# Prove Heisenberg uncertainty principle (measure and integration theory)

Here is a question in measure and integration theory,

Let $f$ be a continuously differentiable complex function on $\mathbf{R}$ s.t. the functions $x \mapsto xf(x)$ and $f'$ are in $L_{\mathbf{C}}^2(\mathbf{R},\lambda)$.

Show that $$x|f(x)|^2dx\leq 4(\int\limits_x^\infty t^2|f(t)|^2dt)^{1/2}(\int\limits_x^\infty |f'(t)|^2dt)^{1/2}$$

and use that to prove Heisenberg uncertainty inequality:

$$\int\limits_{-\infty}^\infty|f(x)|^2\leq 2(\int\limits_{-\infty}^\infty x^2|f(x)|^2dx)^{1/2}(\int\limits_{-\infty}^\infty |f'(x)|^2)^{1/2}dx$$

Well I'm reading through complex measures on Hilbert spaces and I'm not able to solve this. Any help appreciated.

Regards, Raxel.

• Better suited to Physics.StackExchange.com perhaps? – Pieter Geerkens Apr 15 '13 at 3:08
• Well it would be suitable there as well. Since this is just the mathematical perspective I think this fits also well in here and I want to solve this purely by measure and integration theory so I guess this is the right place. – Raxel Apr 15 '13 at 3:13
• I would not post it in PSE. Questions focussing on mathematical methods or even asking for rigourous mathematical proofs of laws of physics tend either to be migrated to MSE or to become the target of unpolite comments by people who do not even know that a null denominator at a point can well make a function discontinuous and unbounded... (ex. of both things here) – Self-teaching worker Mar 6 '16 at 13:46