponding to the classical result in Fourier analysis (the Heisenberg uncertainty relation). On the other hand, with the exponents as above, the best we could come up with is a result by user Frédéric Grosshans that $$ F[\psi]\ge \frac{\pi^2}{4e}\sim 0.9077 $$

Numerical analysis seems to suggest that the true bound is actually closer to $0.95$. Has this problem been addressed in the mathematical literature? What is the correct minimum for $F$?

  • $\begingroup$ Just an idea: In view of the general uncertainty reation (e.g. en.wikipedia.org/wiki/…) one could consider the operators $\sqrt{p}$ and $\sqrt{x}$ and then their commutator $[\sqrt{p}, \sqrt{x}]$ (e.g. by using results form Transtrum, Commutation relations for functions of operators, scholarsarchive.byu.edu/cgi/viewcontent.cgi?referer=https://…) $\endgroup$ – Andreas Rüdinger Mar 4 '18 at 18:50
  • $\begingroup$ @AndreasRüdinger Thanks. That's kind of what one of the users (Wolpertinger) tried to do on physics.SE, but they didn't push the calculation very far. It is in any case a promising approach. $\endgroup$ – AccidentalFourierTransform Mar 4 '18 at 19:00
  • $\begingroup$ Another idea: For symmetry reasons it seems one can restrict the problem without loss of generality to integration over $[0, \infty]$. If one futher expands a normalized $\psi(x)$ (i.e. both denominators constant) in terms of Hermite functions, then because of Hermite functions being eigenvalues of the Fourier transformation the functional to minimized is simplified considerably and one ends up with something like minimizing $\sum_{n,m} \int_0^{\infty} a_n a_m x H_n(x) H_m(x) dx$ under the constraint $\sum a_n^2 =1$ or the like, which could be feasable analytically. $\endgroup$ – Andreas Rüdinger Mar 4 '18 at 20:37
  • $\begingroup$ @AndreasRüdinger Thanks again. That's kind of what another user (in this case, me) tried to do on physics.SE, but I only got some numeric resultsr. I do believe this is a very natural approach and I do hope one could obtain some analytic results, but I didn't get anything useful myself. $\endgroup$ – AccidentalFourierTransform Mar 4 '18 at 20:43
  • $\begingroup$ It is reasonable that the minimizer should be be symmetric (an eigenfunction of the Fourier transform perhaps). There is a discrete conjecture for ONBs (Balian-Low uncertainty principle) by Lammers where this type of behavior is expected. $\endgroup$ – Josiah Park Nov 16 '18 at 3:54

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