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

# Explanation of 16.5 #56

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