Integration seems gaussian but the answer does not match

[tfhub]

Homework Statement

-h^2/2m (sqrt(2b/pi)) e^(-bx^2) d^2/dx^2 (e^(-bx^2)) dx from - to + infinity

Homework Equations

I tried differentiating e^(-bx^2) twice and it came up weird , I positioned the values and finally cam up with (-2b sqrt(pi/2b)...is there any other way to do it ?

The Attempt at a Solution

I tried with gaussian integration and my final answer is h^2b/m but it should be h^2b/2m... how am i missing the 1/2 factor?

 

[Orodruin]

It is difficult to say where you are going wrong if you do not show us exactly what you did step by step.

 

[Ray Vickson]

Homework Statement

-h^2/2m (sqrt(2b/pi)) e^(-bx^2) d^2/dx^2 (e^(-bx^2)) dx from - to + infinity

Homework Equations

I tried differentiating e^(-bx^2) twice and it came up weird , I positioned the values and finally cam up with (-2b sqrt(pi/2b)...is there any other way to do it ?

The Attempt at a Solution

I tried with gaussian integration and my final answer is h^2b/m but it should be h^2b/2m... how am i missing the 1/2 factor?

If you mean that you came up with -2b sqrt(pi/2b) for the integral--that is, that
$$\int_{-\infty}^{\infty} e^{-bx^2} \frac{d^2}{dx^2} e^{-b x^2} \, dx =- 2b \sqrt{\frac{\pi}{2b}},$$
then you are off by a factor or 2: you should have $-b \sqrt{\pi/2b}$. You need to show your work in detail.