Heisenberg’s inequality

Question

I have tried to write a proof for the following inequality, but I couldn't constitute a rigorous one yet.

Suppose $f$ is absolutely continuous on $[−a,a]$ for all $a \in \mathbb R$ so that $f′$ exists a.e. on $\mathbb R$. Then $$\int_{- \infty}^{\infty} |f(x)|^2 dx \leq 2 \Big ( \int_{- \infty}^{\infty} |xf(x)|^2 dx \Big )^{1/2} \Big ( \int_{- \infty}^{\infty} |f'(x)|^2 dx \Big )^{1/2}$$

Any help will be appreciated.

Answer 1

To get started, you can apply integration by parts to $\displaystyle \int_{-a}^a 1 f(x)^2 \, dx$ to obtain boundary terms and the integral $$ \int_{-a}^a x 2 f(x) f'(x) \, dx.$$ Now apply Cauchy-Schwarz, deal with the boundary terms, and send $a \to \infty$.