Question in variational method (QM)

[Safinaz]

Homework Statement

Hi, in this book " [Nouredine_Zettili]_Quantum_Mechanics_Concepts ", Eq. (9.133)

Homework Equations

I don't know how the second line

had come from the first line:

The Attempt at a Solution

I got only two terms such that:

$$ < \psi_0| H | \psi_0 > = A^2 \int^{+\infty}_{-\infty} \Big( - \frac{2\alpha^2 h^2}{m} e^{-2\alpha x^2} + \frac{1}{2} m \omega^2 x^2 e^{-2\alpha x^2} \Big) dx $$So what I missed ?

 

[mfb]

I fixed the images in your post.

The first derivative of $e^{-\alpha x^2}$ gives $-2\alpha x e^{-\alpha x^2}$, the second derivative leads to $(-2\alpha + 4\alpha^2 x^2) e^{-\alpha x^2}$ and the second summand should give the missing term.

 

[Safinaz]

Yap ..