# Uncertainty relation

1. Jan 20, 2009

### Ohekatos

1. The problem statement, all variables and given/known data
Could someone please have a look at this?
I am to show that from the inequation
$$\langle\left \psi | \hbar^2D^2 | \psi\right\rangle + mk\langle \left\psi | x^2 | \psi\right\rangle\geq\hbar\sqrt{mk}$$
you can get the Heisenberg uncertainty relation
$$\langle\psi | \hbar^2D^2 | \psi\rangle\langle \left\psi | x^2 | \psi\right\rangle\geq\frac{1}{4}\hbar^2$$
for all normalized functions $$\psi \in S(\mathbb{R})$$

2. Relevant equations
I know that
$$H_0=\frac{\hbar^2}{2m}D^2+\frac{1}{2}kx^2$$
and that
$$H\psi=\hbar\omega\sum_{n=0}^{\infty}(n+1/2)\langle\Omega_n | \psi\rangle\Omega_n$$
and that $$H_0\psi=H\psi$$ for $$\omega=\sqrt{k/m}$$

3. The attempt at a solution
I tried to square on both sides:
$$\langle\psi | \hbar^2D^2 | \psi\rangle^2+m^2k^2\langle \left\psi | x^2 | \psi\right\rangle^2+2mk\langle \left\psi | x^2 | \psi\right\rangle\langle\psi | \hbar^2D^2 | \psi\rangle \geq\hbar^2 mk$$

But that doesn't seem to work right