Expectation Value for a Function with Cusp

[Septim]

Homework Statement

In my homework assignment I have a wavefunction defined as \Psi(x)=N\exp(-|x|/a) and I am asked to find the expectation value of momentum squared in configuration space.

Homework Equations

\int\Psi*(x)\hat{p^2}\Psi(x)dx

The Attempt at a Solution

N is 1/\sqrt{a} due to the normalization requirement.The first derivative of the wavefunction is piecewise defined hence the second derivative is discontinuous at x=0. I am having difficulties in expressing the first derivative and second derivative in terms of signum function and Dirac delta function. I would like to express them as analytically as possible but due to my unfamiliarity with these functions I am unable to do so. If I disregard the discontinuity of the first derivative function at x=0 the expectation value turns out to be negative. How can I overcome this situation?

Thanks for the replies.
Edit: I have attached the original worksheet in case you would like to look at it.

 

[clamtrox]

Hang on, isn't the second derivative continuous? Differentiating twice brings down two signs of x, so they cancel and limit is same from up and down. If the calculation gives you trouble, just do it in two parts, first for positive and then negative x.

 

[Septim]

The first derivative has a jump at x=0 and the second derivative is infinite at x=0. My question is that how to deal with this situation analytically by using Dirac Delta function and Unit step function.

 

[clamtrox]

Can you reproduce your differentiation steps here? You are making some mistake, because the second derivative of psi is clearly a^{-2} \Psi