Normalization

Modify the driver routine given in Problem 1.1 so that it computes the normalization of the textbook ground-state (``1s state'') wave function for the hydrogen atom,

$$\psi_{1s}(r)=\frac{1}{\sqrt{\pi}} e^{-r}.$$

Specifically, use your routines to check that the total probability ${\cal P}(\vec x)=\psi^*(\vec x)\,\psi(\vec x)$ of finding the electron adds up to one,

$$\int {\cal P}_{1s}(\vec x) \, dV = \int \psi_{1s}^*(\vec x) ... ...) \, dV = \int_0^\infty \psi_{1s}^2(r) \, 4 \pi r^2 \, dr = 1.$$

Hint: Some thought is required to handle properly the endpoints of the integration. This can be done by modifying the code block in the driver routine labelled ``Special cases at end points''.