Function with a discontinuity in its first derivative

Question

I want to compute this integral, $I = \int_{-\infty}^{+\infty} \psi(x)\frac{d^2}{dx^2} \psi(x) dx$. Now, consider a function which has a discontinuity in its first derivative. For an example I'll consider a triangular wave centered at $\frac \Delta 2$. So,

$$ \begin{equation} \psi(x) = \begin{cases} A x, & 0 \le x \le \frac \Delta 2 \\ A (\Delta - x), & \frac \Delta 2 \le x \le \Delta \end{cases} \end{equation} \tag 1 $$

where $A = \sqrt{\frac{12}{\Delta^3}}$ and,

$$ \begin{equation} \frac{d}{dx}\psi(x) = \begin{cases} A , & 0 \le x \le \frac \Delta 2 \\ -A , & \frac \Delta 2 \le x \le \Delta \end{cases} \end{equation} \tag 2 $$

Now Eq. $(2)$ can we written in a more illuminating form using the step function, $\theta(x)$. So,

$$ \frac{d}{dx} \psi(x) = A(\theta(x) - 2\theta(x - \frac \Delta 2) + \theta(x - \Delta)), \forall x \tag 3 $$

Now if I had simply considered the two regions in Eq. $(2)$ then I would have got,

$$ \begin{equation} \frac{d}{dx}\psi(x) = \begin{cases} A(\theta(x) - \theta(x - \frac \Delta 2)) , & 0 \le x \le \frac \Delta 2 \\ -A(\theta(x - \frac \Delta 2) - \theta(x - \Delta)) , & \frac \Delta 2 \le x \le \Delta \end{cases} \end{equation} \tag 4 $$

Since my objective is to compute the second derivative in $\psi(x)$ and then computing the above mentioned integral $I$ then I would have got using Eq. $(3)$, $\frac{12}{\Delta^2}$; and using Eq. $(4)$, $\frac {6} {\Delta^2}$.

Now I know for a fact that using $(4)$ wrong. Could someone explain why? Is $(4)$ counting just half the contribution of the discontinuities than $(3)$? Where am I missing the justification?

Answer 1

A discontinuous function is not differantiable in context of elementary function.The calculation seem like quantum mechanics calculation.So, we are working with generalized functions. Derivative of a function at a jumb gives a dirac delta.Second derivative of the function you are given is dirac delta function. $2A\delta (x)$. For practical purpose you can calculate derivative of discontinuos function adding a $A\delta (x-x_0)$ where $x_0$ is where the jump occurs and A is amount of jump. If you want to understand reason you should study theory of generalized function. If your calculation are regarding physics you may also check some "mathematical methods of physics" books.