I've been solving a problem in quantum mechanics, and I was deriving the standard deviation of $P$, knowing that $\langle P\rangle=0$. Because $\Delta P=\sqrt{\langle P^2 \rangle - \langle P \rangle ^2} = \sqrt{\langle P^2 \rangle}$, I was trying to calculate the expectation value of the square of the momentum. The wave function was given by $\psi(x)=\sqrt{\alpha}e^{-\alpha|x|}$ where $\alpha>0$.
Here is what I've done. $$\langle P^2\rangle = \int_{-\infty}^{\infty}\psi^*(x) \left(-\hbar^2 \frac{d^2}{dx^2}\psi(x)\right)dx = -\hbar^2\alpha^2$$ Now, we have negative expectation value of the square of the momentum, which I think is wierd, and we have to take square root of that value. That's impossible. I couldn't find out what's wrong with my idea. Can somebody help me with this?