# Explanation of 16.5 #56

By popular demand, I present here an explanation of problem 16.5 #56 from the homework due on Monday. The problem gives the equation of the wave function for an electron in a hydrogen atom
$\psi_{S1}(\rho) = -\frac{1}{\sqrt{\pi a_0^3}} e^{-\rho / a_0}.$
Here, $$\rho$$ is the distance from the origin, so that the equation for $$\psi_{1S}$$ is in spherical coordinates. The probability that the electron is in some region $$W$$ can be computed by integrating the function
$p(\rho) = \left|\psi_{S1}(\rho)\right|^2 = \frac{1}{\pi a_0^3} e^{-2 \rho / a_0}$
over the region $$W$$. The problem asks you to compute the probability that an electron is found outside of the Bohr radius, $$a_0$$. In this case, the region $$W$$ is everything in $$\mathbf{R}^3$$ outside of the ball of radius $$a_0$$ centered at the origin. In polar coordinates, $$W$$ is easily described by the following inequalities:
\[
\begin{align*}
a_0 \leq &\rho